Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 425102 6 pageshttpdxdoiorg1011552013425102

Research ArticleOn the Stability of Nonautonomous Linear ImpulsiveDifferential Equations

JinRong Wang12 and Xuezhu Li2

1 School of Mathematics and Computer Science Guizhou Normal College Guiyang Guizhou 550018 China2Department of Mathematics Guizhou University Guiyang Guizhou 550025 China

Correspondence should be addressed to JinRong Wang wjr9668126com

Received 2 May 2013 Accepted 21 June 2013

Academic Editor Gestur Olafsson

Copyright copy 2013 J Wang and X Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce two Ulamrsquos type stability concepts for nonautonomous linear impulsive ordinary differential equations Ulam-Hyersand Ulam-Hyers-Rassias stability results on compact and unbounded intervals are presented respectively

1 Introduction

During the past decades the impulsive differential equationshave attracted many authors since it is better to describedynamics of populations subject to abrupt changes as well asother phenomena such as harvesting and diseases than thecorresponding differential equations without impulses Forthe basic theory on the impulsive differential equations andimpulsive controls the reader can refer to the monographsof Baınov and Simeonov [1] Lakshmikantham et al [2] Yang[3] and Benchohra et al [4] and references therein In partic-ular exponential asymptotical strong weak and Lyapunovstability of all kinds of impulsive differential equations hasbeen studied extensively in the previous monographs andreferences therein

In addition to the previously mentioned stability theoryUlam stability of functional equation which was formulatedby Ulam on a talk given to a conference at WisconsinUniversity in 1940 is one of the central subjects in themathematical analysis area Many researchers paid muchattention to discuss the stability properties of all kinds ofequations In fact Ulamrsquos type stability problems have beentaken up by a large number of mathematicians and the studyof this area has grown to be one of the most importantsubjects in the mathematical analysis area For the advancedcontribution on such problems we refer the reader to AndrasandKolumban [5] Andras andMeszaros [6] Burger et al [7]Cadariu [8] Castro and Ramos [9] Cieplinski [10] Cimpean

and Popa [11] Hyers et al [12] Hegyi and Jung [13] Jung[14 15] Lungu and Popa [16] Miura et al [17 18] Moslehianand Rassias [19] Rassias [20 21] Rus [22 23] Takahasi et al[24] and Wang et al [25ndash27]

As far as we know there are few results on Ulamrsquos typestability of nonautonomous impulsive differential equationsMotivated by recent works [23 25 27] we study Ulamrsquostype stability of nonautonomous linear impulsive differentialequations

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905)

119905 isin 1198691015840= 119869 119905

1 119905

119898

119869 = [0 119871)

0 lt 119871 le +infin

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

(1)

where 119909(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879 119891(119905) = (119891

1(119905)

1198912(119905) 119891

119899(119905))119879isin 119862(119869R119899) 119860(119905) = diag(119860

1(119905) 1198602(119905)

119860119899(119905)) is 119899-order real diagonal matrix and 119861

119896= diag(119861119896

1

119861119896

2 119861

119896

119899) and 119887

119896= (119887119896

1 119887119896

2 119887

119896

119899)119879 are 119899-order bounded

diagonal matrix and 119899-dimensional bounded vector respec-tively Impulsive sequence 119905

119896satisfy 0 = 119905

0lt 1199051lt sdot sdot sdot lt 119905

119898lt

119905119898+1

=119871 Δ119909(119905119896) =119909(119905

+

119896)minus119909(119905

minus

119896) and 119909(119905+

119896) = lim

120598rarr0+119909(119905119896+120598)

2 Journal of Function Spaces and Applications

and119909(119905minus119896) = lim

120598rarr0minus119909(119905119896+120598) represent the right and left limits

of 119909(119905) at 119905 = 119905119896 119896 = 1 2 119898

Firstly we will modify the Ulamrsquos type stability conceptsin [23] and introduce two Ulamrsquos type stability concepts for(1) Secondly we pay attention to check the Ulam-Hyersand Ulam-Hyers-Rassias stability results on a compact andunbounded intervals respectively

2 Preliminaries

Let 119862(119869R119899) be the Banach space of all continuousfunctions from 119869 into R119899 with the norm 119909 =

max1199091119862 1199092119862 119909

119899119862 for 119909 isin 119862(119869R119899) where

119909119896119862

= sup119905isin119869|119909119896(119905)| Also we use the Banach space

119875119862(119869R119899) = 119909 119869 rarr R119899 119909 isin 119862((119905119896 119905119896+1

)R119899) 119896 = 01 119898 and there exist 119909(119905minus

119896) and 119909(119905

+

119896) 119896 = 1 119898

with 119909(119905minus

119896) = 119909(119905

119896) with the norm 119909

119875119862= max119909

1119875119862

1199092119875119862 119909

119899119875119862 Denote 1198751198621(119869R119899) = 119909 isin 119875119862(119869R119899)

1199091015840isin 119875119862(119869R119899) Set 119909

1198751198621 = max119909

119875119862 1199091015840119875119862 It can be

seen that endowed with the norm sdot 1198751198621 1198751198621(119869R119899) is also

a Banach spaceIf 119909 119910 isin R119899 119909 = (119909

1 1199092 119909119899) 119910 = (119910

1 1199102 119910119899) by

119909 le 119910 we mean that 119909119894le 119910119894for 119894 = 1 2 119899

It follows [3] we introduce the concept of piecewisecontinuous solutions

Definition 1 By a 1198751198621 from solution of the following impul-sive Cauchy problem

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin 1198691015840= 119869 119905

119896

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

119909 (0) = 1199090isin R119899

(2)

we mean that the function 119909 isin 1198751198621(119869R119899) which satisfies

119909 (119905) = Ψ (119905 0) 1199090+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+ sum

0lt119905119896lt119905

Ψ (119905 119905+

119896) 119887119896 119905 isin 119869

(3)

where Ψ is called impulsive evolution matrix which is givenby

Ψ (119905 120579) =

Φ(119905 120579) 119905119896minus1

le 120579 le 119905 le 119905119896

Φ (119905 119905+

119896) (119868 + 119861

119896)Φ (119905119896 120579)

119905119896minus1

le 120579 lt 119905119896lt 119905 le 119905

119896+1

Φ (119905 119905+

119896)[

[

prod

120579lt119905119895lt119905

(119868 + 119861119895)Φ (119905

119895 119905+

119895minus1)]

]

times (119868 + 119861119894)Φ (119905119894 120579)

119905119894minus1

le 120579 lt 119905119894le sdot sdot sdot lt 119905

119896lt 119905 le 119905

119896+1

119896 = 1 2 119898

(4)

Φ is the evolution matrix for the system 1199091015840= 119860(119905)119909 and 119868

denotes the identity matrix

If there exists119872 gt 0 such that 119860(119905) = max119905isin119869|119860119894(119905)|

119894 = 1 2 119899 le 119872 for any 119905 isin 119869 thenΦ satisfy

Φ (119905 119904) le 119890119872(119905minus119904)

forall119904 119905 isin 119869 119904 le 119905 (5)

By proceeding with the same elementary computation inLemma 25(5) of [28] we have

Ψ (119905 119904) le 119890119872(119905minus119904)

119898

prod

119894=1

(1 +1003817100381710038171003817119861119894

1003817100381710038171003817) forall119904 119905 isin 119869 119904 lt 119905 (6)

Next we introduce two Ulamrsquos type stability definitionsfor (1) which can be regarded as the extension of the Ulamrsquostype stability concepts for ordinary differential equations in[23]

Let 120598119894gt 0 120595

119894ge 0 and 120593

119894isin 119875119862(119869R

+) be nondecreasing

functions where 119894 = 1 2 119899 For 119905 isin 119869 denote

120577 (119905) =

1 if 119871 lt +infin

119890minus119872119905

if 119871 = +infin

120585 (119905) =

1 if 119871 lt +infin

119890minus119872(119905minus119905

119895) if 119871 = +infin

119905119895isin 1199051 1199052 119905

119898

(7)

We consider the following inequalities

(

(

100381610038161003816100381610038161199101015840

1(119905) minus 119860

1(119905) 1199101(119905) minus 119891

1(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

2(119905) minus 119860

2(119905) 1199102(119905) minus 119891

2(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

119899(119905) minus 119860

119899(119905) 119910119899(119905) minus 119891

119899(119905)

10038161003816100381610038161003816

)

)

le 120577 (119905) (1205981 1205982 120598

119899)119879

119905 isin 1198691015840

(

(

10038161003816100381610038161003816Δ1199101(119905119896) minus 119861119896

11199101(119905minus

119896) minus 119887119896

1

10038161003816100381610038161003816

10038161003816100381610038161003816Δ1199102(119905119896) minus 119861119896

21199102(119905minus

119896) minus 119887119896

2

10038161003816100381610038161003816

10038161003816100381610038161003816Δ119910119899(119905119896) minus 119861119896

119899119910119899(119905minus

119896) minus 119887119896

119899

10038161003816100381610038161003816

)

)

le 120585 (119905) (1205981 1205982 120598

119899)119879

119896 = 1 2 119898

(8)

Journal of Function Spaces and Applications 3

(

(

100381610038161003816100381610038161199101015840

1(119905) minus 119860

1(119905) 1199101(119905) minus 119891

1(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

2(119905) minus 119860

2(119905) 1199102(119905) minus 119891

2(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

119899(119905) minus 119860

119899(119905) 119910119899(119905) minus 119891

119899(119905)

10038161003816100381610038161003816

)

)

le 120577 (119905) (1205931(119905) 1205981 1205932(119905) 1205982 120593

119899(119905) 120598119899)119879

119905 isin 1198691015840

(

(

10038161003816100381610038161003816Δ1199101(119905119896) minus 119861119896

11199101(119905minus

119896) minus 119887119896

1

10038161003816100381610038161003816

10038161003816100381610038161003816Δ1199102(119905119896) minus 119861119896

21199102(119905minus

119896) minus 119887119896

2

10038161003816100381610038161003816

10038161003816100381610038161003816Δ119910119899(119905119896) minus 119861119896

119899119910119899(119905minus

119896) minus 119887119896

119899

10038161003816100381610038161003816

)

)

le 120585 (119905) (12059511205981 12059521205982 120595

119899120598119899)119879

119896 = 1 2 119898

(9)

Definition 2 Equation (1) is Ulam-Hyers stable if there existconstants 119862119894

119891119898gt 0 119894 = 1 2 119899 such that for each 120598

119894gt 0

and for each solution 119910 isin 1198751198621(119869R119899) of inequality (8) there

exists a solution 119909 isin 1198751198621(119869R119899) of (1) with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (1198621

1198911198981205981 1198622

1198911198981205982 119862

119899

119891119898120598119899)

119879

119905 isin 119869

(10)

Definition 3 Equation (1) is Ulam-Hyers-Rassias stable withrespect to (120593

119894 120595119894) if there exist 119862

119891119898120593119894

gt 0 119894 = 1 2 119899 suchthat for each 120598

119894gt 0 and for each solution 119910 isin 119875119862

1(119869R119899) of

inequality (9) there exists a solution 119909 isin 1198751198621(119869R119899) of (1)

with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 1205981

119862119891119898120593

2

(1205932(119905) + 120595

2) 1205982

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598119899

) 119905 isin 119869

(11)

3 Stability Results in the Case 119871lt+infin

We introduce the following assumptions

(1198671) 119891 isin 119862(119869R119899)

(1198672) 119860119894isin 119862(119869R) 119894 = 1 2 119899

Denote119872119894= max

119905isin119869|119860119894(119905)| and denote119872 = max119872

1

1198722 119872

119899

Now we are ready to state our first Ulam-Hyers stableresult on a compact interval

Theorem 4 Assume that (1198671)-(1198672) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8)

Define

119892 (119905) = (1198921(119905) 1198922(119905) 119892

119899(119905))119879

= 1199101015840

(119905) minus 119860 (119905) 119910 (119905) minus 119891 (119905)

119905 isin 1198691015840

119892119896= (119892119896

1 119892119896

2 119892

119896

119899)

119879

= Δ119910 (119905119896) minus 119861119896119910 (119905minus

119896) minus 119887119896

119896 = 1 2 119898

(12)

Then we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120598max = max 120598

1 1205982 120598

119899 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120598max 119896 = 1 2 119898

(13)

According to Definition 1 for each 119905 isin (119905119896 119905119896+1

] we have

119910 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+ int

119905

0

Ψ (119905 119904) 119892 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) (119887119896+ 119892119896)

(14)

Suppose that 119909 be the unique solution of the impulsiveCauchy problem

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin 1198691015840

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

119909 (0) = 119910 (0) 119910 (0) isin R119899

(15)

Then for each 119905 isin (119905119896 119905119896+1

] we have

119909 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

(16)

4 Journal of Function Spaces and Applications

It follows that for each 119905 isin (119905119896 119905119896+1

] we can derive

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

119910 (119905) minus Ψ (119905 0) 119910 (0)

minus

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

minusint

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817

le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817119892119896

1003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

+ 119871119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

= 119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898 + 119871) 120598max

(17)

Thus

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(18)

where

119862119891119898

= 119890119872119871

(119898 + 119871)

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (19)

So (1) is Ulam-Hyers stable The proof is completed

In order to discuss Ulam-Hyers-Rassias stability we needthe following condition

(1198673) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890119872(119905minus119904)

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (20)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

Theorem 5 Assume that (1198671)ndash(1198673) are satisfied Then (1) is

Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120595119894120598max 119896 = 1 2 119898

(21)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) int

119905

0

119890119872(119905minus119904)

120593119894(119905) 119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898119890119872119871

+ 120582120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(22)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(23)

where

119862119891119898120593

119894

= (119898119890119872119871

+ 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(24)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

4 Stability Results in the Case 119871 = +infin

In this section we will present stability results on anunbounded interval

We change (1198672) to the following strong condition

(1198672

1015840) 119860119894is continuous and uniformly bounded function on

119869 119894 = 1 2 119899 Thus there exists 119872 gt 0 such that119860(119905) le 119872 for any 119905 isin 119869

Journal of Function Spaces and Applications 5

Theorem 6 Assume that (1198671)ndash(1198672

1015840) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8) For

119892 and 119892119896 defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120598max 119896 = 1 2 119898

(25)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] we have

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817100381711989211989610038171003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

119889119904

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) 120598max

(26)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(27)

where

119862119891119898

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) (28)

Thus (1) is Ulam-Hyers stable The proof is completed

Next we suppose the following

(1198673

1015840) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890minus119872119904

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (29)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

We have

Theorem 7 Assume that (1198671)ndash(1198672

1015840) and (119867

3

1015840) are satisfied

Then (1) is Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120595119894120598max 119896 = 1 2 119898

(30)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673

1015840) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

120593119894(119905) 119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 + 120582

120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(31)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(32)

where119862119891119898120593

119894

= (1 + 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(33)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

Acknowledgments

The authors thank the referee for valuable comments andsuggestions which improved their paper This work is sup-ported by the National Natural Science Foundation of China(11201091) Key Projects of Science and Technology Researchin the Chinese Ministry of Education (211169) and KeySupport Subject (Applied Mathematics) of Guizhou NormalCollege

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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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

2 Journal of Function Spaces and Applications

and119909(119905minus119896) = lim

120598rarr0minus119909(119905119896+120598) represent the right and left limits

of 119909(119905) at 119905 = 119905119896 119896 = 1 2 119898

Firstly we will modify the Ulamrsquos type stability conceptsin [23] and introduce two Ulamrsquos type stability concepts for(1) Secondly we pay attention to check the Ulam-Hyersand Ulam-Hyers-Rassias stability results on a compact andunbounded intervals respectively

2 Preliminaries

Let 119862(119869R119899) be the Banach space of all continuousfunctions from 119869 into R119899 with the norm 119909 =

max1199091119862 1199092119862 119909

119899119862 for 119909 isin 119862(119869R119899) where

119909119896119862

= sup119905isin119869|119909119896(119905)| Also we use the Banach space

119875119862(119869R119899) = 119909 119869 rarr R119899 119909 isin 119862((119905119896 119905119896+1

)R119899) 119896 = 01 119898 and there exist 119909(119905minus

119896) and 119909(119905

+

119896) 119896 = 1 119898

with 119909(119905minus

119896) = 119909(119905

119896) with the norm 119909

119875119862= max119909

1119875119862

1199092119875119862 119909

119899119875119862 Denote 1198751198621(119869R119899) = 119909 isin 119875119862(119869R119899)

1199091015840isin 119875119862(119869R119899) Set 119909

1198751198621 = max119909

119875119862 1199091015840119875119862 It can be

seen that endowed with the norm sdot 1198751198621 1198751198621(119869R119899) is also

a Banach spaceIf 119909 119910 isin R119899 119909 = (119909

1 1199092 119909119899) 119910 = (119910

1 1199102 119910119899) by

119909 le 119910 we mean that 119909119894le 119910119894for 119894 = 1 2 119899

It follows [3] we introduce the concept of piecewisecontinuous solutions

Definition 1 By a 1198751198621 from solution of the following impul-sive Cauchy problem

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin 1198691015840= 119869 119905

119896

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

119909 (0) = 1199090isin R119899

(2)

we mean that the function 119909 isin 1198751198621(119869R119899) which satisfies

119909 (119905) = Ψ (119905 0) 1199090+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+ sum

0lt119905119896lt119905

Ψ (119905 119905+

119896) 119887119896 119905 isin 119869

(3)

where Ψ is called impulsive evolution matrix which is givenby

Ψ (119905 120579) =

Φ(119905 120579) 119905119896minus1

le 120579 le 119905 le 119905119896

Φ (119905 119905+

119896) (119868 + 119861

119896)Φ (119905119896 120579)

119905119896minus1

le 120579 lt 119905119896lt 119905 le 119905

119896+1

Φ (119905 119905+

119896)[

[

prod

120579lt119905119895lt119905

(119868 + 119861119895)Φ (119905

119895 119905+

119895minus1)]

]

times (119868 + 119861119894)Φ (119905119894 120579)

119905119894minus1

le 120579 lt 119905119894le sdot sdot sdot lt 119905

119896lt 119905 le 119905

119896+1

119896 = 1 2 119898

(4)

Φ is the evolution matrix for the system 1199091015840= 119860(119905)119909 and 119868

denotes the identity matrix

If there exists119872 gt 0 such that 119860(119905) = max119905isin119869|119860119894(119905)|

119894 = 1 2 119899 le 119872 for any 119905 isin 119869 thenΦ satisfy

Φ (119905 119904) le 119890119872(119905minus119904)

forall119904 119905 isin 119869 119904 le 119905 (5)

By proceeding with the same elementary computation inLemma 25(5) of [28] we have

Ψ (119905 119904) le 119890119872(119905minus119904)

119898

prod

119894=1

(1 +1003817100381710038171003817119861119894

1003817100381710038171003817) forall119904 119905 isin 119869 119904 lt 119905 (6)

Next we introduce two Ulamrsquos type stability definitionsfor (1) which can be regarded as the extension of the Ulamrsquostype stability concepts for ordinary differential equations in[23]

Let 120598119894gt 0 120595

119894ge 0 and 120593

119894isin 119875119862(119869R

+) be nondecreasing

functions where 119894 = 1 2 119899 For 119905 isin 119869 denote

120577 (119905) =

1 if 119871 lt +infin

119890minus119872119905

if 119871 = +infin

120585 (119905) =

1 if 119871 lt +infin

119890minus119872(119905minus119905

119895) if 119871 = +infin

119905119895isin 1199051 1199052 119905

119898

(7)

We consider the following inequalities

(

(

100381610038161003816100381610038161199101015840

1(119905) minus 119860

1(119905) 1199101(119905) minus 119891

1(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

2(119905) minus 119860

2(119905) 1199102(119905) minus 119891

2(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

119899(119905) minus 119860

119899(119905) 119910119899(119905) minus 119891

119899(119905)

10038161003816100381610038161003816

)

)

le 120577 (119905) (1205981 1205982 120598

119899)119879

119905 isin 1198691015840

(

(

10038161003816100381610038161003816Δ1199101(119905119896) minus 119861119896

11199101(119905minus

119896) minus 119887119896

1

10038161003816100381610038161003816

10038161003816100381610038161003816Δ1199102(119905119896) minus 119861119896

21199102(119905minus

119896) minus 119887119896

2

10038161003816100381610038161003816

10038161003816100381610038161003816Δ119910119899(119905119896) minus 119861119896

119899119910119899(119905minus

119896) minus 119887119896

119899

10038161003816100381610038161003816

)

)

le 120585 (119905) (1205981 1205982 120598

119899)119879

119896 = 1 2 119898

(8)

Journal of Function Spaces and Applications 3

(

(

100381610038161003816100381610038161199101015840

1(119905) minus 119860

1(119905) 1199101(119905) minus 119891

1(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

2(119905) minus 119860

2(119905) 1199102(119905) minus 119891

2(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

119899(119905) minus 119860

119899(119905) 119910119899(119905) minus 119891

119899(119905)

10038161003816100381610038161003816

)

)

le 120577 (119905) (1205931(119905) 1205981 1205932(119905) 1205982 120593

119899(119905) 120598119899)119879

119905 isin 1198691015840

(

(

10038161003816100381610038161003816Δ1199101(119905119896) minus 119861119896

11199101(119905minus

119896) minus 119887119896

1

10038161003816100381610038161003816

10038161003816100381610038161003816Δ1199102(119905119896) minus 119861119896

21199102(119905minus

119896) minus 119887119896

2

10038161003816100381610038161003816

10038161003816100381610038161003816Δ119910119899(119905119896) minus 119861119896

119899119910119899(119905minus

119896) minus 119887119896

119899

10038161003816100381610038161003816

)

)

le 120585 (119905) (12059511205981 12059521205982 120595

119899120598119899)119879

119896 = 1 2 119898

(9)

Definition 2 Equation (1) is Ulam-Hyers stable if there existconstants 119862119894

119891119898gt 0 119894 = 1 2 119899 such that for each 120598

119894gt 0

and for each solution 119910 isin 1198751198621(119869R119899) of inequality (8) there

exists a solution 119909 isin 1198751198621(119869R119899) of (1) with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (1198621

1198911198981205981 1198622

1198911198981205982 119862

119899

119891119898120598119899)

119879

119905 isin 119869

(10)

Definition 3 Equation (1) is Ulam-Hyers-Rassias stable withrespect to (120593

119894 120595119894) if there exist 119862

119891119898120593119894

gt 0 119894 = 1 2 119899 suchthat for each 120598

119894gt 0 and for each solution 119910 isin 119875119862

1(119869R119899) of

inequality (9) there exists a solution 119909 isin 1198751198621(119869R119899) of (1)

with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 1205981

119862119891119898120593

2

(1205932(119905) + 120595

2) 1205982

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598119899

) 119905 isin 119869

(11)

3 Stability Results in the Case 119871lt+infin

We introduce the following assumptions

(1198671) 119891 isin 119862(119869R119899)

(1198672) 119860119894isin 119862(119869R) 119894 = 1 2 119899

Denote119872119894= max

119905isin119869|119860119894(119905)| and denote119872 = max119872

1

1198722 119872

119899

Now we are ready to state our first Ulam-Hyers stableresult on a compact interval

Theorem 4 Assume that (1198671)-(1198672) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8)

Define

119892 (119905) = (1198921(119905) 1198922(119905) 119892

119899(119905))119879

= 1199101015840

(119905) minus 119860 (119905) 119910 (119905) minus 119891 (119905)

119905 isin 1198691015840

119892119896= (119892119896

1 119892119896

2 119892

119896

119899)

119879

= Δ119910 (119905119896) minus 119861119896119910 (119905minus

119896) minus 119887119896

119896 = 1 2 119898

(12)

Then we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120598max = max 120598

1 1205982 120598

119899 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120598max 119896 = 1 2 119898

(13)

According to Definition 1 for each 119905 isin (119905119896 119905119896+1

] we have

119910 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+ int

119905

0

Ψ (119905 119904) 119892 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) (119887119896+ 119892119896)

(14)

Suppose that 119909 be the unique solution of the impulsiveCauchy problem

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin 1198691015840

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

119909 (0) = 119910 (0) 119910 (0) isin R119899

(15)

Then for each 119905 isin (119905119896 119905119896+1

] we have

119909 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

(16)

4 Journal of Function Spaces and Applications

It follows that for each 119905 isin (119905119896 119905119896+1

] we can derive

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

119910 (119905) minus Ψ (119905 0) 119910 (0)

minus

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

minusint

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817

le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817119892119896

1003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

+ 119871119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

= 119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898 + 119871) 120598max

(17)

Thus

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(18)

where

119862119891119898

= 119890119872119871

(119898 + 119871)

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (19)

So (1) is Ulam-Hyers stable The proof is completed

In order to discuss Ulam-Hyers-Rassias stability we needthe following condition

(1198673) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890119872(119905minus119904)

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (20)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

Theorem 5 Assume that (1198671)ndash(1198673) are satisfied Then (1) is

Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120595119894120598max 119896 = 1 2 119898

(21)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) int

119905

0

119890119872(119905minus119904)

120593119894(119905) 119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898119890119872119871

+ 120582120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(22)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(23)

where

119862119891119898120593

119894

= (119898119890119872119871

+ 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(24)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

4 Stability Results in the Case 119871 = +infin

In this section we will present stability results on anunbounded interval

We change (1198672) to the following strong condition

(1198672

1015840) 119860119894is continuous and uniformly bounded function on

119869 119894 = 1 2 119899 Thus there exists 119872 gt 0 such that119860(119905) le 119872 for any 119905 isin 119869

Journal of Function Spaces and Applications 5

Theorem 6 Assume that (1198671)ndash(1198672

1015840) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8) For

119892 and 119892119896 defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120598max 119896 = 1 2 119898

(25)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] we have

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817100381711989211989610038171003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

119889119904

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) 120598max

(26)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(27)

where

119862119891119898

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) (28)

Thus (1) is Ulam-Hyers stable The proof is completed

Next we suppose the following

(1198673

1015840) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890minus119872119904

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (29)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

We have

Theorem 7 Assume that (1198671)ndash(1198672

1015840) and (119867

3

1015840) are satisfied

Then (1) is Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120595119894120598max 119896 = 1 2 119898

(30)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673

1015840) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

120593119894(119905) 119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 + 120582

120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(31)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(32)

where119862119891119898120593

119894

= (1 + 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(33)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

Acknowledgments

The authors thank the referee for valuable comments andsuggestions which improved their paper This work is sup-ported by the National Natural Science Foundation of China(11201091) Key Projects of Science and Technology Researchin the Chinese Ministry of Education (211169) and KeySupport Subject (Applied Mathematics) of Guizhou NormalCollege

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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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

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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

Journal of Function Spaces and Applications 3

(

(

100381610038161003816100381610038161199101015840

1(119905) minus 119860

1(119905) 1199101(119905) minus 119891

1(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

2(119905) minus 119860

2(119905) 1199102(119905) minus 119891

2(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161199101015840

119899(119905) minus 119860

119899(119905) 119910119899(119905) minus 119891

119899(119905)

10038161003816100381610038161003816

)

)

le 120577 (119905) (1205931(119905) 1205981 1205932(119905) 1205982 120593

119899(119905) 120598119899)119879

119905 isin 1198691015840

(

(

10038161003816100381610038161003816Δ1199101(119905119896) minus 119861119896

11199101(119905minus

119896) minus 119887119896

1

10038161003816100381610038161003816

10038161003816100381610038161003816Δ1199102(119905119896) minus 119861119896

21199102(119905minus

119896) minus 119887119896

2

10038161003816100381610038161003816

10038161003816100381610038161003816Δ119910119899(119905119896) minus 119861119896

119899119910119899(119905minus

119896) minus 119887119896

119899

10038161003816100381610038161003816

)

)

le 120585 (119905) (12059511205981 12059521205982 120595

119899120598119899)119879

119896 = 1 2 119898

(9)

Definition 2 Equation (1) is Ulam-Hyers stable if there existconstants 119862119894

119891119898gt 0 119894 = 1 2 119899 such that for each 120598

119894gt 0

and for each solution 119910 isin 1198751198621(119869R119899) of inequality (8) there

exists a solution 119909 isin 1198751198621(119869R119899) of (1) with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (1198621

1198911198981205981 1198622

1198911198981205982 119862

119899

119891119898120598119899)

119879

119905 isin 119869

(10)

Definition 3 Equation (1) is Ulam-Hyers-Rassias stable withrespect to (120593

119894 120595119894) if there exist 119862

119891119898120593119894

gt 0 119894 = 1 2 119899 suchthat for each 120598

119894gt 0 and for each solution 119910 isin 119875119862

1(119869R119899) of

inequality (9) there exists a solution 119909 isin 1198751198621(119869R119899) of (1)

with

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 1205981

119862119891119898120593

2

(1205932(119905) + 120595

2) 1205982

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598119899

) 119905 isin 119869

(11)

3 Stability Results in the Case 119871lt+infin

We introduce the following assumptions

(1198671) 119891 isin 119862(119869R119899)

(1198672) 119860119894isin 119862(119869R) 119894 = 1 2 119899

Denote119872119894= max

119905isin119869|119860119894(119905)| and denote119872 = max119872

1

1198722 119872

119899

Now we are ready to state our first Ulam-Hyers stableresult on a compact interval

Theorem 4 Assume that (1198671)-(1198672) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8)

Define

119892 (119905) = (1198921(119905) 1198922(119905) 119892

119899(119905))119879

= 1199101015840

(119905) minus 119860 (119905) 119910 (119905) minus 119891 (119905)

119905 isin 1198691015840

119892119896= (119892119896

1 119892119896

2 119892

119896

119899)

119879

= Δ119910 (119905119896) minus 119861119896119910 (119905minus

119896) minus 119887119896

119896 = 1 2 119898

(12)

Then we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120598max = max 120598

1 1205982 120598

119899 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120598max 119896 = 1 2 119898

(13)

According to Definition 1 for each 119905 isin (119905119896 119905119896+1

] we have

119910 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+ int

119905

0

Ψ (119905 119904) 119892 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) (119887119896+ 119892119896)

(14)

Suppose that 119909 be the unique solution of the impulsiveCauchy problem

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905) 119905 isin 1198691015840

Δ119909 (119905119896) = 119861119896119909 (119905minus

119896) + 119887119896 119896 = 1 2 119898

119909 (0) = 119910 (0) 119910 (0) isin R119899

(15)

Then for each 119905 isin (119905119896 119905119896+1

] we have

119909 (119905) = Ψ (119905 0) 119910 (0)

+ int

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

+

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

(16)

4 Journal of Function Spaces and Applications

It follows that for each 119905 isin (119905119896 119905119896+1

] we can derive

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

119910 (119905) minus Ψ (119905 0) 119910 (0)

minus

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

minusint

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817

le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817119892119896

1003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

+ 119871119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

= 119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898 + 119871) 120598max

(17)

Thus

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(18)

where

119862119891119898

= 119890119872119871

(119898 + 119871)

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (19)

So (1) is Ulam-Hyers stable The proof is completed

In order to discuss Ulam-Hyers-Rassias stability we needthe following condition

(1198673) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890119872(119905minus119904)

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (20)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

Theorem 5 Assume that (1198671)ndash(1198673) are satisfied Then (1) is

Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120595119894120598max 119896 = 1 2 119898

(21)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) int

119905

0

119890119872(119905minus119904)

120593119894(119905) 119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898119890119872119871

+ 120582120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(22)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(23)

where

119862119891119898120593

119894

= (119898119890119872119871

+ 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(24)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

4 Stability Results in the Case 119871 = +infin

In this section we will present stability results on anunbounded interval

We change (1198672) to the following strong condition

(1198672

1015840) 119860119894is continuous and uniformly bounded function on

119869 119894 = 1 2 119899 Thus there exists 119872 gt 0 such that119860(119905) le 119872 for any 119905 isin 119869

Journal of Function Spaces and Applications 5

Theorem 6 Assume that (1198671)ndash(1198672

1015840) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8) For

119892 and 119892119896 defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120598max 119896 = 1 2 119898

(25)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] we have

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817100381711989211989610038171003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

119889119904

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) 120598max

(26)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(27)

where

119862119891119898

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) (28)

Thus (1) is Ulam-Hyers stable The proof is completed

Next we suppose the following

(1198673

1015840) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890minus119872119904

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (29)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

We have

Theorem 7 Assume that (1198671)ndash(1198672

1015840) and (119867

3

1015840) are satisfied

Then (1) is Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120595119894120598max 119896 = 1 2 119898

(30)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673

1015840) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

120593119894(119905) 119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 + 120582

120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(31)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(32)

where119862119891119898120593

119894

= (1 + 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(33)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

Acknowledgments

The authors thank the referee for valuable comments andsuggestions which improved their paper This work is sup-ported by the National Natural Science Foundation of China(11201091) Key Projects of Science and Technology Researchin the Chinese Ministry of Education (211169) and KeySupport Subject (Applied Mathematics) of Guizhou NormalCollege

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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 Journal of Function Spaces and Applications

It follows that for each 119905 isin (119905119896 119905119896+1

] we can derive

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817=

10038171003817100381710038171003817100381710038171003817

119910 (119905) minus Ψ (119905 0) 119910 (0)

minus

119896

sum

119894=1

Ψ (119905 119905+

119894) 119887119896

minusint

119905

0

Ψ (119905 119904) 119891 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817

le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817119892119896

1003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

+ 119871119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120598max

= 119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898 + 119871) 120598max

(17)

Thus

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(18)

where

119862119891119898

= 119890119872119871

(119898 + 119871)

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (19)

So (1) is Ulam-Hyers stable The proof is completed

In order to discuss Ulam-Hyers-Rassias stability we needthe following condition

(1198673) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890119872(119905minus119904)

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (20)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

Theorem 5 Assume that (1198671)ndash(1198673) are satisfied Then (1) is

Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 120595119894120598max 119896 = 1 2 119898

(21)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) int

119905

0

119890119872(119905minus119904)

120593119894(119905) 119889119904

le 119898119890119872119871

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +1003817100381710038171003817119861119896

1003817100381710038171003817) (119898119890119872119871

+ 120582120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(22)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(23)

where

119862119891119898120593

119894

= (119898119890119872119871

+ 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(24)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

4 Stability Results in the Case 119871 = +infin

In this section we will present stability results on anunbounded interval

We change (1198672) to the following strong condition

(1198672

1015840) 119860119894is continuous and uniformly bounded function on

119869 119894 = 1 2 119899 Thus there exists 119872 gt 0 such that119860(119905) le 119872 for any 119905 isin 119869

Journal of Function Spaces and Applications 5

Theorem 6 Assume that (1198671)ndash(1198672

1015840) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8) For

119892 and 119892119896 defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120598max 119896 = 1 2 119898

(25)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] we have

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817100381711989211989610038171003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

119889119904

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) 120598max

(26)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(27)

where

119862119891119898

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) (28)

Thus (1) is Ulam-Hyers stable The proof is completed

Next we suppose the following

(1198673

1015840) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890minus119872119904

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (29)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

We have

Theorem 7 Assume that (1198671)ndash(1198672

1015840) and (119867

3

1015840) are satisfied

Then (1) is Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120595119894120598max 119896 = 1 2 119898

(30)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673

1015840) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

120593119894(119905) 119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 + 120582

120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(31)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(32)

where119862119891119898120593

119894

= (1 + 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(33)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

Acknowledgments

The authors thank the referee for valuable comments andsuggestions which improved their paper This work is sup-ported by the National Natural Science Foundation of China(11201091) Key Projects of Science and Technology Researchin the Chinese Ministry of Education (211169) and KeySupport Subject (Applied Mathematics) of Guizhou NormalCollege

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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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

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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

Journal of Function Spaces and Applications 5

Theorem 6 Assume that (1198671)ndash(1198672

1015840) are satisfied Then (1) is

Ulam-Hyers stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (8) For

119892 and 119892119896 defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120598max 119896 = 1 2 119898

(25)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] we have

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

sum

119894=1

1003817100381710038171003817Ψ (119905 119905

+

119894)1003817100381710038171003817

1003817100381710038171003817100381711989211989610038171003817100381710038171003817

+ int

119905

0

Ψ (119905 119904)1003817100381710038171003817119892 (119904)

1003817100381710038171003817119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

119889119904

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) 120598max

(26)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (119862119891119898

120598max 119862119891119898120598max 119862119891119898120598max)119879

119905 isin 119869

(27)

where

119862119891119898

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 +

1

119872

) (28)

Thus (1) is Ulam-Hyers stable The proof is completed

Next we suppose the following

(1198673

1015840) There exist 120582

120593119894

gt 0 119894 = 1 2 119899 such that

int

119905

0

119890minus119872119904

120593119894(119904) 119889119904 le 120582

120593119894

120593119894(119905) for each 119905 isin 119869 (29)

where 120593119894isin 119875119862(119869R

+) is nondecreasing

We have

Theorem 7 Assume that (1198671)ndash(1198672

1015840) and (119867

3

1015840) are satisfied

Then (1) is Ulam-Hyers-Rassias stable

Proof Let 119910 isin 1198751198621(119869R119899) be a solution of inequality (9) For

119892 and 119892119896defined in (12) we have

1003817100381710038171003817119892 (119905)

1003817100381710038171003817le 119890minus119872119905

120593119894(119905) 120598max 119905 isin 119869

1003817100381710038171003817119892119896

1003817100381710038171003817le 119890minus119872(119905minus119905

119894)120595119894120598max 119896 = 1 2 119898

(30)

Let 119909 be the unique solution of (15) Hence for each 119905 isin(119905119896 119905119896+1

] it follows from (1198673

1015840) that

1003817100381710038171003817119910 (119905) minus 119909 (119905)

1003817100381710038171003817le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) int

119905

0

119890minus119872119904

120593119894(119905) 119889119904

le

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120595119894120598max

+ 120598max

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 120582120593119894

120593119894(119905)

=

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) (1 + 120582

120593119894

)

times (120593119894(119905) + 120595

119894) 120598max 119894 = 1 2 119899

(31)

Thus we obtain

(

10038161003816100381610038161199101(119905) minus 119909

1(119905)1003816100381610038161003816

10038161003816100381610038161199102(119905) minus 119909

2(119905)1003816100381610038161003816

1003816100381610038161003816119910119899(119905) minus 119909

119899(119905)1003816100381610038161003816

)

le (

119862119891119898120593

1

(1205931(119905) + 120595

1) 120598max

119862119891119898120593

2

(1205932(119905) + 120595

2) 120598max

119862119891119898120593

119899

(120593119899(119905) + 120595

119899) 120598max

) 119905 isin 119869

(32)

where119862119891119898120593

119894

= (1 + 120582120593119894

)

119898

prod

119896=1

(1 +

1003817100381710038171003817100381711986111989610038171003817100381710038171003817) 119894 = 1 2 119899

(33)

Thus (1) is Ulam-Hyers-Rassias stable The proof is com-pleted

Acknowledgments

The authors thank the referee for valuable comments andsuggestions which improved their paper This work is sup-ported by the National Natural Science Foundation of China(11201091) Key Projects of Science and Technology Researchin the Chinese Ministry of Education (211169) and KeySupport Subject (Applied Mathematics) of Guizhou NormalCollege

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

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

6 Journal of Function Spaces and Applications

References

[1] D D Baınov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications vol 66 LongmanScientific amp Technical New York NY USA 1993

[2] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing Teaneck NJ USA 1989

[3] T Yang Impulsive Control Theory vol 272 of Lecture Notes inControl and Information Sciences Springer Berlin Germany2001

[4] M Benchohra J Henderson and S K Ntouyas ImpulsiveDifferential Equations and Inclusions vol 2 Hindawi PublishingCorporation New York NY USA 2006

[5] S Andras and J J Kolumban ldquoOn the Ulam-Hyers stability offirst order differential systems with nonlocal initial conditionsrdquoNonlinear Analysis Theory Methods amp Applications A vol 82pp 1ndash11 2013

[6] S Andras and A R Meszaros ldquoUlam-Hyers stability ofdynamic equations on time scales via Picard operatorsrdquoAppliedMathematics and Computation vol 219 no 9 pp 4853ndash48642013

[7] M Burger N Ozawa and A Thom ldquoOn Ulam stabilityrdquo IsraelJournal of Mathematics vol 193 no 1 pp 109ndash129 2013

[8] L Cadariu Stabilitatea Ulam-Hyers-Bourgin Pentru EcuatiiFunctionale Universitatii de Vest Timisoara Timisara Roma-nia 2007

[9] L P Castro and A Ramos ldquoHyers-Ulam-Rassias stability for aclass of nonlinear Volterra integral equationsrdquo Banach Journalof Mathematical Analysis vol 3 no 1 pp 36ndash43 2009

[10] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[11] D S Cimpean and D Popa ldquoHyers-Ulam stability of Eulerrsquosequationrdquo Applied Mathematics Letters vol 24 no 9 pp 1539ndash1543 2011

[12] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[13] B Hegyi and S-M Jung ldquoOn the stability of Laplacersquos equationrdquoApplied Mathematics Letters vol 26 no 5 pp 549ndash552 2013

[14] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Mathematical Analysis Hadronic Press Palm HarborFL USA 2001

[15] S-M Jung ldquoHyers-Ulam stability of linear differential equa-tions of first orderrdquo Applied Mathematics Letters vol 17 no 10pp 1135ndash1140 2004

[16] N Lungu and D Popa ldquoHyers-Ulam stability of a first orderpartial differential equationrdquo Journal of Mathematical Analysisand Applications vol 385 no 1 pp 86ndash91 2012

[17] TMiura SMiyajima and S-E Takahasi ldquoA characterization ofHyers-Ulam stability of first order linear differential operatorsrdquoJournal of Mathematical Analysis and Applications vol 286 no1 pp 136ndash146 2003

[18] TMiura SMiyajima and S-E Takahasi ldquoHyers-Ulam stabilityof linear differential operator with constant coefficientsrdquoMath-ematische Nachrichten vol 258 pp 90ndash96 2003

[19] M S Moslehian and T M Rassias ldquoStability of functionalequations in non-Archimedean spacesrdquoApplicable Analysis andDiscrete Mathematics vol 1 no 2 pp 325ndash334 2007

[20] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[21] T M Rassias ldquoOn the stability of functional equations and aproblem of Ulamrdquo Acta Applicandae Mathematicae vol 62 no1 pp 23ndash130 2000

[22] I A Rus ldquoUlam stability of ordinary differential equationsrdquoStudia Universitatis Babes-Bolyai Mathematica vol 54 no 4pp 125ndash133 2009

[23] I A Rus ldquoUlam stabilities of ordinary differential equations ina Banach spacerdquoCarpathian Journal of Mathematics vol 26 no1 pp 103ndash107 2010

[24] S-E Takahasi T Miura and S Miyajima ldquoOn the Hyers-Ulamstability of the Banach space-valued differential equation 119910

1015840

=

120582119910rdquo Bulletin of the Korean Mathematical Society vol 39 no 2pp 309ndash315 2002

[25] J Wang Y Zhou and M Feckan ldquoNonlinear impulsive prob-lems for fractional differential equations and Ulam stabilityrdquoComputers ampMathematics with Applications vol 64 no 10 pp3389ndash3405 2012

[26] J Wang and Y Zhou ldquoMittag-Leffler-Ulam stabilities of frac-tional evolution equationsrdquo Applied Mathematics Letters vol25 no 4 pp 723ndash728 2012

[27] JWangM Feckan andY Zhou ldquoUlamrsquos type stability of impul-sive ordinary differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 395 no 1 pp 258ndash264 2012

[28] J Wang X Xiang W Wei and Q Chen ldquoExistence and globalasymptotical stability of periodic solution for the 119879-periodiclogistic system with time-varying generating operators and 119879

0-

periodic impulsive perturbations on Banach spacesrdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 52494516 pages 2008

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