Research ArticleOn Asymptotic Behavior of Solutions of GeneralizedEmden-Fowler Differential Equations with Delay Argument

Alexander Domoshnitsky1 and Roman Koplatadze2

1 Department of Mathematics and Computer Sciences Ariel University 40700 Ariel Israel2 Department of Mathematics Tbilisi State University University Street 2 0143 Tbilisi Georgia

Correspondence should be addressed to Alexander Domoshnitsky adomarielacil

Received 4 July 2013 Accepted 8 November 2013 Published 22 January 2014

Academic Editor Josef Diblik

Copyright copy 2014 A Domoshnitsky and R Koplatadze This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The following differential equation 119906(119899)(119905) + 119901(119905)|119906(120590(119905))|

120583(119905) sign 119906(120590(119905)) = 0 is considered Here 119901 isin 119871 loc(119877+ 119877+) 120583 isin

119862(119877+ (0 +infin)) 120590 isin 119862(119877+ 119877+) 120590(119905) le 119905 and lim119905rarr+infin120590(119905) = +infin We say that the equation is almost linear if the conditionlim119905rarr+infin120583(119905) = 1 is fulfilled while if lim sup119905rarr+infin120583(119905) = 1 or lim inf 119905rarr+infin120583(119905) = 1 then the equation is an essentially nonlineardifferential equation In the case of almost linear and essentially nonlinear differential equations with advanced argumentoscillatory properties have been extensively studied but there are no results on delay equations of this sort In this paper newsufficient conditions implying Property A for delay Emden-Fowler equations are obtained

1 Introduction

This work deals with oscillatory properties of solutions of afunctional differential equation of the form

119906(119899)(119905) + 119901 (119905) |119906 (120590 (119905))|

120583(119905) sign 119906 (120590 (119905)) = 0 (1)

where

119901 isin 119871 loc (119877+ 119877) 120583 isin 119862 (119877+ (0 +infin))

120590 isin 119862 (119877+ 119877+) 120590 (119905) le 119905 for 119905 isin 119877+

lim119905rarr+infin

120590 (119905) = +infin

(2)

It will always be assumed that the condition

119901 (119905) ge 0 for 119905 isin 119877+ (3)

is fulfilledLet 1199050 isin 119877+ A function 119906 [1199050 +infin) rarr 119877 is said to be

a proper solution of (1) if it is locally absolutely continuoustogether with its derivatives up to order 119899 minus 1 inclusive

sup |119906 (119904)| 119904 isin [119905 +infin) gt 0 for 119905 ge 1199050 (4)

and there exists a function 119906 isin 119862(119877+ 119877) such that119906(119905) equiv 119906(119905) on [1199050 +infin) and the equality 119906

(119899)(119905) +

119901(119905)|119906(120590(119905))|120583(119905) sign 119906(120590(119905)) = 0 holds for 119905 isin [1199050 +infin)

A proper solution 119906 [1199050 +infin) rarr 119877 of (1) is said tobe oscillatory if it has a sequence of zeros tending to +infinOtherwise the solution 119906 is said to be nonoscillatory

Definition 1 We say that (1) has Property A if any of itsproper solutions is oscillatory when 119899 is even and either isoscillatory or satisfies

10038161003816100381610038161003816119906(119894)(119905)

10038161003816100381610038161003816darr 0 for 119905 uarr +infin (119894 = 0 119899 minus 1) (5)

when 119899 is odd

Definition 2 We say that (1) is almost linear if the conditionlim119905rarr+infin120583(119905) = 1 holds while if lim sup119905rarr+infin120583(119905) = 1 orlim inf 119905rarr+infin120583(119905) = 1 then we say that the equation is anessentially nonlinear differential equation

The Emden-Fowler equation originated from theoriesconcerning gaseous dynamics in astrophysics in the middleof the nineteenth century In the study of stellar structure at

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 168425 13 pageshttpdxdoiorg1011552014168425

2 Abstract and Applied Analysis

that time it was important to investigate the equilibrium con-figuration of the mass of spherical clouds of gas Lord Kelvinin 1862 assumed that the gaseous cloud is under convectiveequilibrium and then Lane [1] studied the equation

1

1199052

119889

119889119905

(1199052 119889119906

119889119905

) + 119906120574= 0 (6)

The Emden-Fowler equations were first considered onlyfor second-order equations and written in the form

119889

119889119905

(119901 (119905)

119889119906

119889119905

) + 119902 (119905) 119906120574= 0 119905 ge 0 (7)

which could be reduced in the case of positive and continuouscoefficients to the equation

11990910158401015840+ 119886 (119905) 119909

120574= 0 119905 ge 0 (8)

To avoid difficulties of defining 119909120574 when 119909(119905) is negative and120574 is not an integer the equation

11990910158401015840(119905) + 119886 (119905) |119909 (119905)|

120574 sign 119909 (119905) = 0 119905 ge 0 (9)

was usually considered The mathematical foundation of thetheory of such equations was built by Fowler [2] and thedescription of the results can be found in Chapter 7 of [3]

We see also the Emden-Fowler equation in gas dynamicsand fluid mechanics (see Sansone [4] page 431 and the paper[5]) Nonoscillation of these equations is important in variousapplications Note that the zero of such solutions correspondsto an equilibrium state in a fluid with spherical distributionof density and under mutual attraction of its particles TheEmden-Fowler equations can be either oscillatory (ie allproper solutions have a sequence of zeros tending to zero) ornonoscillatory if solutions are eventually positive or negativeor in contrast with the case of linear differential equations ofsecond ordermay possess both oscillating and nonoscillatingsolutions For example for the equation

11990910158401015840(119905) + 119905

120583|119909 (119905)|

120574 sign 119909 (119905) = 0 119905 ge 0 (10)

it was proven in [2] that for 120583 ge minus2 gt minus(120574 + 3)2 all solutionsoscillate for 120583 lt minus(120574 + 3)2mdashall solutions nonoscillate andfor minus(120574 + 3)2 le 120583 lt minus2 there are both oscillating andnonoscillating solutions

The Emden-Fowler equation presents one of the classicalobjects in the theory of differential equations Tests foroscillation and nonoscillation of all solutions and existenceof oscillating solutions were obtained in the works [6ndash8] In[9] for the case 0 lt 120574 lt 1 it was obtained that all solutions ofthe equation

11990910158401015840(119905) + 119886 (119905)

1003816100381610038161003816119909120574(119905)1003816100381610038161003816sign 119909 (119905) = 0 119905 ge 0 (11)

oscillate if and only if

int

infin

0

119905120574119886 (119905) 119889119905 = infin (12)

The latest research results in this area are presented in thebook [8] Behavior of solutions to nth order Emden-Fowler

equations can be essentially more complicated Properties Aand B defined by Kiguradze are studied in the abovemen-tioned book

There are essentially less results on oscillation of delayEmden-Fowler equations Oscillation properties of nonlineardelay differential equations where Emden-Fowler equationswere also included as a particular case were studied in [10ndash20] Results of these papers are discussed in [13 15] wherevarious examples demonstrating essentialities of conditionsare also presented Note that for delay differential equationsthere are no results on nonoscillation of all solutions and onlyexistence of nonoscillating solutions is studied Actually theresults on oscillation of delayed equations are based on theapproaches existing for ordinary differential equations withdevelopment in the direction of preventing the obstructiveinfluence of delay In the paper [15] the following equation

119909(119899)(119905) + 119886 (119905) 119891 [119909 (120590 (119905))] = 0 119905 ge 0 119899 even (13)

and its particular case

119909(119899)(119905) + 119886 (119905) |119909 (120590 (119905))|

120574 sign 119909 (120590 (119905)) = 0

119905 ge 0 119899 even(14)

are considered It was obtained for the last equation undersome standard assumptions on the coefficients [15] that in thecase 0 lt 120574 lt 1

int

infin

0

120590(119899minus1)120574

(119905) 119886 (119905) 119889119905 = infin (15)

all solutions oscillate We see that the integral depends ondeviation of argument 120590(119905) and the power of the equation 119899For the equation

119909(119899)(119905) + 119886 (119905)

119898

prod

119894=1

1003816100381610038161003816119909 (120590119894 (119905))

1003816100381610038161003816

120574119894 sign 119909 (120590119894 (119905)) = 0

119905 ge 0 119899 even

(16)

where 120574119894 is the ratio of two positive odd integers 120590(119905) le

120590119894(119905) le 119905 for 119894 = 1 119898 and 120590(119905) rarr infin as 119905 rarr infin each ofthe following conditions (a) (b) and (c) ensures oscillationof all solutions

(a)

int

infin

0

120590(119899minus1)120574

(119905) 119886 (119905) 119889119905 = infin for 120574 =119898

sum

119894=1

120574119894 lt 1 (17)

(b)

int

infin

0

120590(119899minus1)120572

(119905) 119886 (119905) 119889119905 = infin for 120574 = 1 0 lt 120572 lt 1 (18)

(c)

int

infin

0

120590119899minus1

(119905) 119886 (119905) 119889119905 = infin 1205901015840(119905) ge 0 for 120574 gt 1 (19)

Abstract and Applied Analysis 3

Most proofs of results on oscillation of all solutions tosecond order equations utilize the fact that if a nonoscillatingsolution exists the signs of the solution 119909(119905) and its secondderivative 11990910158401015840(119905) are opposite to each other for sufficientlylarge 119905 Then a growth of nonoscillating solution is estimatedand the authors come to contradiction with conditions thatproves oscillation of all solutions Note that delays disturboscillation Instead of 119905120574120590120574(119905) appears The principle isclear for oscillation of all solutions we have to achieve acorresponding smallness of the delay 119905 minus120590(119905) All this is morecomplicated if we study 119899th order equations In this case alsothe fact that 119909(119905) and its 119899th derivative 119909(119899)(119905) have differentsigns for sufficiently large 119905 is used but the technique is morecomplicated

In the papers [21ndash28] a generalization of Emden-Fowlerequations was considered The powers in these papers canbe functions and not constants In many cases it leads toessentially new oscillation properties of such equations Sur-prisingly oscillation behavior of equations with the power120582 and with functional power 120583(119905) such that lim119905rarrinfin120583(119905) =120582 can be quite different The main purpose of our paperis to study conditions under which the generalized (in thissense) equations preserve the known oscillation properties ofEmden-Fowler equations and conditions under which theseproperties are not preserved Oscillatory properties of almostlinear and essentially nonlinear differential equation withadvanced argument have already been studied in [21ndash28] Inthis paper we study oscillation properties of nth order delayEmden-Fowler equations

2 Some Auxiliary Lemmas

In the sequel 119862loc([1199050 +infin)) will denote the set of all func-tions 119906 [1199050 +infin) rarr 119877 absolutely continuous on any finitesubinternal of [1199050 +infin) along with their derivatives of orderup to including 119899 minus 1

Lemma 3 (see [28]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) 119906(119905) gt 0

119906(119899)(119905) le 0 for 119905 ge 1199050 and 119906(119899)(119905) equiv 0 in any neighborhood

of +infin Then there exist 1199051 ge 1199050 and ℓ isin 0 119899 minus 1 suchthat ℓ + 119899 is odd and

119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = 0 ℓ minus 1)

(minus1)119894+ℓ119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = ℓ 119899 minus 1)

(20ℓ)

Remark 4 If 119899 is odd and ℓ = 0 then it means that in (200)only the second inequalities are fulfilled

Lemma 5 (see [29]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) and let (20ℓ) befulfilled for some ℓ isin 0 119899 minus 1 with ℓ + 119899 odd Then

int

+infin

1199050

119905119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 lt +infin (21)

If moreover

int

+infin

1199050

119905119899minusℓ 10038161003816100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 = +infin (22)

then there exists 119905lowast gt 1199050 such that

119906(119894)(119905)

119905ℓminus119894

darr

119906(119894)(119905)

119905ℓminus119894minus1

uarr (119894 = 0 ℓ minus 1) (23119894)

119906 (119905) ge

119905ℓminus1

ℓ

119906(ℓminus1)

(119905) for 119905 ge 119905lowast (24)

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904

+

1

(119899 minus ℓ)

int

119905

119905lowast

119904119899minusℓ 10038161003816100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904 for 119905 ge 119905lowast

(25)

3 Necessary Conditions for the Existence ofa Solution of Type (20ℓ)

The following notation will be used throughout the work

120572 = inf 120583 (119905) 119905 isin 119877+ 120573 = sup 120583 (119905) 119905 isin 119877+ (26)

120590(minus1) (119905) = sup 119904 ge 0 120590 (119904) le 119905

120590(minus119896) = 120590(minus1) ∘ 120590(minus(119896minus1)) 119896 = 2 3

(27)

Clearly 120590(minus1)(119905) ge 119905 and 120590(minus1) is nondecreasing and coincideswith the inverse of 120590 when the latter exists

Definition 6 Let 1199050 isin 119877+ By Uℓ1199050 one denotes the set of allproper solutions 119906 [1199050 +infin) rarr 119877 of (1) satisfying the con-dition (20ℓ) with some 1199051 ge 1199050

Lemma 7 Let the conditions (2) (3) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd and let 119906 isin Uℓ1199050 be a positiveproper solution of (1) If moreover 120572 ge 1 120573 lt +infin

int

+infin

0

119905119899minusℓ(120590 (119905))

(ℓminus1)120583(119905)119901 (119905) 119889119905 = +infin (28ℓ)

then for any119872 isin (1 +infin) there exists 119905lowast gt 1199050 such that for any119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (29)

4 Abstract and Applied Analysis

where 120572 is given by the first equality of (26) and

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872ℓ (120572) int119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

times119901 (120585) 119889120585 119889119904

(30)

120588(120572)

119894ℓ119905lowast(119905) = ℓ

+

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(31)

119872ℓ (120572) =

1

ℓ (119899 minus ℓ)

if 120572 = 1

119872 if 120572 gt 1(32)

Proof Let 1199050 isin 119877+ ℓ isin 1 119899 minus 1 with ℓ + 119899 odd and119906 isin Uℓ1199050 (see Definition 6) is solution of (1) Since 120573 lt +infinaccording to (1) (20ℓ) and (28ℓ) it is clear that condition(22) holds Thus by Lemma 5 there exists 1199052 gt 1199051 such thatthe conditions (23119894)ndash(25) with 119905lowast = 1199052 are fulfilled and

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

+

1

(119899 minus ℓ)

int

119905

1199052

119904119899minusℓ119901 (119904) (119906 (120590 (119904)))

120583(119904)119889119904

for 119905 ge 1199052

(33)

Observe that there exists 1199053 gt 1199052 such that 120590(119905) ge 1199052 for 119905 ge 1199053Thus by (24) for any 119905 ge 1199053 we get

119906(ℓminus1)

(119905)

ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

minus

1

(119899 minus ℓ)

int

119905

1199052

119904119889int

+infin

119904

120585119899minusℓminus1

119901 (120585)

times (119906 (120590 (120585)))120583(120585)119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(34)

According to (28ℓ) and (23ℓminus1) choose 119905lowast gt 1199053 such that

1

(119899 minus ℓ)

int

119905lowast

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119901 (120585) 119889120585 119889119904 gt ℓ

(35)

By (34) and (35) we have

119906(ℓminus1)

(119905) ge ℓ

+

1

(119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119889120585 119889119904

for 119905 ge 119905lowast(36)

Let 120572 = 1 Since 119906(ℓminus1)(119905) rarr +infin as 119905 rarr +infin without lossof generally we can assume that 119906(ℓminus1)(120590(120585)) ge ℓ for 120585 ge 1199053Then by (23ℓ) from (36) we get

119906(ℓminus1)

(119905)

ge ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times(120590 (120585))1+(ℓminus1)120583(120585)

119906(ℓminus1)

(120585) 119901 (120585) 119889120585 119889119904

for 119905ge 119905lowast(37)

It is obvious that

1199091015840(119905) ge

119906(ℓminus1)

(119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

(38)

where

119909 (119905) = ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119906(ℓminus1)

(120585) 119889120585 119889119904

(39)

Abstract and Applied Analysis 5

Thus according to (23ℓminus1) (37) and (39) from (38) we get

1199091015840(119905)

ge

119909 (119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

for 119905 ge 119905lowast(40)

Therefore

119909 (119905)

ge ℓ exp 1

ℓ (119899 minus ℓ)

timesint

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

for 119905 ge 119905lowast(41)

Hence according to (37) and (39)

119906(ℓminus1)

(119905) ge 120588(1)

1ℓ119905lowast(119905) for 119905 ge 120590(minus1) (119905lowast) (42)

where

120588(1)

1ℓ119905lowast(119905)

= ℓ exp 1

ℓ (119899 minus ℓ)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

(43)

Thus according to (36) and (42)

119906(ℓminus1)

(119905) ge 120588(1)

119894ℓ119905lowast(119905) for 119905 ge 120590(minus119894) (119905lowast) (119894 = 1 119896)

(44)

where

120588(1)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(45)

Now assume that 120572 gt 1 and119872 isin (1 +infin) Since 119906(ℓminus1)(119905) uarr+infin for 119905 uarr +infin without loss of generality we can assume that

(119906(ℓminus1)

(120590(120585))ℓ)120572minus1

ge ℓ(119899minusℓ) 119872 for 120585 ge 119905lowastTherefore from(36) we have

119906(ℓminus1)

(119905)

ge ℓ + 119872int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585) 119906(ℓminus1)

(120590 (120585)) 119889120585 119889119904

for 119905 ge 119905lowast

(46)

Taking into account (46) as above we can find that if 120572 gt 1then

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (47)

where

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

times 119901 (120585) 119889120585 119889119904

for 119905 ge 120590(minus1) (119905lowast) (48)

120588(120572)

119894ℓ119905lowast(119905) = ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

for 119905 ge 120590(minus119894) (119905lowast) (119894 = 2 119896) (49)

According to (43)ndash(45) and (47)ndash(49) it is obvious that forany 120572 ge 1 119896 isin 119873 and119872 gt 1 there exists 119905lowast isin 119877+ such that(29)ndash(31) hold where 119872ℓ(120572) is defined by (32) This provesthe validity of the lemma

Analogously we can prove

Lemma 8 Let conditions (2) (3) (28ℓ) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd 1 le 120573 lt +infin and let 119906 isin Uℓ1199050be a positive proper solution of (1) Then for any119872 isin (1 +infin)

there exists 119905lowast gt 1199050 such that for any 119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120573)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (50)

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

that time it was important to investigate the equilibrium con-figuration of the mass of spherical clouds of gas Lord Kelvinin 1862 assumed that the gaseous cloud is under convectiveequilibrium and then Lane [1] studied the equation

1

1199052

119889

119889119905

(1199052 119889119906

119889119905

) + 119906120574= 0 (6)

The Emden-Fowler equations were first considered onlyfor second-order equations and written in the form

119889

119889119905

(119901 (119905)

119889119906

119889119905

) + 119902 (119905) 119906120574= 0 119905 ge 0 (7)

which could be reduced in the case of positive and continuouscoefficients to the equation

11990910158401015840+ 119886 (119905) 119909

120574= 0 119905 ge 0 (8)

To avoid difficulties of defining 119909120574 when 119909(119905) is negative and120574 is not an integer the equation

11990910158401015840(119905) + 119886 (119905) |119909 (119905)|

120574 sign 119909 (119905) = 0 119905 ge 0 (9)

was usually considered The mathematical foundation of thetheory of such equations was built by Fowler [2] and thedescription of the results can be found in Chapter 7 of [3]

We see also the Emden-Fowler equation in gas dynamicsand fluid mechanics (see Sansone [4] page 431 and the paper[5]) Nonoscillation of these equations is important in variousapplications Note that the zero of such solutions correspondsto an equilibrium state in a fluid with spherical distributionof density and under mutual attraction of its particles TheEmden-Fowler equations can be either oscillatory (ie allproper solutions have a sequence of zeros tending to zero) ornonoscillatory if solutions are eventually positive or negativeor in contrast with the case of linear differential equations ofsecond ordermay possess both oscillating and nonoscillatingsolutions For example for the equation

11990910158401015840(119905) + 119905

120583|119909 (119905)|

120574 sign 119909 (119905) = 0 119905 ge 0 (10)

it was proven in [2] that for 120583 ge minus2 gt minus(120574 + 3)2 all solutionsoscillate for 120583 lt minus(120574 + 3)2mdashall solutions nonoscillate andfor minus(120574 + 3)2 le 120583 lt minus2 there are both oscillating andnonoscillating solutions

The Emden-Fowler equation presents one of the classicalobjects in the theory of differential equations Tests foroscillation and nonoscillation of all solutions and existenceof oscillating solutions were obtained in the works [6ndash8] In[9] for the case 0 lt 120574 lt 1 it was obtained that all solutions ofthe equation

11990910158401015840(119905) + 119886 (119905)

1003816100381610038161003816119909120574(119905)1003816100381610038161003816sign 119909 (119905) = 0 119905 ge 0 (11)

oscillate if and only if

int

infin

0

119905120574119886 (119905) 119889119905 = infin (12)

The latest research results in this area are presented in thebook [8] Behavior of solutions to nth order Emden-Fowler

equations can be essentially more complicated Properties Aand B defined by Kiguradze are studied in the abovemen-tioned book

There are essentially less results on oscillation of delayEmden-Fowler equations Oscillation properties of nonlineardelay differential equations where Emden-Fowler equationswere also included as a particular case were studied in [10ndash20] Results of these papers are discussed in [13 15] wherevarious examples demonstrating essentialities of conditionsare also presented Note that for delay differential equationsthere are no results on nonoscillation of all solutions and onlyexistence of nonoscillating solutions is studied Actually theresults on oscillation of delayed equations are based on theapproaches existing for ordinary differential equations withdevelopment in the direction of preventing the obstructiveinfluence of delay In the paper [15] the following equation

119909(119899)(119905) + 119886 (119905) 119891 [119909 (120590 (119905))] = 0 119905 ge 0 119899 even (13)

and its particular case

119909(119899)(119905) + 119886 (119905) |119909 (120590 (119905))|

120574 sign 119909 (120590 (119905)) = 0

119905 ge 0 119899 even(14)

are considered It was obtained for the last equation undersome standard assumptions on the coefficients [15] that in thecase 0 lt 120574 lt 1

int

infin

0

120590(119899minus1)120574

(119905) 119886 (119905) 119889119905 = infin (15)

all solutions oscillate We see that the integral depends ondeviation of argument 120590(119905) and the power of the equation 119899For the equation

119909(119899)(119905) + 119886 (119905)

119898

prod

119894=1

1003816100381610038161003816119909 (120590119894 (119905))

1003816100381610038161003816

120574119894 sign 119909 (120590119894 (119905)) = 0

119905 ge 0 119899 even

(16)

where 120574119894 is the ratio of two positive odd integers 120590(119905) le

120590119894(119905) le 119905 for 119894 = 1 119898 and 120590(119905) rarr infin as 119905 rarr infin each ofthe following conditions (a) (b) and (c) ensures oscillationof all solutions

(a)

int

infin

0

120590(119899minus1)120574

(119905) 119886 (119905) 119889119905 = infin for 120574 =119898

sum

119894=1

120574119894 lt 1 (17)

(b)

int

infin

0

120590(119899minus1)120572

(119905) 119886 (119905) 119889119905 = infin for 120574 = 1 0 lt 120572 lt 1 (18)

(c)

int

infin

0

120590119899minus1

(119905) 119886 (119905) 119889119905 = infin 1205901015840(119905) ge 0 for 120574 gt 1 (19)

Abstract and Applied Analysis 3

Most proofs of results on oscillation of all solutions tosecond order equations utilize the fact that if a nonoscillatingsolution exists the signs of the solution 119909(119905) and its secondderivative 11990910158401015840(119905) are opposite to each other for sufficientlylarge 119905 Then a growth of nonoscillating solution is estimatedand the authors come to contradiction with conditions thatproves oscillation of all solutions Note that delays disturboscillation Instead of 119905120574120590120574(119905) appears The principle isclear for oscillation of all solutions we have to achieve acorresponding smallness of the delay 119905 minus120590(119905) All this is morecomplicated if we study 119899th order equations In this case alsothe fact that 119909(119905) and its 119899th derivative 119909(119899)(119905) have differentsigns for sufficiently large 119905 is used but the technique is morecomplicated

In the papers [21ndash28] a generalization of Emden-Fowlerequations was considered The powers in these papers canbe functions and not constants In many cases it leads toessentially new oscillation properties of such equations Sur-prisingly oscillation behavior of equations with the power120582 and with functional power 120583(119905) such that lim119905rarrinfin120583(119905) =120582 can be quite different The main purpose of our paperis to study conditions under which the generalized (in thissense) equations preserve the known oscillation properties ofEmden-Fowler equations and conditions under which theseproperties are not preserved Oscillatory properties of almostlinear and essentially nonlinear differential equation withadvanced argument have already been studied in [21ndash28] Inthis paper we study oscillation properties of nth order delayEmden-Fowler equations

2 Some Auxiliary Lemmas

In the sequel 119862loc([1199050 +infin)) will denote the set of all func-tions 119906 [1199050 +infin) rarr 119877 absolutely continuous on any finitesubinternal of [1199050 +infin) along with their derivatives of orderup to including 119899 minus 1

Lemma 3 (see [28]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) 119906(119905) gt 0

119906(119899)(119905) le 0 for 119905 ge 1199050 and 119906(119899)(119905) equiv 0 in any neighborhood

of +infin Then there exist 1199051 ge 1199050 and ℓ isin 0 119899 minus 1 suchthat ℓ + 119899 is odd and

119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = 0 ℓ minus 1)

(minus1)119894+ℓ119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = ℓ 119899 minus 1)

(20ℓ)

Remark 4 If 119899 is odd and ℓ = 0 then it means that in (200)only the second inequalities are fulfilled

Lemma 5 (see [29]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) and let (20ℓ) befulfilled for some ℓ isin 0 119899 minus 1 with ℓ + 119899 odd Then

int

+infin

1199050

119905119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 lt +infin (21)

If moreover

int

+infin

1199050

119905119899minusℓ 10038161003816100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 = +infin (22)

then there exists 119905lowast gt 1199050 such that

119906(119894)(119905)

119905ℓminus119894

darr

119906(119894)(119905)

119905ℓminus119894minus1

uarr (119894 = 0 ℓ minus 1) (23119894)

119906 (119905) ge

119905ℓminus1

ℓ

119906(ℓminus1)

(119905) for 119905 ge 119905lowast (24)

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904

+

1

(119899 minus ℓ)

int

119905

119905lowast

119904119899minusℓ 10038161003816100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904 for 119905 ge 119905lowast

(25)

3 Necessary Conditions for the Existence ofa Solution of Type (20ℓ)

The following notation will be used throughout the work

120572 = inf 120583 (119905) 119905 isin 119877+ 120573 = sup 120583 (119905) 119905 isin 119877+ (26)

120590(minus1) (119905) = sup 119904 ge 0 120590 (119904) le 119905

120590(minus119896) = 120590(minus1) ∘ 120590(minus(119896minus1)) 119896 = 2 3

(27)

Clearly 120590(minus1)(119905) ge 119905 and 120590(minus1) is nondecreasing and coincideswith the inverse of 120590 when the latter exists

Definition 6 Let 1199050 isin 119877+ By Uℓ1199050 one denotes the set of allproper solutions 119906 [1199050 +infin) rarr 119877 of (1) satisfying the con-dition (20ℓ) with some 1199051 ge 1199050

Lemma 7 Let the conditions (2) (3) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd and let 119906 isin Uℓ1199050 be a positiveproper solution of (1) If moreover 120572 ge 1 120573 lt +infin

int

+infin

0

119905119899minusℓ(120590 (119905))

(ℓminus1)120583(119905)119901 (119905) 119889119905 = +infin (28ℓ)

then for any119872 isin (1 +infin) there exists 119905lowast gt 1199050 such that for any119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (29)

4 Abstract and Applied Analysis

where 120572 is given by the first equality of (26) and

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872ℓ (120572) int119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

times119901 (120585) 119889120585 119889119904

(30)

120588(120572)

119894ℓ119905lowast(119905) = ℓ

+

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(31)

119872ℓ (120572) =

1

ℓ (119899 minus ℓ)

if 120572 = 1

119872 if 120572 gt 1(32)

Proof Let 1199050 isin 119877+ ℓ isin 1 119899 minus 1 with ℓ + 119899 odd and119906 isin Uℓ1199050 (see Definition 6) is solution of (1) Since 120573 lt +infinaccording to (1) (20ℓ) and (28ℓ) it is clear that condition(22) holds Thus by Lemma 5 there exists 1199052 gt 1199051 such thatthe conditions (23119894)ndash(25) with 119905lowast = 1199052 are fulfilled and

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

+

1

(119899 minus ℓ)

int

119905

1199052

119904119899minusℓ119901 (119904) (119906 (120590 (119904)))

120583(119904)119889119904

for 119905 ge 1199052

(33)

Observe that there exists 1199053 gt 1199052 such that 120590(119905) ge 1199052 for 119905 ge 1199053Thus by (24) for any 119905 ge 1199053 we get

119906(ℓminus1)

(119905)

ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

minus

1

(119899 minus ℓ)

int

119905

1199052

119904119889int

+infin

119904

120585119899minusℓminus1

119901 (120585)

times (119906 (120590 (120585)))120583(120585)119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(34)

According to (28ℓ) and (23ℓminus1) choose 119905lowast gt 1199053 such that

1

(119899 minus ℓ)

int

119905lowast

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119901 (120585) 119889120585 119889119904 gt ℓ

(35)

By (34) and (35) we have

119906(ℓminus1)

(119905) ge ℓ

+

1

(119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119889120585 119889119904

for 119905 ge 119905lowast(36)

Let 120572 = 1 Since 119906(ℓminus1)(119905) rarr +infin as 119905 rarr +infin without lossof generally we can assume that 119906(ℓminus1)(120590(120585)) ge ℓ for 120585 ge 1199053Then by (23ℓ) from (36) we get

119906(ℓminus1)

(119905)

ge ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times(120590 (120585))1+(ℓminus1)120583(120585)

119906(ℓminus1)

(120585) 119901 (120585) 119889120585 119889119904

for 119905ge 119905lowast(37)

It is obvious that

1199091015840(119905) ge

119906(ℓminus1)

(119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

(38)

where

119909 (119905) = ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119906(ℓminus1)

(120585) 119889120585 119889119904

(39)

Abstract and Applied Analysis 5

Thus according to (23ℓminus1) (37) and (39) from (38) we get

1199091015840(119905)

ge

119909 (119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

for 119905 ge 119905lowast(40)

Therefore

119909 (119905)

ge ℓ exp 1

ℓ (119899 minus ℓ)

timesint

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

for 119905 ge 119905lowast(41)

Hence according to (37) and (39)

119906(ℓminus1)

(119905) ge 120588(1)

1ℓ119905lowast(119905) for 119905 ge 120590(minus1) (119905lowast) (42)

where

120588(1)

1ℓ119905lowast(119905)

= ℓ exp 1

ℓ (119899 minus ℓ)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

(43)

Thus according to (36) and (42)

119906(ℓminus1)

(119905) ge 120588(1)

119894ℓ119905lowast(119905) for 119905 ge 120590(minus119894) (119905lowast) (119894 = 1 119896)

(44)

where

120588(1)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(45)

Now assume that 120572 gt 1 and119872 isin (1 +infin) Since 119906(ℓminus1)(119905) uarr+infin for 119905 uarr +infin without loss of generality we can assume that

(119906(ℓminus1)

(120590(120585))ℓ)120572minus1

ge ℓ(119899minusℓ) 119872 for 120585 ge 119905lowastTherefore from(36) we have

119906(ℓminus1)

(119905)

ge ℓ + 119872int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585) 119906(ℓminus1)

(120590 (120585)) 119889120585 119889119904

for 119905 ge 119905lowast

(46)

Taking into account (46) as above we can find that if 120572 gt 1then

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (47)

where

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

times 119901 (120585) 119889120585 119889119904

for 119905 ge 120590(minus1) (119905lowast) (48)

120588(120572)

119894ℓ119905lowast(119905) = ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

for 119905 ge 120590(minus119894) (119905lowast) (119894 = 2 119896) (49)

According to (43)ndash(45) and (47)ndash(49) it is obvious that forany 120572 ge 1 119896 isin 119873 and119872 gt 1 there exists 119905lowast isin 119877+ such that(29)ndash(31) hold where 119872ℓ(120572) is defined by (32) This provesthe validity of the lemma

Analogously we can prove

Lemma 8 Let conditions (2) (3) (28ℓ) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd 1 le 120573 lt +infin and let 119906 isin Uℓ1199050be a positive proper solution of (1) Then for any119872 isin (1 +infin)

there exists 119905lowast gt 1199050 such that for any 119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120573)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (50)

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Most proofs of results on oscillation of all solutions tosecond order equations utilize the fact that if a nonoscillatingsolution exists the signs of the solution 119909(119905) and its secondderivative 11990910158401015840(119905) are opposite to each other for sufficientlylarge 119905 Then a growth of nonoscillating solution is estimatedand the authors come to contradiction with conditions thatproves oscillation of all solutions Note that delays disturboscillation Instead of 119905120574120590120574(119905) appears The principle isclear for oscillation of all solutions we have to achieve acorresponding smallness of the delay 119905 minus120590(119905) All this is morecomplicated if we study 119899th order equations In this case alsothe fact that 119909(119905) and its 119899th derivative 119909(119899)(119905) have differentsigns for sufficiently large 119905 is used but the technique is morecomplicated

In the papers [21ndash28] a generalization of Emden-Fowlerequations was considered The powers in these papers canbe functions and not constants In many cases it leads toessentially new oscillation properties of such equations Sur-prisingly oscillation behavior of equations with the power120582 and with functional power 120583(119905) such that lim119905rarrinfin120583(119905) =120582 can be quite different The main purpose of our paperis to study conditions under which the generalized (in thissense) equations preserve the known oscillation properties ofEmden-Fowler equations and conditions under which theseproperties are not preserved Oscillatory properties of almostlinear and essentially nonlinear differential equation withadvanced argument have already been studied in [21ndash28] Inthis paper we study oscillation properties of nth order delayEmden-Fowler equations

2 Some Auxiliary Lemmas

In the sequel 119862loc([1199050 +infin)) will denote the set of all func-tions 119906 [1199050 +infin) rarr 119877 absolutely continuous on any finitesubinternal of [1199050 +infin) along with their derivatives of orderup to including 119899 minus 1

Lemma 3 (see [28]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) 119906(119905) gt 0

119906(119899)(119905) le 0 for 119905 ge 1199050 and 119906(119899)(119905) equiv 0 in any neighborhood

of +infin Then there exist 1199051 ge 1199050 and ℓ isin 0 119899 minus 1 suchthat ℓ + 119899 is odd and

119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = 0 ℓ minus 1)

(minus1)119894+ℓ119906(119894)(119905) gt 0 for 119905 ge 1199051 (119894 = ℓ 119899 minus 1)

(20ℓ)

Remark 4 If 119899 is odd and ℓ = 0 then it means that in (200)only the second inequalities are fulfilled

Lemma 5 (see [29]) Let 119906 isin 119862119899minus1loc ([1199050 +infin)) and let (20ℓ) befulfilled for some ℓ isin 0 119899 minus 1 with ℓ + 119899 odd Then

int

+infin

1199050

119905119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 lt +infin (21)

If moreover

int

+infin

1199050

119905119899minusℓ 10038161003816100381610038161003816119906(119899)(119905)

10038161003816100381610038161003816119889119905 = +infin (22)

then there exists 119905lowast gt 1199050 such that

119906(119894)(119905)

119905ℓminus119894

darr

119906(119894)(119905)

119905ℓminus119894minus1

uarr (119894 = 0 ℓ minus 1) (23119894)

119906 (119905) ge

119905ℓminus1

ℓ

119906(ℓminus1)

(119905) for 119905 ge 119905lowast (24)

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1 10038161003816

100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904

+

1

(119899 minus ℓ)

int

119905

119905lowast

119904119899minusℓ 10038161003816100381610038161003816119906(119899)(119904)

10038161003816100381610038161003816119889119904 for 119905 ge 119905lowast

(25)

3 Necessary Conditions for the Existence ofa Solution of Type (20ℓ)

The following notation will be used throughout the work

120572 = inf 120583 (119905) 119905 isin 119877+ 120573 = sup 120583 (119905) 119905 isin 119877+ (26)

120590(minus1) (119905) = sup 119904 ge 0 120590 (119904) le 119905

120590(minus119896) = 120590(minus1) ∘ 120590(minus(119896minus1)) 119896 = 2 3

(27)

Clearly 120590(minus1)(119905) ge 119905 and 120590(minus1) is nondecreasing and coincideswith the inverse of 120590 when the latter exists

Definition 6 Let 1199050 isin 119877+ By Uℓ1199050 one denotes the set of allproper solutions 119906 [1199050 +infin) rarr 119877 of (1) satisfying the con-dition (20ℓ) with some 1199051 ge 1199050

Lemma 7 Let the conditions (2) (3) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd and let 119906 isin Uℓ1199050 be a positiveproper solution of (1) If moreover 120572 ge 1 120573 lt +infin

int

+infin

0

119905119899minusℓ(120590 (119905))

(ℓminus1)120583(119905)119901 (119905) 119889119905 = +infin (28ℓ)

then for any119872 isin (1 +infin) there exists 119905lowast gt 1199050 such that for any119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (29)

4 Abstract and Applied Analysis

where 120572 is given by the first equality of (26) and

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872ℓ (120572) int119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

times119901 (120585) 119889120585 119889119904

(30)

120588(120572)

119894ℓ119905lowast(119905) = ℓ

+

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(31)

119872ℓ (120572) =

1

ℓ (119899 minus ℓ)

if 120572 = 1

119872 if 120572 gt 1(32)

Proof Let 1199050 isin 119877+ ℓ isin 1 119899 minus 1 with ℓ + 119899 odd and119906 isin Uℓ1199050 (see Definition 6) is solution of (1) Since 120573 lt +infinaccording to (1) (20ℓ) and (28ℓ) it is clear that condition(22) holds Thus by Lemma 5 there exists 1199052 gt 1199051 such thatthe conditions (23119894)ndash(25) with 119905lowast = 1199052 are fulfilled and

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

+

1

(119899 minus ℓ)

int

119905

1199052

119904119899minusℓ119901 (119904) (119906 (120590 (119904)))

120583(119904)119889119904

for 119905 ge 1199052

(33)

Observe that there exists 1199053 gt 1199052 such that 120590(119905) ge 1199052 for 119905 ge 1199053Thus by (24) for any 119905 ge 1199053 we get

119906(ℓminus1)

(119905)

ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

minus

1

(119899 minus ℓ)

int

119905

1199052

119904119889int

+infin

119904

120585119899minusℓminus1

119901 (120585)

times (119906 (120590 (120585)))120583(120585)119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(34)

According to (28ℓ) and (23ℓminus1) choose 119905lowast gt 1199053 such that

1

(119899 minus ℓ)

int

119905lowast

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119901 (120585) 119889120585 119889119904 gt ℓ

(35)

By (34) and (35) we have

119906(ℓminus1)

(119905) ge ℓ

+

1

(119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119889120585 119889119904

for 119905 ge 119905lowast(36)

Let 120572 = 1 Since 119906(ℓminus1)(119905) rarr +infin as 119905 rarr +infin without lossof generally we can assume that 119906(ℓminus1)(120590(120585)) ge ℓ for 120585 ge 1199053Then by (23ℓ) from (36) we get

119906(ℓminus1)

(119905)

ge ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times(120590 (120585))1+(ℓminus1)120583(120585)

119906(ℓminus1)

(120585) 119901 (120585) 119889120585 119889119904

for 119905ge 119905lowast(37)

It is obvious that

1199091015840(119905) ge

119906(ℓminus1)

(119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

(38)

where

119909 (119905) = ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119906(ℓminus1)

(120585) 119889120585 119889119904

(39)

Abstract and Applied Analysis 5

Thus according to (23ℓminus1) (37) and (39) from (38) we get

1199091015840(119905)

ge

119909 (119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

for 119905 ge 119905lowast(40)

Therefore

119909 (119905)

ge ℓ exp 1

ℓ (119899 minus ℓ)

timesint

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

for 119905 ge 119905lowast(41)

Hence according to (37) and (39)

119906(ℓminus1)

(119905) ge 120588(1)

1ℓ119905lowast(119905) for 119905 ge 120590(minus1) (119905lowast) (42)

where

120588(1)

1ℓ119905lowast(119905)

= ℓ exp 1

ℓ (119899 minus ℓ)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

(43)

Thus according to (36) and (42)

119906(ℓminus1)

(119905) ge 120588(1)

119894ℓ119905lowast(119905) for 119905 ge 120590(minus119894) (119905lowast) (119894 = 1 119896)

(44)

where

120588(1)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(45)

Now assume that 120572 gt 1 and119872 isin (1 +infin) Since 119906(ℓminus1)(119905) uarr+infin for 119905 uarr +infin without loss of generality we can assume that

(119906(ℓminus1)

(120590(120585))ℓ)120572minus1

ge ℓ(119899minusℓ) 119872 for 120585 ge 119905lowastTherefore from(36) we have

119906(ℓminus1)

(119905)

ge ℓ + 119872int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585) 119906(ℓminus1)

(120590 (120585)) 119889120585 119889119904

for 119905 ge 119905lowast

(46)

Taking into account (46) as above we can find that if 120572 gt 1then

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (47)

where

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

times 119901 (120585) 119889120585 119889119904

for 119905 ge 120590(minus1) (119905lowast) (48)

120588(120572)

119894ℓ119905lowast(119905) = ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

for 119905 ge 120590(minus119894) (119905lowast) (119894 = 2 119896) (49)

According to (43)ndash(45) and (47)ndash(49) it is obvious that forany 120572 ge 1 119896 isin 119873 and119872 gt 1 there exists 119905lowast isin 119877+ such that(29)ndash(31) hold where 119872ℓ(120572) is defined by (32) This provesthe validity of the lemma

Analogously we can prove

Lemma 8 Let conditions (2) (3) (28ℓ) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd 1 le 120573 lt +infin and let 119906 isin Uℓ1199050be a positive proper solution of (1) Then for any119872 isin (1 +infin)

there exists 119905lowast gt 1199050 such that for any 119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120573)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (50)

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

where 120572 is given by the first equality of (26) and

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872ℓ (120572) int119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

times119901 (120585) 119889120585 119889119904

(30)

120588(120572)

119894ℓ119905lowast(119905) = ℓ

+

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(31)

119872ℓ (120572) =

1

ℓ (119899 minus ℓ)

if 120572 = 1

119872 if 120572 gt 1(32)

Proof Let 1199050 isin 119877+ ℓ isin 1 119899 minus 1 with ℓ + 119899 odd and119906 isin Uℓ1199050 (see Definition 6) is solution of (1) Since 120573 lt +infinaccording to (1) (20ℓ) and (28ℓ) it is clear that condition(22) holds Thus by Lemma 5 there exists 1199052 gt 1199051 such thatthe conditions (23119894)ndash(25) with 119905lowast = 1199052 are fulfilled and

119906(ℓminus1)

(119905) ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

+

1

(119899 minus ℓ)

int

119905

1199052

119904119899minusℓ119901 (119904) (119906 (120590 (119904)))

120583(119904)119889119904

for 119905 ge 1199052

(33)

Observe that there exists 1199053 gt 1199052 such that 120590(119905) ge 1199052 for 119905 ge 1199053Thus by (24) for any 119905 ge 1199053 we get

119906(ℓminus1)

(119905)

ge

119905

(119899 minus ℓ)

int

+infin

119905

119904119899minusℓminus1

119901 (119904) (119906 (120590 (119904)))120583(119904)119889119904

minus

1

(119899 minus ℓ)

int

119905

1199052

119904119889int

+infin

119904

120585119899minusℓminus1

119901 (120585)

times (119906 (120590 (120585)))120583(120585)119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(34)

According to (28ℓ) and (23ℓminus1) choose 119905lowast gt 1199053 such that

1

(119899 minus ℓ)

int

119905lowast

1199053

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119901 (120585) 119889120585 119889119904 gt ℓ

(35)

By (34) and (35) we have

119906(ℓminus1)

(119905) ge ℓ

+

1

(119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

119906(ℓminus1)

(120590 (120585))

ℓ

)

120583(120585)

119889120585 119889119904

for 119905 ge 119905lowast(36)

Let 120572 = 1 Since 119906(ℓminus1)(119905) rarr +infin as 119905 rarr +infin without lossof generally we can assume that 119906(ℓminus1)(120590(120585)) ge ℓ for 120585 ge 1199053Then by (23ℓ) from (36) we get

119906(ℓminus1)

(119905)

ge ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times(120590 (120585))1+(ℓminus1)120583(120585)

119906(ℓminus1)

(120585) 119901 (120585) 119889120585 119889119904

for 119905ge 119905lowast(37)

It is obvious that

1199091015840(119905) ge

119906(ℓminus1)

(119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

(38)

where

119909 (119905) = ℓ +

1

ℓ (119899 minus ℓ)

int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119906(ℓminus1)

(120585) 119889120585 119889119904

(39)

Abstract and Applied Analysis 5

Thus according to (23ℓminus1) (37) and (39) from (38) we get

1199091015840(119905)

ge

119909 (119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

for 119905 ge 119905lowast(40)

Therefore

119909 (119905)

ge ℓ exp 1

ℓ (119899 minus ℓ)

timesint

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

for 119905 ge 119905lowast(41)

Hence according to (37) and (39)

119906(ℓminus1)

(119905) ge 120588(1)

1ℓ119905lowast(119905) for 119905 ge 120590(minus1) (119905lowast) (42)

where

120588(1)

1ℓ119905lowast(119905)

= ℓ exp 1

ℓ (119899 minus ℓ)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

(43)

Thus according to (36) and (42)

119906(ℓminus1)

(119905) ge 120588(1)

119894ℓ119905lowast(119905) for 119905 ge 120590(minus119894) (119905lowast) (119894 = 1 119896)

(44)

where

120588(1)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(45)

Now assume that 120572 gt 1 and119872 isin (1 +infin) Since 119906(ℓminus1)(119905) uarr+infin for 119905 uarr +infin without loss of generality we can assume that

(119906(ℓminus1)

(120590(120585))ℓ)120572minus1

ge ℓ(119899minusℓ) 119872 for 120585 ge 119905lowastTherefore from(36) we have

119906(ℓminus1)

(119905)

ge ℓ + 119872int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585) 119906(ℓminus1)

(120590 (120585)) 119889120585 119889119904

for 119905 ge 119905lowast

(46)

Taking into account (46) as above we can find that if 120572 gt 1then

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (47)

where

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

times 119901 (120585) 119889120585 119889119904

for 119905 ge 120590(minus1) (119905lowast) (48)

120588(120572)

119894ℓ119905lowast(119905) = ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

for 119905 ge 120590(minus119894) (119905lowast) (119894 = 2 119896) (49)

According to (43)ndash(45) and (47)ndash(49) it is obvious that forany 120572 ge 1 119896 isin 119873 and119872 gt 1 there exists 119905lowast isin 119877+ such that(29)ndash(31) hold where 119872ℓ(120572) is defined by (32) This provesthe validity of the lemma

Analogously we can prove

Lemma 8 Let conditions (2) (3) (28ℓ) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd 1 le 120573 lt +infin and let 119906 isin Uℓ1199050be a positive proper solution of (1) Then for any119872 isin (1 +infin)

there exists 119905lowast gt 1199050 such that for any 119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120573)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (50)

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Thus according to (23ℓminus1) (37) and (39) from (38) we get

1199091015840(119905)

ge

119909 (119905)

ℓ (119899 minus ℓ)

int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585

for 119905 ge 119905lowast(40)

Therefore

119909 (119905)

ge ℓ exp 1

ℓ (119899 minus ℓ)

timesint

119905

119905lowast

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

for 119905 ge 119905lowast(41)

Hence according to (37) and (39)

119906(ℓminus1)

(119905) ge 120588(1)

1ℓ119905lowast(119905) for 119905 ge 120590(minus1) (119905lowast) (42)

where

120588(1)

1ℓ119905lowast(119905)

= ℓ exp 1

ℓ (119899 minus ℓ)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904

(43)

Thus according to (36) and (42)

119906(ℓminus1)

(119905) ge 120588(1)

119894ℓ119905lowast(119905) for 119905 ge 120590(minus119894) (119905lowast) (119894 = 1 119896)

(44)

where

120588(1)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

(119894 = 2 119896)

(45)

Now assume that 120572 gt 1 and119872 isin (1 +infin) Since 119906(ℓminus1)(119905) uarr+infin for 119905 uarr +infin without loss of generality we can assume that

(119906(ℓminus1)

(120590(120585))ℓ)120572minus1

ge ℓ(119899minusℓ) 119872 for 120585 ge 119905lowastTherefore from(36) we have

119906(ℓminus1)

(119905)

ge ℓ + 119872int

119905

119905lowast

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times 119901 (120585) 119906(ℓminus1)

(120590 (120585)) 119889120585 119889119904

for 119905 ge 119905lowast

(46)

Taking into account (46) as above we can find that if 120572 gt 1then

119906(ℓminus1)

(119905) ge 120588(120572)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (47)

where

120588(120572)

1ℓ119905lowast(119905)

= ℓ exp119872int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

times 119901 (120585) 119889120585 119889119904

for 119905 ge 120590(minus1) (119905lowast) (48)

120588(120572)

119894ℓ119905lowast(119905) = ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904

for 119905 ge 120590(minus119894) (119905lowast) (119894 = 2 119896) (49)

According to (43)ndash(45) and (47)ndash(49) it is obvious that forany 120572 ge 1 119896 isin 119873 and119872 gt 1 there exists 119905lowast isin 119877+ such that(29)ndash(31) hold where 119872ℓ(120572) is defined by (32) This provesthe validity of the lemma

Analogously we can prove

Lemma 8 Let conditions (2) (3) (28ℓ) be fulfilled let ℓ isin

1 119899 minus 1 with ℓ + 119899 odd 1 le 120573 lt +infin and let 119906 isin Uℓ1199050be a positive proper solution of (1) Then for any119872 isin (1 +infin)

there exists 119905lowast gt 1199050 such that for any 119896 isin 119873

119906(ℓminus1)

(119905) ge 120588(120573)

119896ℓ119905lowast(119905) for 119905 ge 120590(minus119896) (119905lowast) (50)

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

where 120573 is defined by the second equality of (26) and

120588(120573)

1ℓ119905lowast(119905) = ℓ exp

times 119872(120573)

times int

119905

120590(minus1)(119905lowast)

int

+infin

119904

120585119899minusℓminus2

times (120590 (120585))1+ℓ120583(120585)minus120573

times119901 (120585) 119889120585 119889119904

(51)

120588(120573)

119894ℓ119905lowast(119905)

= ℓ +

1

(119899 minus ℓ)

times int

119905

120590(minus119894)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588119894minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904

(119894 = 2 119896)

(52)

119872(120573) =

1

ℓ (119899 minus ℓ)

if 120573 = 1

119872 if 120573 gt 1(53)

Remark 9 In Lemma 7 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 1)Consider (1) where 119899 is even and

120590 (119905) equiv 119905 119901 (119905) =

119899 119905log1119888119905

119905119899+1(119888119905 minus 1)

log1119888119905

120573 (119905) = log1119888119905 119905 ge

2

119888

(54)

It is obvious that the function 119906(119905) = 119888 minus (1119905) is thesolution of (1) and it satisfies the condition (201) for 119905 ge

(2119888) On the other hand the condition (281) holds but thecondition (22) is not fulfilled

Theorem 10 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let 120573 lt+infin and the conditions (2) (3) (28ℓ) and let

int

+infin

0

119905119899minusℓminus1

(120590 (119905))ℓ120583(119905)

119901 (119905) 119889119905 = +infin (55ℓ)

be fulfilled and for some 1199050 isin 119877+Uℓ1199050 = 0Then for any119872 gt 1

there exists 119905lowast gt 1199050 such that if 120572 = 1

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 = 0

(56)

and if 120572 gt 1 then for any 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))(ℓminus1)120583(120585)+120575

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(57)

where 120572 is defined by first equality of (26) and 120588(120572)119896ℓ119905lowast

is givenby (30)ndash(32)

Proof Let 119872 gt 1 and 1199050 isin 119877+ such that Uℓ1199050 = 0 Bydefinition (1) has a proper solution 119906 isin Uℓ1199050 satisfyingthe condition (20ℓ) with some 119905ℓ ge 1199050 Due to (1) (20ℓ)and (28ℓ) it is obvious that condition (22) holds Thus byLemma 5 there exists 1199052 gt 1199051 such that conditions (23119894)-(24)with 119905lowast = 1199052 are fulfilled On the other hand according toLemma 7 (and its proof) we see that

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

1199052

int

+infin

119904

120585119899minusℓminus1

119901 (120585) (119906 (120590 (120585)))120583(120585)119889120585 119889119904

for 119905 ge 1199052

(58)

and there exists 119905lowast gt 1199052 such that relation (30) is fulfilledWithout loss of generality we can assume that 120590(119905) ge 1199052 for119905 ge 119905lowast Therefore by (24) from (58) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)

119889120585 119889119904

(59)

Assume that 120572 = 1 Then by (44) and (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896minus1ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 for 119905 ge 120590(minus119896) (119905lowast) (60)

On the other hand according to (23ℓminus1) and (55ℓ) it is obviousthat

119906(ℓminus1)

(119905)

119905

darr 0 for 119905 uarr +infin (61)

Therefore from (60) we get

lim119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119889120585 119889119904 = 0

(62)

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Now assume that 120572 gt 1 and 120575 isin (1 120572] Then according to(47) (23ℓminus1) and (61) from (59) we have

119906(ℓminus1)

(119905)

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120583(120585)minus120575

times (

1

ℓ

119906(ℓminus1)

(120590 (120585)))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

times(

1

ℓ

119906(ℓminus1)

(120585))

120575

119889120585 119889119904

ge

1

(119899 minus ℓ)

int

119905

120590(minus119896)(119905lowast)

(

119906ℓminus1(119904)

ℓ

)

120575

times int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(63)

Thus we obtain

(V (119905))120575 ge1

(ℓ (119899 minus ℓ))120575

times (int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904)

120575

(64)

where V(119905) = (1ℓ)119906(ℓminus1)(119905)It is obvious that there exist 1199051 gt 120590(minus119896)(119905lowast) such that

int

119905

120590(minus119896)(119905lowast)

V120575 (119904) int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 gt 0

for 119905 ge 1199051(65)

Therefore from (64)

int

119905

1199051

1205931015840(119904)

(120593 (119904))120575119889119904

ge

1

(ℓ (119899 minus ℓ))120575

times int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(66)

where

120593 (119905)

= int

119905

120590(minus119896)(119905lowast)

(V (119904))120575 int+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times 119901 (120585) (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

(67)

From the last inequality we get

int

119905

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

[1205931minus120575

(1199051) minus 1205931minus120575

(119905)]

le

(ℓ (119899 minus ℓ))120575

120575 minus 1

1205931minus120575

(1199051) for 119905 ge 1199051

(68)

Passing to the limit in the latter inequality we get

int

+infin

1199051

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119889120585 119889119904 lt +infin

(69)

that is according to (62) and (69) (56) and (57) hold whichproves the validity of the theorem

Using Lemma 8 in a similar manner one can prove thefollowing

Theorem11 Let ℓ isin 1 119899minus1with ℓ+119899 be odd let (2) (3)(28ℓ) and (55ℓ) be fulfilled and for some 1199050 isin 119877+ Uℓ1199050 = 0Then there exists 119905lowast gt 1199050 such that if 120573 = 1 for any 119896 isin 119873

lim sup119905rarr+infin

1

119905

times int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)119901 (120585) 119889120585 119889119904 = 0

(70)

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

and if 1 lt 120573 lt +infin then for any 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 lt +infin

(71)

where 120573 is defined by the second equality of (26) and 120588 (120573)119896ℓ119905lowast

isgiven by (51)ndash(53)

4 Sufficient Conditions for Nonexistence ofSolutions of the Type (20ℓ)

Theorem 12 Let 120573 lt +infin ℓ isin 1 119899 minus 1 with ℓ + 119899 oddlet the conditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if120572 = 1 for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(72ℓ)

or if 120572 gt 1 for same 119896 isin 119873 and 120575 isin (1 120572]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

times (

1

ℓ

120588(120572)

119896ℓ119905lowast(120590 (120585)))

120583(120585)minus120575

119901 (120585) 119889120585 119889119904 = +infin

(73ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by (26) and 120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Assume the contrary Let there exist 1199050 isin 119877+ suchthatUℓ1199050 = 0 (seeDefinition 6)Then (1) has a proper solution119906 [1199050 +infin) rarr 119877 satisfying the condition (20ℓ) Since thecondition of Theorem 10 is fulfilled there exists 119905lowast gt 1199050 suchthat if 120572 = 1 (if 120572 gt 1) the condition (56) (the condition(57)) holds which contradicts (72ℓ) and (73ℓ) The obtainedcontradiction proves the validity of the theorem

Using Theorem 11 analogously we can prove the follow-ing

Theorem 13 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled and if 120573 = 1for any large 119905lowast isin 119877+ and for some 119896 isin 119873

lim sup119905rarr+infin

1

119905

int

119905

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

times (

1

ℓ

120588(1)

119896ℓ119905lowast(120590 (120585)))

120583(120585)

119901 (120585) 119889120585 119889119904 gt 0

(74ℓ)

or if 1 lt 120573 lt +infin for same 119896 isin 119873 and 120575 isin (1 120573]

int

+infin

120590(minus119896)(119905lowast)

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

times (120588(120573)

119896ℓ119905lowast(120590 (120585)))

120573minus120575

119901 (120585) 119889120585 119889119904 = +infin

(75ℓ)

Then for any 1199050 isin 119877+ we have Uℓ1199050 = 0 where 120573 is defined bythe second equality of (26) and 120588119896ℓ119905lowast is given by (51)ndash(53)

Corollary 14 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (55ℓ) be fulfilled 120572 = 1 120573 lt +infin and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (76ℓ)

Then Uℓ1199050 = 0 for any 1199050 isin 119877+

Proof Since

120588(120572)

1ℓ119905 (120590 (119905lowast)) ge ℓ for 119905 ge 120590(minus1) (119905lowast) (77)

it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ)are fulfilled for 119896 = 1

Corollary 15 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) and (3) be fulfilled 120572 = 1 120573 lt +infin and

lim inf119905rarr+infin

119905 int

+infin

119905

119904119899minusℓminus2

(120590 (119904))1+(ℓminus1)120583(119904)

119901 (119904) 119889119904 = 120574 gt 0 (78ℓ)

If moreover for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

timesint

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)(ℓ(119899minusℓ))))

119901 (120585) 119889120585 119889119904

gt 0

(79ℓ)

then for any 1199050 isin 119877+ Uℓ1199050 = 0

Proof Clearly by virtue of (78ℓ) conditions (28ℓ) and (55ℓ)are fulfilled Let 120576 isin (0 120574) According to (78ℓ) and (79ℓ) it isobvious that for large 119905 120588(1)

1ℓ119905lowast(119905) ge ℓ119905

(120574minus120576)(ℓ(119899minusℓ))Thereforeby (79ℓ) for 119896 = 1 (72ℓ) holds which proves the validity ofthe corollary

Corollary 16 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 120573 lt

+infin and for some 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin (80ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by thefirst condition of (3)

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Proof By (80ℓ) and (77) for 119896 = 1 the condition (73ℓ) holdswhich proves the validity of the corollary

Corollary 17 Let ℓ isin 1 119899 minus 1 with ℓ+ 119899 odd and let theconditions (2) (3) and (78ℓ) be fulfilled Moreover if 120572 gt 1120573 lt +infin and there exists119898 isin 119873 such that

lim inf119905rarr+infin

120590119898(119905)

119905

gt 0 (81)

then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120572 is defined by the firstcondition of (3)

Proof By (78ℓ) there exist 119888 gt 0 and 1199051 isin 119877+ such that

119905 int

+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 ge 119888 for 119905 ge 1199051 (82ℓ)

Let 120575 = (1 + 120572)2 and119872 = 119898(1 + 120572)119888(120572 minus 120575) Then by (82ℓ)and (30) for large 119905lowast gt 1199051

120588(120572)

1ℓ119905lowast(119905) ge 119905

119872119888119905 ge 119905lowast (83)

Therefore

(

120590 (119905)

119905

)

120575

(

1

ℓ

120588(120572)

1ℓ119905lowast(120590 (119905)))

120583(119905)minus120575

ge (

120590 (119905)

119905

)

120575

(

1

ℓ

120590119872119888(119905))

120572minus120575

=

1

(ℓ)120572minus120575

(

(120590 (119905))1+(119872119888(120572minus120575)120575)

119905

)

120575

gt

1

(ℓ)120572minus120575

(

(120590 (119905))119898

119905

)

120575

(84)

Thus by (81) and (78ℓ) it is obvious that (73ℓ) holds whichproves the corollary

Quite similarly one can prove the following

Corollary 18 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 120572 gt 1 and120573 lt +infin Moreover if

lim inf119905rarr+infin

119905 ln 119905 int+infin

119905

120585119899minusℓminus2

(120590 (120585))1+(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (85ℓ)

and for some 120575 isin (1 120572] and119898 isin 119873

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))120575+(ℓminus1)120583(120585)

(ln 120590 (120585))119898119889120585 119889119904 = +infin

(86ℓ)

then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 and 120573 aredefined by the condition of (26)

Corollary 19 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 be odd let theconditions (2) (3) and (28ℓ) be fulfilled 120572 gt 1 and 120573 lt +infinMoreover let there exist 120574 isin (0 1) and 119903 isin (0 1] such that

lim inf119905rarr+infin

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 0 (87ℓ)

lim inf119905rarr+infin

120590 (119905)

119905119903gt 0 (88)

and let at least one of the conditions

119903120572 ge 1 (89)

or 119903120572 lt 1 and for some 120576 gt 0 and 120575 isin (1 120572]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575+(120572119903(1minus120574)(1minus120572119903))minus120576

times (120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 = +infin

(90ℓ)

be fulfilled Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120572 isdefined by (26)

Proof It suffices to show that the condition (73ℓ) is satisfiedfor some 119896 isin 119873 and 120575 = (1+120572)2 Indeed according to (87ℓ)and (88) there exist 120574 isin (0 1) 119903 isin (0 1] 119888 gt 0 and 1199051 isin 119877+such that

119905120574int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 gt 119888 for 119905 ge 1199051 (91)

120590 (119905) ge 119888119905119903 for 119905 ge 1199051 (92)

By (77) (31) and (91) from (31) we get

120588(120572)

2ℓ119905lowast(119905)

ge

119888

(119899 minus ℓ)

int

119905

120590(minus1)(119905lowast)

119904minus120574119889119904

=

119888 (1199051minus120574

minus 1205901minus120574

(minus1)(119905lowast))

(119899 minus ℓ) (1 minus 120574)

for 119905 ge 120590(minus1) (119905lowast)

(93)

Let 1205741 isin (120574 1) Choose 1199052 gt 120590(minus1)(119905lowast) such that

120588(120572)

2ℓ119905lowast(119905) ge 119905

1minus1205741 for 119905 ge 1199052 (94)

Therefore by (91) from (31) we can find 1199053 gt 1199052 such that

120588(120572)

3ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903) for 119905 ge 1199053 (95)

Hence for any 1198960 isin 119873 there exists 1199051198960 such that

120588(120572)

1198960 ℓ119905lowast(119905) ge 119905

(1minus1205741)(1+120572119903+sdotsdotsdot+(120572119903)119896120572minus2) for 119905 ge 1199051198960 (96)

Assume that (89) is fulfilled Choose 1198960 isin 119873 such that1198960 minus 1 ge ((1 minus 119903)(1 + 120572))((1 minus 1205741)(120572 minus 1)) Then according

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

to (92) (96) and (28ℓ) the condition (73ℓ) holds for 119896 = 1198960and 120575 = (1 + 120572)2 In this case the validity of the corollaryhas already been proven

Assume now that (90ℓ) is fulfilled Let 120576 gt 0 and choose1198960 isin 119873 and 1205741 isin (120574 1) such that

(1 minus 1205741) (1 + 120572119903 + sdot sdot sdot + (120572119903)1198960minus2

) gt

(1 minus 120574) 120572119903

1 minus 120572119903

minus 120576 (97)

Then according to (96) (92) and (90ℓ) it is obvious that (73ℓ)holds for 119896 = 1198960 The proof of the corollary is complete

Using Theorem 13 in a manner similar to above we canprove the following

Corollary 20 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 119889119904 gt 0 (98ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0 where 120573 is given by (26)

Corollary 21 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) and (28ℓ) be fulfilled 120573 = 1 and

lim inf119905rarr+infin

119905 int

+infin

119905

120585119899minusℓminus1

(120590 (120585))(ℓminus1)120583(120585)

119901 (120585) 119889120585 = 120574 gt 0 (99ℓ)

Moreover let for some 120576 isin (0 120574)

lim sup119905rarr+infin

1

119905

int

119905

0

int

+infin

119904

120585119899minusℓminus1

(120590 (120585))120583(120585)(ℓminus1+((120574minus120576)ℓ(119899minusℓ)))

times 119901 (120585) 119889120585 119889119904 gt 0

(100ℓ)

Then for any 1199050 isin 119877+ one has Uℓ1199050 = 0 where 120573 is given by(26)

Corollary 22 Let ℓ isin 1 119899 minus 1 with ℓ + 119899 odd let theconditions (2) (3) (28ℓ) and (55ℓ) be fulfilled 1 lt 120573 lt +infinand for some 120575 isin (1 120573]

int

+infin

0

int

+infin

119904

120585119899minusℓminus1minus120575

(120590 (120585))ℓ120583(120585)+120575minus120573

119901 (120585) 119889120585 119889119904 = +infin

(101ℓ)

Then for any 1199050 isin 119877+ Uℓ1199050 = 0

5 Differential Equation with Property A

Theorem 23 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (74ℓ)and (75ℓ) hold Moreover for any large 119905lowast isin 119877+ if 120572 = 1 and120573 lt +infin for some 119896 isin 119873 let (72ℓ) be fulfilled or if 120572 gt 1 and120573 lt +infin for some 119896 isin 119873 119872 isin (1 +infin) and 120575 isin (1 120572]let (72ℓ) be fulfilled Then if for odd 119899

int

+infin

0

119905119899minus1119901 (119905) 119889119905 = +infin (102)

then (1) has PropertyA where 120572 and 120573 are defined by (26) and120588(120572)

119896ℓ119905lowastis given by (30)ndash(32)

Proof Let (1) have a proper nonoscillatory solution 119906

[1199050 +infin) rarr (0 +infin) (the case 119906(119905) lt 0 is similar) Thenby (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1 suchthat ℓ+119899 is odd and conditions (20ℓ) hold Since for any ℓ isin1 119899minus1with ℓ+119899 odd the conditions ofTheorem 12 arefulfilled we have ℓ notin 1 119899 minus 1 Now assume that ℓ = 0119899 is odd and there exists 119888 isin (0 1) such that 119906(119905) ge 119888 forsufficiently large 119905 According to (200) since 120573 lt +infin from(1) we have

119899minus1

sum

119894=0

(119899 minus 119894 minus 1)119905119894

1

10038161003816100381610038161003816119906(119894)(1199051)

10038161003816100381610038161003816

ge int

119905

1199051

119904119899minus1119901 (119904) 119888

120583(119904)119889119904

ge 119888120573int

119905

1199051

119904119899minus1119901 (119904) 119889119904 for 119905 ge 1199051

(103)

where 1199051 is a sufficiently large number The last inequalitycontradicts the condition (102) The obtained contradictionproves that (1) has Property A

Theorem 24 Let the conditions (2) (3) be fulfilled and forany ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditions (28ℓ) and(55ℓ) hold Let moreover if 120573 = 1 for some 119896 isin 119873 (74ℓ) holdor if 1 lt 120573 lt +infin and for some 119896 isin 119873119872 isin (1 +infin) and120575 isin (1 120573] (75ℓ) hold Then if for odd 119899 (102) is fulfilled then(1) has PropertyA where 120573 is defined by the second equality of(26) and 120588 (120573)

119896ℓ119905lowastis given by (51)ndash(53)

Proof The proof of the theorem is analogous to thatof Theorem 23 We simply use Theorem 13 instead ofTheorem 12

Theorem 25 Let 120572 gt 1 120573 lt +infin let the conditions (2) (3)(281) and (551) be fulfilled and

lim inf119905rarr+infin

(120590 (119905))120583(119905)

119905

gt 0 (104)

If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572](731) holds and for odd n (102) is fulfilled then (1) has PropertyA where 120572 and 120573 are defined by (26) and 120588(120572)

1198961119905lowastis given by

(30)ndash(32)

Proof It suffices to note that by (104) and (721) for any ℓ isin2 119899 minus 1 there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572) suchthat condition (72ℓ) is fulfilled

Theorem 26 Let 1 lt 120573 lt +infin and conditions (2) (3) (281)(29) and (104) be fulfilled If moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (751) holds and for odd n (102)is fulfilled then (1) has Property A where 120573 is defined by thesecond condition of (26) and 120588 (120573)

1198961119905lowastis given by (51)ndash(55)

Proof The theorem is proved similarly to Theorem 25 if wereplace the condition (731) by the condition (751)

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

Corollary 27 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(281) (761) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Proof By (281) (761) and (104) condition (102) and forany ℓ isin 1 119899 minus 1 (76ℓ) holds Now assume that (1) hasa proper nonoscillatory solution 119906 [1199050 +infin) rarr (0 +infin)Then by (2) (3) and Lemma 3 there exists ℓ isin 0 119899 minus 1such that ℓ + 119899 is odd and the condition (20ℓ) holds Sincefor any ℓ isin 1 119899 minus 1 with ℓ + 119899 odd the conditionsof Corollary 14 are fulfilled we have ℓ notin 1 119899 minus 1Therefore 119899 is odd and ℓ = 0 According to (102) and (200)it is obvious that the condition (5) holds Therefore (1) hasProperty A

Using Corollaries 15ndash19 the validity of Corollaries 28ndash32can be proven similarly to Corollary 27

Corollary 28 Let 120572 = 1 120573 lt +infin and conditions (2) (3)(781) (791) and (104) be fulfilled Then (1) has Property Awhere 120572 and 120573 are given by (26)

Corollary 29 Let 120572 gt 1 120573 lt +infin conditions (2) (3) (104)be fulfilled and (801) for some 120575 isin (1 120572] hold and if 119899 is oddcondition (102) holds Then (1) has Property A

Corollary 30 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (781) (104) be fulfilled and (81) for some119898 isin 119873 holdsThen (1) has Property A

Corollary 31 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(281) (104) and (851) and for some 120575 isin (1 120572] and 119898 isin

119873 (861) be fulfilled Then (1) has Property A where 120572 and 120573are given by (26)

Corollary 32 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (281) and (104) be fulfilled Let moreover there exist 120574 isin(0 1) and 119903 isin (0 1] such that (871) and (88) hold Then eithercondition (89) or condition (901) is sufficient for (1) to haveProperty A

Corollary 33 Let 120573 = 1 and the conditions (2) (3) (281)(104) and (981) be fulfilled Then (1) has Property A where 120573is defined by the second condition of (26)

Proof By (281) (104) and (981) the condition (102) and forany ℓ isin 1 119899minus1 (98ℓ) holdsTherefore by Corollary 20for any 1199050 isin 119877+ and ℓ isin 1 119899minus1with ℓ+119899 is oddUℓ1199050 =0 On the other hand if 119899 is odd and ℓ = 0 according to (102)it is obvious that the condition (5) holds which proves that(1) has Property A

Using Corollaries 21 and 22 we can analogously prove thefollowing corollaries

Corollary 34 Let 120573 = 1 and the conditions (2) (3) (104)(991) and (1001) be fulfilledThen (1) has PropertyA where 120573is given by (26)

Corollary 35 Let 1 lt 120573 lt +infin the conditions (2) (3)(104) (281) and (291) be fulfilled and if 119899 is odd (102) holds

Moreover if for some 120575 isin (1 120573) (1011) holds then (1) hasProperty A

Theorem 36 Let 120572 gt 1 120573 lt +infin the conditions (2) (3)(28119899minus1) and (29119899minus1) be fulfilled and

lim sup119905rarr+infin

(120590 (119905))120583(119905)

119905

lt +infin (105)

Moreover for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120572] let(73119899minus1) be fulfilledThen (1) has PropertyA where 120572 and 120573 aredefined by (26)

Proof It suffices to note that by (28119899minus1) (29119899minus1) (105) and(73119899minus1) there exist119872 gt 1 119896 isin 119873 and 120575 isin (1 120572] such that(28ℓ) (29ℓ) and (73ℓ) hold for any ℓ isin 1 119899 minus 2

Theorem 37 Let 1 lt 120573 lt +infin and the conditions (2) (3)(28119899minus1) (29119899minus1) and (105) be fulfilled and for some 119896 isin 119873119872 isin (1 +infin) and 120575 isin (1 120573] (75119899minus1) holds Then (1) hasProperty A where 120573 is given by the second condition of (26)

Proof The proof is similar to that of Theorem 36 we replacethe condition (73119899minus1) by the condition (75119899minus1)

Corollary 38 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (76119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Proof By (28119899minus1) (76119899minus1) and (105) the condition (102) andfor any ℓ isin 1 119899minus1 the condition (76ℓ)holds it is obviousthat (1) has Property A

Using Corollaries 15ndash19 the validity of Corollaries 39ndash43below can be proven similarly to Corollary 38

Corollary 39 Let 120572 = 1 120573 lt +infin and the conditions (2) (3)(78119899minus1) (79119899minus1) and (105) be fulfilled Then (1) has PropertyA where 120572 and 120573 are given by (26)

Corollary 40 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(105) and for some 120575 isin (1 120572] (80119899minus1) be fulfilledThen (1) hasProperty A where 120572 is given by (26)

Corollary 41 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (78119899minus1) (105) and for some 119898 isin 119873 (81) be fulfilledThen (1) has Property A where 120572 and 120573 are given by (26)

Corollary 42 Let 120572 gt 1 120573 lt +infin and the conditions (2) (3)(28119899minus1) (85119899minus1) and (105) be fulfilled and for some 120575 isin (1 120572]and 119898 isin 119873 (86119899minus1) holds Then (1) has Property A where 120572and 120573 are given by (26)

Corollary 43 Let 120572 gt 1 120573 lt +infin and the conditions (2)(3) (28119899minus1) and (105) be fulfilled Let moreover there exist120574 isin (0 1) and 119903 isin (0 1) such that (87119899minus1) and (88) hold Theneither condition (89) or condition (90119899minus1) is sufficient for (1) tohave Property A

Corollary 44 Let 120573 = 1 and the conditions (2) (3) (28119899minus1)(105) and (98119899minus1) be fulfilled Then (1) has Property A

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

Proof By (28119899minus1) (105) and (98119899minus1) the conditions (102)and for any ℓ isin 1 119899 minus 1 (98ℓ) holds Therefore byCorollary 20 it is clear that (1) has Property A

Using Corollaries 21 and 22 analogously we can proveCorollaries 45 and 518

Corollary 45 Let 120573 = 1 and the conditions (2) (3) (105)(99119899minus1) and (100119899minus1) be fulfilled Then (1) has Property Awhere 120573 is given by (26)

Corollary 46 Let 1 lt 120573 lt +infin and the conditions (2) (3)(105) and (29119899minus1) be fulfilled If for some 120575 isin (1 120573] (101119899minus1)holds then (1) has Property A

6 Necessary and Sufficient Conditions

Theorem 47 Let 120572 gt 1 and 120573 lt +infin let the conditions (2)and (3) be fulfilled and

lim inf119905rarr+infin

120590 (119905)

119905

gt 0 (106)

Then the condition (102) is necessary and sufficient for (1) tohave Property A where 120572 and 120573 are given by (26)

Proof Necessity Assume that (1) has Property A and

int

+infin

0

119905119899minus1119901 (119905) 119889119905 lt +infin (107)

Therefore by Lemma 41 from [28] there exists 119888 = 0 such that(1) has a proper solution 119906 [0 +infin) rarr 119877 satisfying thecondition lim119905rarr+infin119906(119905) = 119888 But this contradicts the fact that(1) has Property ASufficiency By (106) and (102) it is obvious that the con-dition (801) holds Therefore the sufficiency follows fromCorollary 29

Remark 48 InTheorem 47 the condition 120573 lt +infin cannot bereplaced by the condition 120573 = +infin Indeed let 119888 isin (0 12)120572 = 12119888 and

119901 (119905) =

119899119905lg120572119905

1199051+119899(1 + 119888119905)

lg120572119905

119905 ge 1 (108)

It is obvious that the condition (102) is fulfilled but equation

119906(119899)(119905) + 119901 (119905) |119906 (119905)|

lg120572119905 sign 119906 (119905) = 0 (109)

has solution 119906(119905) = (1119905 + 119888) Therefore (109) does not haveProperty A

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theworkwas supported by the Sh Rustaveli National ScienceFoundation Grant no 3109

References

[1] I J H Lane ldquoOn the theoretical temperature of the sun underthe hypothesis of a gaseous mass maintaining its volume byits internal heat and depending on the laws of gases known toterrestial experimentrdquoAmerican Journal of Science andArts vol4 pp 57ndash74 1870

[2] R H Fowler ldquoFurther studies of Emdenrsquos and similar differen-tial equationsrdquo Quarterly Journal of Mathematics vol 2 no 1pp 259ndash288 1931

[3] R Bellman Stability Theory of Differential Equations MaGraw-Hill New York NY USA 1953

[4] G Sansone Equazioni Differenziali nel Campo Reale IIZanichelli Bologna Italy 3rd edition 1963

[5] R Conti D Graffi and G Sansone ldquoThe Italian contribution tothe theory of non-linear ordinary differential equations and tononlinear mechanics during the years 1951ndash1961rdquo inQualitativeMethods in the Theory of Nonlinear Vibrations II pp 172ndash1891961

[6] F V Atkinson ldquoOn second order nonlinear oscillationrdquo PacificJournal of Mathematics vol 5 no 5 pp 643ndash647 1955

[7] J Kurzweil ldquoA note on oscillatory solutions of the equation 11991010158401015840+119891(119909)119910

2119899minus1= 0rdquo Casopis Pro Pestovanı Matematiky vol 85 pp

357ndash358 1960[8] I T Kiguradze and T A Chanturia Asymptotic Properties of

Solutions of Nonautonomous Ordinary Differential EquationsNauka Moscow Russia 1990

[9] S Belohorec ldquoOscillatory solutions of certain nonlinear dif-ferential equations of second orderrdquo Matematicko-FyzikalnyCasopis vol 11 pp 250ndash255 1961

[10] R S Dahiya and B Singh ldquoOn oscillatory behavior of evenorder delay equationsrdquo Journal of Mathematical Analysis andApplications vol 42 no 1 pp 183ndash190 1973

[11] R G Koplatadze ldquoOn oscillatory solutions of second orderdelay differential inequalitiesrdquo Journal of Mathematical Analysisand Applications vol 42 no 1 pp 148ndash157 1973

[12] R G Koplatadze ldquoOn some properties of solutions of non-linear differential inequalities and equations with a delayedargumentrdquoDifferentsialrsquonye Uravneniya vol 12 no 11 pp 1971ndash1984 1976 (Russian)

[13] R G Koplatadze and T A Chanturia On Some Properties ofSolutions of Nonlinear Differential Inequalities and Equationswith a Delayed Argument Tbilisi State University Press TbilisiGeorgia 1977 (Russian)

[14] S R Grace and B S Lalli ldquoOscillation theorems for certainsecond-order perturbed nonlinear differential equationsrdquo Jour-nal of Mathematical Analysis and Applications vol 77 no 1 pp205ndash214 1980

[15] S R Grace and B S Lalli ldquoOscillation theorems for nth-orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 91 no 2 pp 352ndash366 1983

[16] A G Kartsatos ldquoRecent results on oscillation of solutions offorced and perturbed nonlinear differential equations of evenorderrdquo in Stability of Dynamic SystemsTheory and Applicationsvol 28 of Lecture Notes in Pure and AppliedMathematics pp 17ndash72 Dekker New York NY USA 1977

[17] T Kusano and H Onose ldquoOscillations of functional differen-tial equations with retarded argumentrdquo Journal of DifferentialEquations vol 15 no 2 pp 269ndash277 1974

[18] W E Mahfoud ldquoOscillation theorems for a second order delaydifferential equationrdquo Journal of Mathematical Analysis andApplications vol 63 no 2 pp 339ndash346 1978

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 13

[19] W E Mahfoud ldquoOscillation and asymptotic behavior of solu-tions of Nth order nonlinear delay differential equationsrdquoJournal of Differential Equations vol 24 no 1 pp 75ndash98 1977

[20] U Sztaba ldquoNote on theDahiya and Singh paper ldquoOn oscillatorybehavior of even-order delay equationsrdquordquo Journal of Mathemat-ical Analysis and Applications vol 63 no 2 pp 313ndash318 1978

[21] J R Graef R Koplatadze and G Kvinikadze ldquoNonlinearfunctional differential equations with Properties A and BrdquoJournal of Mathematical Analysis and Applications vol 306 no1 pp 136ndash160 2005

[22] R Koplatadze ldquoQuasi-linear functional differential equationswith Property Ardquo Journal of Mathematical Analysis and Appli-cations vol 330 no 1 pp 483ndash510 2007

[23] R Koplatadze ldquoOn oscillatory properties of solutions of gener-alized Emden-Fowler type differential equationsrdquoProceedings ofA Razmadze Mathematical Institute vol 145 pp 117ndash121 2007

[24] R Koplatadze ldquoOn asymptotic behaviors of solutions of ldquoalmostlinearrdquo and essential nonlinear functional differential equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 71 no 12 pp e396ndashe400 2009

[25] R Koplatadze and E Litsyn ldquoOscillation criteria for higherorder ldquoalmost linearrdquo functional differential equationrdquo Func-tional Differential Equations vol 16 no 3 pp 387ndash434 2009

[26] R Koplatadze ldquoOn asymptotic behavior of solutions of n-th order Emden-Fowler differential equations with advancedargumentrdquo Czechoslovak Mathematical Journal vol 60 no 3pp 817ndash833 2010

[27] R Koplatadze ldquoOscillation criteria for higher order nonlinearfunctional differential equations with advanced argumentrdquoNonlinear Oscillations vol 16 no 1 pp 44ndash64 2013

[28] R Koplatadze ldquoOn oscillatory properties of solutions of func-tional differential equationsrdquoMemoirs on Differential Equationsand Mathematical Physics vol 3 pp 3ndash179 1994

[29] M K Grammatikopulos R Koplatadze and G KvinikadzeldquoLinear functional differential equations with Property ArdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 294ndash314 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of