Research ArticleSome Notes on the Existence of Solution forOrdinary Differential Equations via Fixed Point Theory

Fei He

School of Mathematical Sciences Inner Mongolia University Hohhot 010021 China

Correspondence should be addressed to Fei He hefeiimueducn

Received 27 June 2014 Accepted 25 July 2014 Published 14 October 2014

Academic Editor Wei-Shih Du

Copyright copy 2014 Fei HeThis is an open access article distributed under theCreativeCommonsAttribution License which permitsunrestricted use distribution and reproduction in any medium provided the original work is properly cited

We establish a fixed point theorem with w-distance for nonlinear contractive mappings in complete metric spaces As applicationsof our results we derive the existence and uniqueness of solution for a first-order ordinary differential equation with periodicboundary conditions Here we need not assume that the equation has a lower solution

1 Introduction and Preliminaries

The main purpose of this paper is to obtain the existenceand uniqueness of solution for a periodic boundary valueproblem This topic has been considered in [1ndash11] and thereferences therein A powerful tool to solve the problem is thefixed point theorem in partially orderedmetric spacesOn theexisting research results admitting the existence of a lowersolution is necessary In this paper we establish a fixed pointtheorem for w-distance contraction type maps in completemetric spaces without partially ordered structure From thiswe obtain some results on the existence and uniqueness forordinary differential equations We will see that the assump-tion of the existence of a lower solution can be removed Inparticular we also show that under assumptions of a recentresult in [10] the equation has a unique zero solution

In this note we consider the following periodic boundaryvalue problem

1199061015840

(119905) = 119891 (119905 119906 (119905)) if 119905 isin 119868 = [0 119879]

119906 (0) = 119906 (119879)

(1)

where 119879 gt 0 and 119891 119868 times R is a continuous function

Definition 1 A lower solution for (1) is a function 120572 isin 1198621(119868)such that

1205721015840

(119905) le 119891 (119905 120572 (119905)) for 119905 isin 119868

120572 (0) le 120572 (119879)

(2)

Let 119883 = 119862(119868) be the set of all real continuous functionson a closed interval 119868 We endow119883 with the norm

119906 = max119905isin119868

|119906 (119905)| (3)

for all 119906 isin 119883 Obviously this space is a Banach space and thenorm induces a complete metric on119883 as follows

119889 (119906 V) = 119906 minus V = max119905isin119868

|119906 (119905) minus V (119905)| (4)

for all 119906 V isin 119883Let B denote the class of those functions 120573 [0infin) rarr

[0 1) which satisfy the condition

120573 (119905119899) 997888rarr 1 implies 119905

119899997888rarr 0 (5)

LetA denote the class of the functions 120601 [0infin) rarr [0infin)

which have the following properties

(i) 120601 is increasing

(ii) for each 119909 gt 0 120601(119909) lt 119909

(iii) 120573(119909) = 120601(119909)119909 isinB

For example 120601(119905) = 120583119905 where 0 le 120583 lt 1 120601(119905) = 119905(1 + 119905) and120601(119905) = ln(1 + 119905) are inA

Recently Amini-Harandi and Emami [4] proved thefollowing existence theorem

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 309613 6 pageshttpdxdoiorg1011552014309613

2 Abstract and Applied Analysis

Theorem 2 (see [4 Theorem 31]) Consider problem (1) with119891 119868 times R rarr R continuous and suppose that there exists 120582 gt 0such that for 119909 119910 isin R with 119910 ge 119909

0 le 119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909] le 120582120601 (119910 minus 119909) (6)

where 120601 isin A Then the existence of a lower solution of (1)provides the existence of a unique solution of (1)

Then Caballero et al [6] also give an existence theorem

Theorem 3 (see [6Theorem 32]) Consider problem (1) with119891 119868timesR rarr R continuous and suppose that there exist 120582 120572 gt 0with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(7)

such that for 119909 119910 isin R with 119910 ge 119909

0 le 119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909] le 120572radic(119910 minus 119909) 120601 (119910 minus 119909)

(8)

where 120601 isin A Then the existence of a lower solution of (1)provides the existence of a solution of (1)

Very recently Hussain et al [10] obtain the followingresult

Theorem 4 (see [10 Section 3]) Consider problem (1) with119891 119868 times R rarr R continuous and suppose that there exists 120582 gt 0such that for 119909 119910 isin 119862(119868)

1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln( 119886

2119901minus1(|119909 (119905)| +

1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(9)

where 0 le 119886 lt 1 and 119901 gt 1 Then the existence of a lowersolution of (1) provides the existence of a solution of (1)

Remark 5 If we take 119909(119905) = 119910(119905) = 120579(119905) = 0 for all 119905 isin 119868 in thecondition (9) then we deduce that

119891 (119905 120579 (119905)) = 0 forall119905 isin 119868 (10)

This means that 120579(119905) is a solution of (1) (of course a lowersolution of (1)) In Section 3 we will show this fact again seeRemark 17

Now let us recall the concept of w-distance which wasintroduced by Kada et al [12]

Definition 6 Let 119883 be a metric space with metric 119889 Then afunction 119901 119883 times 119883 rarr [0infin) is called a w-distance on 119883 ifthe following are satisfied

(w1) 119901(119909 119911) le 119901(119909 119910) + 119901(119910 119911) for any 119909 119910 119911 isin 119883(w2) for any 119909 isin 119883 119901(119909 sdot) 119883 rarr [0infin) is lower semi-

continuous that is 119901(119909 1199100) le lim inf

119899rarrinfin119901(119909 119910

119899)

whenever 119910119899isin 119883 and 119910

119899rarr 1199100

(w3) for any 120576 gt 0 there exists 120575 gt 0 such that 119901(119911 119909) le 120575and 119901(119911 119910) le 120575 imply 119889(119909 119910) le 120576

Let us give some basic examples of w-distances (see [12])

Example 7 Let (119883 119889) be a metric space Then the metric 119889 isa w-distance on119883

Example 8 Let 119883 be a normed space with norm sdot Thenthe function 119901 119883 times 119883 rarr [0infin) defined by

119901 (119909 119910) = 119909 +10038171003817100381710038171199101003817100381710038171003817

(11)

for every 119909 119910 isin 119883 is a w-distance on119883

Example 9 Let 119883 be a normed space with norm sdot Thenthe function 119901 119883 times 119883 rarr [0infin) defined by

119901 (119909 119910) =10038171003817100381710038171199101003817100381710038171003817

(12)

for every 119909 119910 isin 119883 is a w-distance on119883

We also need the following example in Section 3

Example 10 Let119883 = 119862(119868) be endowed with the norm

119906 = max119905isin119868

|119906 (119905)| (13)

for all 119906 isin 119883 We easily deduce that the function 119901 119883times119883 rarr

[0infin) defined by

119901 (119909 119910) = max119905isin119868

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816) (14)

for every 119909 119910 isin 119883 is a w-distance on119883

The following lemma has been proved in [12]

Lemma 11 Let 119883 be a metric space with metric 119889 and let 119901be a w-distance on 119883 Let 119909

119899 and 119910

119899 be sequences in 119883 let

120572119899 and 120573

119899 be sequences in [0infin) converging to 0 and let

119909 119910 119911 isin 119883 Then the following hold

(i) If 119901(119909119899 119910) le 120572

119899and 119901(119909

119899 119911) le 120573

119899for any 119899 isin N then

119910 = 119911 In particular if 119901(119909 119910) = 0 and 119901(119909 119911) = 0then 119910 = 119911

(ii) If 119901(119909119899 119910119899) le 120572119899and 119901(119909

119899 119911) le 120573

119899for any 119899 isin N then

119910119899 converges to 119911

(iii) If 119901(119909119899 119909119898) le 120572119899for any 119899119898 isin N with 119898 gt 119899 then

119909119899 is a Cauchy sequence

2 Fixed Point Results with 119908-Distance

We are now ready to state and prove our main theorem

Theorem 12 Let (119883 119889) be a complete metric space and let 119901be a w-distance on 119883 Let 119891 119883 rarr 119883 be a map and supposethat there exists 120573 isinB such that

119901 (119891119909 119891119910) le 120573 (119901 (119909 119910)) 119901 (119909 119910) (15)

for all 119909 119910 isin 119883 Then 119891 has a unique fixed point 119911 in 119883 and119901(119911 119911) = 0

Abstract and Applied Analysis 3

Proof Starting with an arbitrary 1199090isin 119883 we construct by

induction a sequence 119891119899(1199090) in 119883 Put 119909

119899= 119891119899(1199090) 119899 =

1 2 Applying the condition (15) with 119909 = 119909

119899and 119910 = 119909

119899+1 we

have

119901 (119909119899+1 119909119899+2) = 119901 (119891119909

119899 119891119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) 119901 (119909

119899 119909119899+1)

le 119901 (119909119899 119909119899+1)

(16)

Therefore 119901(119909119899 119909119899+1) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119909119899+1) = 119903 ge 0

Let us prove that 119903 = 0 Assume that 119903 gt 0 From (15) weobtain

119901 (119909119899+1 119909119899+2)

119901 (119909119899 119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) lt 1 119899 = 1 2 (17)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119909119899+1)) = 1 and since 120573 isin B this implies

119903 = 0 Thus lim119899rarrinfin

119901(119909119899 119909119899+1) = 0 Analogously it can be

proved that lim119899rarrinfin

119901(119909119899+1 119909119899) = 0

Now we show that

119901 (119909119899 119909119898) 997888rarr 0 as 119898 gt 119899 997888rarr infin (18)

Suppose that this is not true Then there exists 1205760gt 0 for

which we can find subsequences 119909119899119896 and 119909

119898119896 of 119909

119899 with

119898119896gt 119899119896gt 119896 such that

119901 (119909119899119896 119909119898119896) ge 1205760 119896 = 1 2 (19)

By the triangle inequality and the condition (15) for every119896 isin N

119901 (119909119899119896 119909119898119896) le 119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119899119896+1

119909119898119896+1

)

+ 119901 (119909119898119896+1

119909119898119896)

le 119901 (119909119899119896 119909119899119896+1

) + 120573 (119901 (119909119899119896 119909119898119896)) 119901 (119909

119899119896 119909119898119896)

+ 119901 (119909119898119896+1

119909119898119896)

(20)

From this we obtain

1 minus 120573 (119901 (119909119899119896 119909119898119896))

le 119901(119909119899119896 119909119898119896)

minus1

sdot [119901 (119909119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

le 120576minus1

0sdot [119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

(21)

Letting 119896 rarr infin in the above inequality using lim119899rarrinfin

119901(119909119899

119909119899+1) = 0 and lim

119899rarrinfin119901(119909119899+1 119909119899) = 0 we deduce that

lim119896rarrinfin

120573 (119901 (119909119899119896 119909119898119896)) = 1 (22)

But since 120573 isin B we get lim119896rarrinfin

119901(119909119899119896 119909119898119896) = 0 This is a

contradiction and hence we have shown that (18) holdsFrom (18) and Lemma 11(iii) it follows that 119909

119899 is a

Cauchy sequence in 119883 Since (119883 119889) is a complete metricspace there exists a 119911 isin 119883 such that lim

119899rarrinfin119909119899= 119911

Next we prove that 119911 is a fixed point of 119891 To the end weprove that lim

119899rarrinfin119901(119909119899 119911) = 0 and lim

119899rarrinfin119901(119909119899 119891119911) = 0

Let 120576 gt 0 be given Using (18) there exists 1198990isin N such that

119901(119909119899 119909119898) lt 120576 for all 119898 gt 119899 ge 119899

0 Letting 119898 rarr infin by (w2)

we have 119901(119909119899 119911) le lim inf

119898rarrinfin119901(119909119899 119909119898) le 120576 for all 119899 ge 119899

0

Thismeans that lim119899rarrinfin

119901(119909119899 119911) = 0 Applying the condition

(15) with 119909 = 119909119899and 119910 = 119911 we have

119901 (119909119899+1 119891119911) = 119901 (119891119909

119899 119891119911) le 120573 (119901 (119909

119899 119911)) 119901 (119909

119899 119911)

le 119901 (119909119899 119911)

(23)

Due to lim119899rarrinfin

119901(119909119899 119911) = 0 it follows that lim

119899rarrinfin119901(119909119899

119891119911) = 0 According to Lemma 11(i) we obtain 119891119911 = 119911 thatis 119911 is a fixed point of 119891

To prove the uniqueness of the fixed point of 119891 let ussuppose that119910 is another fixed point of119891 Using the condition(15) we get

119901 (119909119899+1 119910) = 119901 (119891119909

119899 119891119910) le 120573 (119901 (119909

119899 119910)) 119901 (119909

119899 119910)

le 119901 (119909119899 119910)

(24)

Therefore 119901(119909119899 119910) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119910) = 119904 ge 0 Suppose that 119904 gt

0 Using the condition (15) we have

119901 (119909119899+1 119910)

119901 (119909119899 119910)

le 120573 (119901 (119909119899 119910)) lt 1 119899 = 1 2 (25)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119910)) = 1 and since 120573 isin B this implies

119904 = 0 Then lim119899rarrinfin

119901(119909119899 119910) = 0 Combining this and

lim119899rarrinfin

119901(119909119899 119911) = 0 we get by Lemma 11(i) that 119910 = 119911

Finally we prove 119901(119911 119911) = 0 when 119911 is a fixed point of 119891Applying the condition (15) with 119909 = 119911 and 119910 = 119911 we obtain

119901 (119911 119911) = 119901 (119891119911 119891119911) le 120573 (119901 (119911 119911)) 119901 (119911 119911) (26)

and hence (1 minus 120573(119901(119911 119911))) sdot 119901(119911 119911) = 0 Now that 120573 isin B itfollows that 119901(119911 119911) = 0

The following corollary is immediate result from Theo-rem 12 and Example 7

Corollary 13 Let (119883 119889) be a complete metric space Let 119891 119883 rarr 119883 be a map and suppose that there exists 120573 isin B suchthat

119889 (119891119909 119891119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (27)

for all 119909 119910 isin 119883 Then 119891 has a unique fixed point 119911 in119883

3 Application to OrdinaryDifferential Equations

Now we prove some results on the existence of solution forproblem (1)

4 Abstract and Applied Analysis

Theorem 14 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120582120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (28)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) can be rewritten as

1199061015840

(119905) + 120582119906 (119905) = 119891 (119905 119906 (119905)) + 120582119906 (119905) if 119905 isin 119868 = [0 119879]

119906 (0) = 119906 (119879)

(29)

Using variation of parameters formula we can easily deducethat problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (30)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879minus 1

0 le 119904 lt 119905 le 119879

119890120582(119904minus119905)

119890120582119879minus 1

0 le 119905 lt 119904 le 119879

(31)

Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (32)

Note that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) is asolution of (1)

Now we check that hypotheses of Corollary 13 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (33)

Then by (28) for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

(34)

As the function 120601(119909) is increasing we obtain

119889 (119865119906 119865V) le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

le 120582120601 (119889 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119889 (119906 V)) sdotmax119905isin119868

1

119890120582119879minus 1

times (

1

120582

119890120582(119879+119904minus119905)

]

119905

0

+

1

120582

119890120582(119904minus119905)

]

119879

119905

)

= 120582120601 (119889 (119906 V)) sdot1

119890120582119879minus 1

sdot

1

120582

(119890120582119879minus 1)

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (35)

FromCorollary 13 we see that119865 has a unique fixed point

Remark 15 Clearly we see that the condition (6) ofTheorem2 implies the condition (28) of Theorem 14 In additionTheorem 14 need not assume the existence of a lower solutionfor (1) Hence this result is an improvement of Theorem 2

Theorem 16 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119909) + 120582119909

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910) + 120582119910

1003816100381610038161003816le 120582120601 (|119909| +

10038161003816100381610038161199101003816100381610038161003816) (36)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (37)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (38)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Theorem 12 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (39)

We take a w-distance on 119883 = 119862(119868) by 119901(119909 119910) =

max119905isin119868(|119909(119905)| + |119910(119905)|) see Example 10 Using the monotene

property of the function 120601(119909) and condition (36) for any119906 V isin 119883

119901 (119865119906 119865V) = max119905isin119868

(|(119865119906) (119905)| + |(119865V) (119905)|)

le max119905isin119868

(int

119879

0

119866 (119905 119904) (1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

1003816100381610038161003816

+1003816100381610038161003816119891 (119905 V (119904)) + 120582V (119904)100381610038161003816

1003816) 119889119904)

Abstract and Applied Analysis 5

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904)| + |V (119904)|) 119889119904

le 120582120601 (119901 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119901 (119906 V)) sdot1

120582

= 120601 (119901 (119906 V)) =120601 (119901 (119906 V))119901 (119906 V)

sdot 119901 (119906 V)

= 120573 (119901 (119906 V)) 119901 (119906 V) (40)

FromTheorem 12 we deduce that 119865 has a unique fixed point

Remark 17 Let 0 le 119886 lt 1 and 119901 gt 1 If we denote

120601 (119905) =

1

2119901minus1

119901radicln( 119886

2119901minus1119905119901+ 1) (41)

then 120601 isin A and 120601(119905) le ln(119905 + 1) Thus Theorem 4 is asepical case ofTheorem 16 However fromTheorem 12 we get119901(119906 119906) = max

119905isin119868(2|119906(119905)|) = 2119906 = 0 if 119906 is a solution of

(1) This means 119906(119905) = 0 for all 119905 isin 119868 In other words theunique solution for (1) fromTheorem 16 (in particular fromTheorem 4) can only be a zero function

Theorem 18 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 and 119901 gt 1 with

120572 le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

(42)

such that for 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (43)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (44)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (45)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Corollary 13 By (43)for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120572120601 (|119906 (119904) minus V (119904)|) 119889119904

= 120572max119905isin119868

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

(46)

Using the Holder inequality in the last integral we get

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

le (int

119879

0

119866(119905 119904)119901119889119904)

1119901

(int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

(47)

where 119902 gt 0 with 1119901 + 1119902 = 1 The first integral gives us

int

119879

0

119866(119905 119904)119901119889119904 = int

119905

0

119866(119905 119904)119901119889119904 + int

119879

119905

119866(119905 119904)119901119889119904

= int

119905

0

119890119901120582(119879+119904minus119905)

(119890120582119879minus 1)119901119889119904 + int

119879

119905

119890119901120582(119904minus119905)

(119890120582119879minus 1)119901119889119904

=

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901

(48)

As the function 120601(119909) is increasing and 120572 le (119901120582(119890120582119879 minus 1)119901119879119901minus1(119890119901120582119879

minus 1))1119901 from (47) we obtain

119889 (119865119906 119865V) le 120572max119905isin119868

[(int

119879

0

119866(119905 119904)119901119889119904)

1119901

times (int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

]

le 120572(

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

(int

119879

0

120601(119906 minus V)119902119889119904)1119902

= 120572 sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119906 minus V)

le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119889 (119906 V))

6 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Theorem 2 (see [4 Theorem 31]) Consider problem (1) with119891 119868 times R rarr R continuous and suppose that there exists 120582 gt 0such that for 119909 119910 isin R with 119910 ge 119909

0 le 119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909] le 120582120601 (119910 minus 119909) (6)

where 120601 isin A Then the existence of a lower solution of (1)provides the existence of a unique solution of (1)

Then Caballero et al [6] also give an existence theorem

Theorem 3 (see [6Theorem 32]) Consider problem (1) with119891 119868timesR rarr R continuous and suppose that there exist 120582 120572 gt 0with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(7)

such that for 119909 119910 isin R with 119910 ge 119909

0 le 119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909] le 120572radic(119910 minus 119909) 120601 (119910 minus 119909)

(8)

where 120601 isin A Then the existence of a lower solution of (1)provides the existence of a solution of (1)

Very recently Hussain et al [10] obtain the followingresult

Theorem 4 (see [10 Section 3]) Consider problem (1) with119891 119868 times R rarr R continuous and suppose that there exists 120582 gt 0such that for 119909 119910 isin 119862(119868)

1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln( 119886

2119901minus1(|119909 (119905)| +

1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(9)

where 0 le 119886 lt 1 and 119901 gt 1 Then the existence of a lowersolution of (1) provides the existence of a solution of (1)

Remark 5 If we take 119909(119905) = 119910(119905) = 120579(119905) = 0 for all 119905 isin 119868 in thecondition (9) then we deduce that

119891 (119905 120579 (119905)) = 0 forall119905 isin 119868 (10)

This means that 120579(119905) is a solution of (1) (of course a lowersolution of (1)) In Section 3 we will show this fact again seeRemark 17

Now let us recall the concept of w-distance which wasintroduced by Kada et al [12]

Definition 6 Let 119883 be a metric space with metric 119889 Then afunction 119901 119883 times 119883 rarr [0infin) is called a w-distance on 119883 ifthe following are satisfied

(w1) 119901(119909 119911) le 119901(119909 119910) + 119901(119910 119911) for any 119909 119910 119911 isin 119883(w2) for any 119909 isin 119883 119901(119909 sdot) 119883 rarr [0infin) is lower semi-

continuous that is 119901(119909 1199100) le lim inf

119899rarrinfin119901(119909 119910

119899)

whenever 119910119899isin 119883 and 119910

119899rarr 1199100

(w3) for any 120576 gt 0 there exists 120575 gt 0 such that 119901(119911 119909) le 120575and 119901(119911 119910) le 120575 imply 119889(119909 119910) le 120576

Let us give some basic examples of w-distances (see [12])

Example 7 Let (119883 119889) be a metric space Then the metric 119889 isa w-distance on119883

Example 8 Let 119883 be a normed space with norm sdot Thenthe function 119901 119883 times 119883 rarr [0infin) defined by

119901 (119909 119910) = 119909 +10038171003817100381710038171199101003817100381710038171003817

(11)

for every 119909 119910 isin 119883 is a w-distance on119883

Example 9 Let 119883 be a normed space with norm sdot Thenthe function 119901 119883 times 119883 rarr [0infin) defined by

119901 (119909 119910) =10038171003817100381710038171199101003817100381710038171003817

(12)

for every 119909 119910 isin 119883 is a w-distance on119883

We also need the following example in Section 3

Example 10 Let119883 = 119862(119868) be endowed with the norm

119906 = max119905isin119868

|119906 (119905)| (13)

for all 119906 isin 119883 We easily deduce that the function 119901 119883times119883 rarr

[0infin) defined by

119901 (119909 119910) = max119905isin119868

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816) (14)

for every 119909 119910 isin 119883 is a w-distance on119883

The following lemma has been proved in [12]

Lemma 11 Let 119883 be a metric space with metric 119889 and let 119901be a w-distance on 119883 Let 119909

119899 and 119910

119899 be sequences in 119883 let

120572119899 and 120573

119899 be sequences in [0infin) converging to 0 and let

119909 119910 119911 isin 119883 Then the following hold

(i) If 119901(119909119899 119910) le 120572

119899and 119901(119909

119899 119911) le 120573

119899for any 119899 isin N then

119910 = 119911 In particular if 119901(119909 119910) = 0 and 119901(119909 119911) = 0then 119910 = 119911

(ii) If 119901(119909119899 119910119899) le 120572119899and 119901(119909

119899 119911) le 120573

119899for any 119899 isin N then

119910119899 converges to 119911

(iii) If 119901(119909119899 119909119898) le 120572119899for any 119899119898 isin N with 119898 gt 119899 then

119909119899 is a Cauchy sequence

2 Fixed Point Results with 119908-Distance

We are now ready to state and prove our main theorem

Theorem 12 Let (119883 119889) be a complete metric space and let 119901be a w-distance on 119883 Let 119891 119883 rarr 119883 be a map and supposethat there exists 120573 isinB such that

119901 (119891119909 119891119910) le 120573 (119901 (119909 119910)) 119901 (119909 119910) (15)

for all 119909 119910 isin 119883 Then 119891 has a unique fixed point 119911 in 119883 and119901(119911 119911) = 0

Abstract and Applied Analysis 3

Proof Starting with an arbitrary 1199090isin 119883 we construct by

induction a sequence 119891119899(1199090) in 119883 Put 119909

119899= 119891119899(1199090) 119899 =

1 2 Applying the condition (15) with 119909 = 119909

119899and 119910 = 119909

119899+1 we

have

119901 (119909119899+1 119909119899+2) = 119901 (119891119909

119899 119891119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) 119901 (119909

119899 119909119899+1)

le 119901 (119909119899 119909119899+1)

(16)

Therefore 119901(119909119899 119909119899+1) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119909119899+1) = 119903 ge 0

Let us prove that 119903 = 0 Assume that 119903 gt 0 From (15) weobtain

119901 (119909119899+1 119909119899+2)

119901 (119909119899 119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) lt 1 119899 = 1 2 (17)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119909119899+1)) = 1 and since 120573 isin B this implies

119903 = 0 Thus lim119899rarrinfin

119901(119909119899 119909119899+1) = 0 Analogously it can be

proved that lim119899rarrinfin

119901(119909119899+1 119909119899) = 0

Now we show that

119901 (119909119899 119909119898) 997888rarr 0 as 119898 gt 119899 997888rarr infin (18)

Suppose that this is not true Then there exists 1205760gt 0 for

which we can find subsequences 119909119899119896 and 119909

119898119896 of 119909

119899 with

119898119896gt 119899119896gt 119896 such that

119901 (119909119899119896 119909119898119896) ge 1205760 119896 = 1 2 (19)

By the triangle inequality and the condition (15) for every119896 isin N

119901 (119909119899119896 119909119898119896) le 119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119899119896+1

119909119898119896+1

)

+ 119901 (119909119898119896+1

119909119898119896)

le 119901 (119909119899119896 119909119899119896+1

) + 120573 (119901 (119909119899119896 119909119898119896)) 119901 (119909

119899119896 119909119898119896)

+ 119901 (119909119898119896+1

119909119898119896)

(20)

From this we obtain

1 minus 120573 (119901 (119909119899119896 119909119898119896))

le 119901(119909119899119896 119909119898119896)

minus1

sdot [119901 (119909119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

le 120576minus1

0sdot [119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

(21)

Letting 119896 rarr infin in the above inequality using lim119899rarrinfin

119901(119909119899

119909119899+1) = 0 and lim

119899rarrinfin119901(119909119899+1 119909119899) = 0 we deduce that

lim119896rarrinfin

120573 (119901 (119909119899119896 119909119898119896)) = 1 (22)

But since 120573 isin B we get lim119896rarrinfin

119901(119909119899119896 119909119898119896) = 0 This is a

contradiction and hence we have shown that (18) holdsFrom (18) and Lemma 11(iii) it follows that 119909

119899 is a

Cauchy sequence in 119883 Since (119883 119889) is a complete metricspace there exists a 119911 isin 119883 such that lim

119899rarrinfin119909119899= 119911

Next we prove that 119911 is a fixed point of 119891 To the end weprove that lim

119899rarrinfin119901(119909119899 119911) = 0 and lim

119899rarrinfin119901(119909119899 119891119911) = 0

Let 120576 gt 0 be given Using (18) there exists 1198990isin N such that

119901(119909119899 119909119898) lt 120576 for all 119898 gt 119899 ge 119899

0 Letting 119898 rarr infin by (w2)

we have 119901(119909119899 119911) le lim inf

119898rarrinfin119901(119909119899 119909119898) le 120576 for all 119899 ge 119899

0

Thismeans that lim119899rarrinfin

119901(119909119899 119911) = 0 Applying the condition

(15) with 119909 = 119909119899and 119910 = 119911 we have

119901 (119909119899+1 119891119911) = 119901 (119891119909

119899 119891119911) le 120573 (119901 (119909

119899 119911)) 119901 (119909

119899 119911)

le 119901 (119909119899 119911)

(23)

Due to lim119899rarrinfin

119901(119909119899 119911) = 0 it follows that lim

119899rarrinfin119901(119909119899

119891119911) = 0 According to Lemma 11(i) we obtain 119891119911 = 119911 thatis 119911 is a fixed point of 119891

To prove the uniqueness of the fixed point of 119891 let ussuppose that119910 is another fixed point of119891 Using the condition(15) we get

119901 (119909119899+1 119910) = 119901 (119891119909

119899 119891119910) le 120573 (119901 (119909

119899 119910)) 119901 (119909

119899 119910)

le 119901 (119909119899 119910)

(24)

Therefore 119901(119909119899 119910) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119910) = 119904 ge 0 Suppose that 119904 gt

0 Using the condition (15) we have

119901 (119909119899+1 119910)

119901 (119909119899 119910)

le 120573 (119901 (119909119899 119910)) lt 1 119899 = 1 2 (25)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119910)) = 1 and since 120573 isin B this implies

119904 = 0 Then lim119899rarrinfin

119901(119909119899 119910) = 0 Combining this and

lim119899rarrinfin

119901(119909119899 119911) = 0 we get by Lemma 11(i) that 119910 = 119911

Finally we prove 119901(119911 119911) = 0 when 119911 is a fixed point of 119891Applying the condition (15) with 119909 = 119911 and 119910 = 119911 we obtain

119901 (119911 119911) = 119901 (119891119911 119891119911) le 120573 (119901 (119911 119911)) 119901 (119911 119911) (26)

and hence (1 minus 120573(119901(119911 119911))) sdot 119901(119911 119911) = 0 Now that 120573 isin B itfollows that 119901(119911 119911) = 0

The following corollary is immediate result from Theo-rem 12 and Example 7

Corollary 13 Let (119883 119889) be a complete metric space Let 119891 119883 rarr 119883 be a map and suppose that there exists 120573 isin B suchthat

119889 (119891119909 119891119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (27)

for all 119909 119910 isin 119883 Then 119891 has a unique fixed point 119911 in119883

3 Application to OrdinaryDifferential Equations

Now we prove some results on the existence of solution forproblem (1)

4 Abstract and Applied Analysis

Theorem 14 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120582120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (28)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) can be rewritten as

1199061015840

(119905) + 120582119906 (119905) = 119891 (119905 119906 (119905)) + 120582119906 (119905) if 119905 isin 119868 = [0 119879]

119906 (0) = 119906 (119879)

(29)

Using variation of parameters formula we can easily deducethat problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (30)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879minus 1

0 le 119904 lt 119905 le 119879

119890120582(119904minus119905)

119890120582119879minus 1

0 le 119905 lt 119904 le 119879

(31)

Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (32)

Note that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) is asolution of (1)

Now we check that hypotheses of Corollary 13 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (33)

Then by (28) for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

(34)

As the function 120601(119909) is increasing we obtain

119889 (119865119906 119865V) le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

le 120582120601 (119889 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119889 (119906 V)) sdotmax119905isin119868

1

119890120582119879minus 1

times (

1

120582

119890120582(119879+119904minus119905)

]

119905

0

+

1

120582

119890120582(119904minus119905)

]

119879

119905

)

= 120582120601 (119889 (119906 V)) sdot1

119890120582119879minus 1

sdot

1

120582

(119890120582119879minus 1)

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (35)

FromCorollary 13 we see that119865 has a unique fixed point

Remark 15 Clearly we see that the condition (6) ofTheorem2 implies the condition (28) of Theorem 14 In additionTheorem 14 need not assume the existence of a lower solutionfor (1) Hence this result is an improvement of Theorem 2

Theorem 16 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119909) + 120582119909

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910) + 120582119910

1003816100381610038161003816le 120582120601 (|119909| +

10038161003816100381610038161199101003816100381610038161003816) (36)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (37)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (38)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Theorem 12 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (39)

We take a w-distance on 119883 = 119862(119868) by 119901(119909 119910) =

max119905isin119868(|119909(119905)| + |119910(119905)|) see Example 10 Using the monotene

property of the function 120601(119909) and condition (36) for any119906 V isin 119883

119901 (119865119906 119865V) = max119905isin119868

(|(119865119906) (119905)| + |(119865V) (119905)|)

le max119905isin119868

(int

119879

0

119866 (119905 119904) (1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

1003816100381610038161003816

+1003816100381610038161003816119891 (119905 V (119904)) + 120582V (119904)100381610038161003816

1003816) 119889119904)

Abstract and Applied Analysis 5

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904)| + |V (119904)|) 119889119904

le 120582120601 (119901 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119901 (119906 V)) sdot1

120582

= 120601 (119901 (119906 V)) =120601 (119901 (119906 V))119901 (119906 V)

sdot 119901 (119906 V)

= 120573 (119901 (119906 V)) 119901 (119906 V) (40)

FromTheorem 12 we deduce that 119865 has a unique fixed point

Remark 17 Let 0 le 119886 lt 1 and 119901 gt 1 If we denote

120601 (119905) =

1

2119901minus1

119901radicln( 119886

2119901minus1119905119901+ 1) (41)

then 120601 isin A and 120601(119905) le ln(119905 + 1) Thus Theorem 4 is asepical case ofTheorem 16 However fromTheorem 12 we get119901(119906 119906) = max

119905isin119868(2|119906(119905)|) = 2119906 = 0 if 119906 is a solution of

(1) This means 119906(119905) = 0 for all 119905 isin 119868 In other words theunique solution for (1) fromTheorem 16 (in particular fromTheorem 4) can only be a zero function

Theorem 18 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 and 119901 gt 1 with

120572 le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

(42)

such that for 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (43)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (44)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (45)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Corollary 13 By (43)for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120572120601 (|119906 (119904) minus V (119904)|) 119889119904

= 120572max119905isin119868

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

(46)

Using the Holder inequality in the last integral we get

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

le (int

119879

0

119866(119905 119904)119901119889119904)

1119901

(int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

(47)

where 119902 gt 0 with 1119901 + 1119902 = 1 The first integral gives us

int

119879

0

119866(119905 119904)119901119889119904 = int

119905

0

119866(119905 119904)119901119889119904 + int

119879

119905

119866(119905 119904)119901119889119904

= int

119905

0

119890119901120582(119879+119904minus119905)

(119890120582119879minus 1)119901119889119904 + int

119879

119905

119890119901120582(119904minus119905)

(119890120582119879minus 1)119901119889119904

=

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901

(48)

As the function 120601(119909) is increasing and 120572 le (119901120582(119890120582119879 minus 1)119901119879119901minus1(119890119901120582119879

minus 1))1119901 from (47) we obtain

119889 (119865119906 119865V) le 120572max119905isin119868

[(int

119879

0

119866(119905 119904)119901119889119904)

1119901

times (int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

]

le 120572(

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

(int

119879

0

120601(119906 minus V)119902119889119904)1119902

= 120572 sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119906 minus V)

le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119889 (119906 V))

6 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Proof Starting with an arbitrary 1199090isin 119883 we construct by

induction a sequence 119891119899(1199090) in 119883 Put 119909

119899= 119891119899(1199090) 119899 =

1 2 Applying the condition (15) with 119909 = 119909

119899and 119910 = 119909

119899+1 we

have

119901 (119909119899+1 119909119899+2) = 119901 (119891119909

119899 119891119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) 119901 (119909

119899 119909119899+1)

le 119901 (119909119899 119909119899+1)

(16)

Therefore 119901(119909119899 119909119899+1) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119909119899+1) = 119903 ge 0

Let us prove that 119903 = 0 Assume that 119903 gt 0 From (15) weobtain

119901 (119909119899+1 119909119899+2)

119901 (119909119899 119909119899+1)

le 120573 (119901 (119909119899 119909119899+1)) lt 1 119899 = 1 2 (17)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119909119899+1)) = 1 and since 120573 isin B this implies

119903 = 0 Thus lim119899rarrinfin

119901(119909119899 119909119899+1) = 0 Analogously it can be

proved that lim119899rarrinfin

119901(119909119899+1 119909119899) = 0

Now we show that

119901 (119909119899 119909119898) 997888rarr 0 as 119898 gt 119899 997888rarr infin (18)

Suppose that this is not true Then there exists 1205760gt 0 for

which we can find subsequences 119909119899119896 and 119909

119898119896 of 119909

119899 with

119898119896gt 119899119896gt 119896 such that

119901 (119909119899119896 119909119898119896) ge 1205760 119896 = 1 2 (19)

By the triangle inequality and the condition (15) for every119896 isin N

119901 (119909119899119896 119909119898119896) le 119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119899119896+1

119909119898119896+1

)

+ 119901 (119909119898119896+1

119909119898119896)

le 119901 (119909119899119896 119909119899119896+1

) + 120573 (119901 (119909119899119896 119909119898119896)) 119901 (119909

119899119896 119909119898119896)

+ 119901 (119909119898119896+1

119909119898119896)

(20)

From this we obtain

1 minus 120573 (119901 (119909119899119896 119909119898119896))

le 119901(119909119899119896 119909119898119896)

minus1

sdot [119901 (119909119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

le 120576minus1

0sdot [119901 (119909

119899119896 119909119899119896+1

) + 119901 (119909119898119896+1

119909119898119896)]

(21)

Letting 119896 rarr infin in the above inequality using lim119899rarrinfin

119901(119909119899

119909119899+1) = 0 and lim

119899rarrinfin119901(119909119899+1 119909119899) = 0 we deduce that

lim119896rarrinfin

120573 (119901 (119909119899119896 119909119898119896)) = 1 (22)

But since 120573 isin B we get lim119896rarrinfin

119901(119909119899119896 119909119898119896) = 0 This is a

contradiction and hence we have shown that (18) holdsFrom (18) and Lemma 11(iii) it follows that 119909

119899 is a

Cauchy sequence in 119883 Since (119883 119889) is a complete metricspace there exists a 119911 isin 119883 such that lim

119899rarrinfin119909119899= 119911

Next we prove that 119911 is a fixed point of 119891 To the end weprove that lim

119899rarrinfin119901(119909119899 119911) = 0 and lim

119899rarrinfin119901(119909119899 119891119911) = 0

Let 120576 gt 0 be given Using (18) there exists 1198990isin N such that

119901(119909119899 119909119898) lt 120576 for all 119898 gt 119899 ge 119899

0 Letting 119898 rarr infin by (w2)

we have 119901(119909119899 119911) le lim inf

119898rarrinfin119901(119909119899 119909119898) le 120576 for all 119899 ge 119899

0

Thismeans that lim119899rarrinfin

119901(119909119899 119911) = 0 Applying the condition

(15) with 119909 = 119909119899and 119910 = 119911 we have

119901 (119909119899+1 119891119911) = 119901 (119891119909

119899 119891119911) le 120573 (119901 (119909

119899 119911)) 119901 (119909

119899 119911)

le 119901 (119909119899 119911)

(23)

Due to lim119899rarrinfin

119901(119909119899 119911) = 0 it follows that lim

119899rarrinfin119901(119909119899

119891119911) = 0 According to Lemma 11(i) we obtain 119891119911 = 119911 thatis 119911 is a fixed point of 119891

To prove the uniqueness of the fixed point of 119891 let ussuppose that119910 is another fixed point of119891 Using the condition(15) we get

119901 (119909119899+1 119910) = 119901 (119891119909

119899 119891119910) le 120573 (119901 (119909

119899 119910)) 119901 (119909

119899 119910)

le 119901 (119909119899 119910)

(24)

Therefore 119901(119909119899 119910) is a decreasing sequence and bounded

below and hence lim119899rarrinfin

119901(119909119899 119910) = 119904 ge 0 Suppose that 119904 gt

0 Using the condition (15) we have

119901 (119909119899+1 119910)

119901 (119909119899 119910)

le 120573 (119901 (119909119899 119910)) lt 1 119899 = 1 2 (25)

Taking limit when 119899 rarr infin the above inequality yieldslim119899rarrinfin

120573(119901(119909119899 119910)) = 1 and since 120573 isin B this implies

119904 = 0 Then lim119899rarrinfin

119901(119909119899 119910) = 0 Combining this and

lim119899rarrinfin

119901(119909119899 119911) = 0 we get by Lemma 11(i) that 119910 = 119911

Finally we prove 119901(119911 119911) = 0 when 119911 is a fixed point of 119891Applying the condition (15) with 119909 = 119911 and 119910 = 119911 we obtain

119901 (119911 119911) = 119901 (119891119911 119891119911) le 120573 (119901 (119911 119911)) 119901 (119911 119911) (26)

and hence (1 minus 120573(119901(119911 119911))) sdot 119901(119911 119911) = 0 Now that 120573 isin B itfollows that 119901(119911 119911) = 0

The following corollary is immediate result from Theo-rem 12 and Example 7

Corollary 13 Let (119883 119889) be a complete metric space Let 119891 119883 rarr 119883 be a map and suppose that there exists 120573 isin B suchthat

119889 (119891119909 119891119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (27)

for all 119909 119910 isin 119883 Then 119891 has a unique fixed point 119911 in119883

3 Application to OrdinaryDifferential Equations

Now we prove some results on the existence of solution forproblem (1)

4 Abstract and Applied Analysis

Theorem 14 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120582120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (28)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) can be rewritten as

1199061015840

(119905) + 120582119906 (119905) = 119891 (119905 119906 (119905)) + 120582119906 (119905) if 119905 isin 119868 = [0 119879]

119906 (0) = 119906 (119879)

(29)

Using variation of parameters formula we can easily deducethat problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (30)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879minus 1

0 le 119904 lt 119905 le 119879

119890120582(119904minus119905)

119890120582119879minus 1

0 le 119905 lt 119904 le 119879

(31)

Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (32)

Note that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) is asolution of (1)

Now we check that hypotheses of Corollary 13 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (33)

Then by (28) for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

(34)

As the function 120601(119909) is increasing we obtain

119889 (119865119906 119865V) le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

le 120582120601 (119889 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119889 (119906 V)) sdotmax119905isin119868

1

119890120582119879minus 1

times (

1

120582

119890120582(119879+119904minus119905)

]

119905

0

+

1

120582

119890120582(119904minus119905)

]

119879

119905

)

= 120582120601 (119889 (119906 V)) sdot1

119890120582119879minus 1

sdot

1

120582

(119890120582119879minus 1)

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (35)

FromCorollary 13 we see that119865 has a unique fixed point

Remark 15 Clearly we see that the condition (6) ofTheorem2 implies the condition (28) of Theorem 14 In additionTheorem 14 need not assume the existence of a lower solutionfor (1) Hence this result is an improvement of Theorem 2

Theorem 16 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119909) + 120582119909

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910) + 120582119910

1003816100381610038161003816le 120582120601 (|119909| +

10038161003816100381610038161199101003816100381610038161003816) (36)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (37)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (38)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Theorem 12 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (39)

We take a w-distance on 119883 = 119862(119868) by 119901(119909 119910) =

max119905isin119868(|119909(119905)| + |119910(119905)|) see Example 10 Using the monotene

property of the function 120601(119909) and condition (36) for any119906 V isin 119883

119901 (119865119906 119865V) = max119905isin119868

(|(119865119906) (119905)| + |(119865V) (119905)|)

le max119905isin119868

(int

119879

0

119866 (119905 119904) (1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

1003816100381610038161003816

+1003816100381610038161003816119891 (119905 V (119904)) + 120582V (119904)100381610038161003816

1003816) 119889119904)

Abstract and Applied Analysis 5

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904)| + |V (119904)|) 119889119904

le 120582120601 (119901 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119901 (119906 V)) sdot1

120582

= 120601 (119901 (119906 V)) =120601 (119901 (119906 V))119901 (119906 V)

sdot 119901 (119906 V)

= 120573 (119901 (119906 V)) 119901 (119906 V) (40)

FromTheorem 12 we deduce that 119865 has a unique fixed point

Remark 17 Let 0 le 119886 lt 1 and 119901 gt 1 If we denote

120601 (119905) =

1

2119901minus1

119901radicln( 119886

2119901minus1119905119901+ 1) (41)

then 120601 isin A and 120601(119905) le ln(119905 + 1) Thus Theorem 4 is asepical case ofTheorem 16 However fromTheorem 12 we get119901(119906 119906) = max

119905isin119868(2|119906(119905)|) = 2119906 = 0 if 119906 is a solution of

(1) This means 119906(119905) = 0 for all 119905 isin 119868 In other words theunique solution for (1) fromTheorem 16 (in particular fromTheorem 4) can only be a zero function

Theorem 18 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 and 119901 gt 1 with

120572 le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

(42)

such that for 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (43)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (44)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (45)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Corollary 13 By (43)for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120572120601 (|119906 (119904) minus V (119904)|) 119889119904

= 120572max119905isin119868

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

(46)

Using the Holder inequality in the last integral we get

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

le (int

119879

0

119866(119905 119904)119901119889119904)

1119901

(int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

(47)

where 119902 gt 0 with 1119901 + 1119902 = 1 The first integral gives us

int

119879

0

119866(119905 119904)119901119889119904 = int

119905

0

119866(119905 119904)119901119889119904 + int

119879

119905

119866(119905 119904)119901119889119904

= int

119905

0

119890119901120582(119879+119904minus119905)

(119890120582119879minus 1)119901119889119904 + int

119879

119905

119890119901120582(119904minus119905)

(119890120582119879minus 1)119901119889119904

=

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901

(48)

As the function 120601(119909) is increasing and 120572 le (119901120582(119890120582119879 minus 1)119901119879119901minus1(119890119901120582119879

minus 1))1119901 from (47) we obtain

119889 (119865119906 119865V) le 120572max119905isin119868

[(int

119879

0

119866(119905 119904)119901119889119904)

1119901

times (int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

]

le 120572(

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

(int

119879

0

120601(119906 minus V)119902119889119904)1119902

= 120572 sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119906 minus V)

le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119889 (119906 V))

6 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

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Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Theorem 14 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120582120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (28)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) can be rewritten as

1199061015840

(119905) + 120582119906 (119905) = 119891 (119905 119906 (119905)) + 120582119906 (119905) if 119905 isin 119868 = [0 119879]

119906 (0) = 119906 (119879)

(29)

Using variation of parameters formula we can easily deducethat problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (30)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879minus 1

0 le 119904 lt 119905 le 119879

119890120582(119904minus119905)

119890120582119879minus 1

0 le 119905 lt 119904 le 119879

(31)

Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (32)

Note that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) is asolution of (1)

Now we check that hypotheses of Corollary 13 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (33)

Then by (28) for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

(34)

As the function 120601(119909) is increasing we obtain

119889 (119865119906 119865V) le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904) minus V (119904)|) 119889119904

le 120582120601 (119889 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119889 (119906 V)) sdotmax119905isin119868

1

119890120582119879minus 1

times (

1

120582

119890120582(119879+119904minus119905)

]

119905

0

+

1

120582

119890120582(119904minus119905)

]

119879

119905

)

= 120582120601 (119889 (119906 V)) sdot1

119890120582119879minus 1

sdot

1

120582

(119890120582119879minus 1)

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (35)

FromCorollary 13 we see that119865 has a unique fixed point

Remark 15 Clearly we see that the condition (6) ofTheorem2 implies the condition (28) of Theorem 14 In additionTheorem 14 need not assume the existence of a lower solutionfor (1) Hence this result is an improvement of Theorem 2

Theorem 16 Consider problem (1) with 119891 continuous andsuppose that there exists 120582 gt 0 such that for any 119909 119910 isin R

1003816100381610038161003816119891 (119905 119909) + 120582119909

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910) + 120582119910

1003816100381610038161003816le 120582120601 (|119909| +

10038161003816100381610038161199101003816100381610038161003816) (36)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (37)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (38)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Theorem 12 Considerthe space119883 = 119862(119868) with the metric

119889 (119909 119910) =1003817100381710038171003817119909 minus 119910

1003817100381710038171003817= max119905isin119868

1003816100381610038161003816119909 (119905) minus 119910 (119905)

1003816100381610038161003816 (39)

We take a w-distance on 119883 = 119862(119868) by 119901(119909 119910) =

max119905isin119868(|119909(119905)| + |119910(119905)|) see Example 10 Using the monotene

property of the function 120601(119909) and condition (36) for any119906 V isin 119883

119901 (119865119906 119865V) = max119905isin119868

(|(119865119906) (119905)| + |(119865V) (119905)|)

le max119905isin119868

(int

119879

0

119866 (119905 119904) (1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

1003816100381610038161003816

+1003816100381610038161003816119891 (119905 V (119904)) + 120582V (119904)100381610038161003816

1003816) 119889119904)

Abstract and Applied Analysis 5

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904)| + |V (119904)|) 119889119904

le 120582120601 (119901 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119901 (119906 V)) sdot1

120582

= 120601 (119901 (119906 V)) =120601 (119901 (119906 V))119901 (119906 V)

sdot 119901 (119906 V)

= 120573 (119901 (119906 V)) 119901 (119906 V) (40)

FromTheorem 12 we deduce that 119865 has a unique fixed point

Remark 17 Let 0 le 119886 lt 1 and 119901 gt 1 If we denote

120601 (119905) =

1

2119901minus1

119901radicln( 119886

2119901minus1119905119901+ 1) (41)

then 120601 isin A and 120601(119905) le ln(119905 + 1) Thus Theorem 4 is asepical case ofTheorem 16 However fromTheorem 12 we get119901(119906 119906) = max

119905isin119868(2|119906(119905)|) = 2119906 = 0 if 119906 is a solution of

(1) This means 119906(119905) = 0 for all 119905 isin 119868 In other words theunique solution for (1) fromTheorem 16 (in particular fromTheorem 4) can only be a zero function

Theorem 18 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 and 119901 gt 1 with

120572 le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

(42)

such that for 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (43)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (44)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (45)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Corollary 13 By (43)for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120572120601 (|119906 (119904) minus V (119904)|) 119889119904

= 120572max119905isin119868

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

(46)

Using the Holder inequality in the last integral we get

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

le (int

119879

0

119866(119905 119904)119901119889119904)

1119901

(int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

(47)

where 119902 gt 0 with 1119901 + 1119902 = 1 The first integral gives us

int

119879

0

119866(119905 119904)119901119889119904 = int

119905

0

119866(119905 119904)119901119889119904 + int

119879

119905

119866(119905 119904)119901119889119904

= int

119905

0

119890119901120582(119879+119904minus119905)

(119890120582119879minus 1)119901119889119904 + int

119879

119905

119890119901120582(119904minus119905)

(119890120582119879minus 1)119901119889119904

=

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901

(48)

As the function 120601(119909) is increasing and 120572 le (119901120582(119890120582119879 minus 1)119901119879119901minus1(119890119901120582119879

minus 1))1119901 from (47) we obtain

119889 (119865119906 119865V) le 120572max119905isin119868

[(int

119879

0

119866(119905 119904)119901119889119904)

1119901

times (int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

]

le 120572(

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

(int

119879

0

120601(119906 minus V)119902119889119904)1119902

= 120572 sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119906 minus V)

le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119889 (119906 V))

6 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

le max119905isin119868

int

119879

0

119866 (119905 119904) 120582120601 (|119906 (119904)| + |V (119904)|) 119889119904

le 120582120601 (119901 (119906 V)) sdotmax119905isin119868

int

119879

0

119866 (119905 119904) 119889119904

= 120582120601 (119901 (119906 V)) sdot1

120582

= 120601 (119901 (119906 V)) =120601 (119901 (119906 V))119901 (119906 V)

sdot 119901 (119906 V)

= 120573 (119901 (119906 V)) 119901 (119906 V) (40)

FromTheorem 12 we deduce that 119865 has a unique fixed point

Remark 17 Let 0 le 119886 lt 1 and 119901 gt 1 If we denote

120601 (119905) =

1

2119901minus1

119901radicln( 119886

2119901minus1119905119901+ 1) (41)

then 120601 isin A and 120601(119905) le ln(119905 + 1) Thus Theorem 4 is asepical case ofTheorem 16 However fromTheorem 12 we get119901(119906 119906) = max

119905isin119868(2|119906(119905)|) = 2119906 = 0 if 119906 is a solution of

(1) This means 119906(119905) = 0 for all 119905 isin 119868 In other words theunique solution for (1) fromTheorem 16 (in particular fromTheorem 4) can only be a zero function

Theorem 18 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 and 119901 gt 1 with

120572 le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

(42)

such that for 119909 119910 isin R

1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (43)

where 120601 isin A Then there exists a unique solution for (1)

Proof Problem (1) is equivalent to the integral equation

119906 (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (44)

where 119866(119905 119904) is the same as (31) Define 119865 119862(119868) rarr 119862(119868) by

(119865119906) (119905) = int

119879

0

119866 (119905 119904) [119891 (119904 119906 (119904)) + 120582119906 (119904)] 119889119904 (45)

Notice that if 119906 isin 119862(119868) is a fixed point of 119865 then 119906 isin 1198621(119868) isa solution of (1)

Now we check that hypotheses of Corollary 13 By (43)for any 119906 V isin 119883

119889 (119865119906 119865V) = max119905isin119868

|(119865119906) (119905) minus (119865V) (119905)|

le max119905isin119868

int

119879

0

119866 (119905 119904)1003816100381610038161003816119891 (119905 119906 (119904)) + 120582119906 (119904)

minus [119891 (119905 V (119904)) + 120582V (119904)]1003816100381610038161003816119889119904

le max119905isin119868

int

119879

0

119866 (119905 119904) 120572120601 (|119906 (119904) minus V (119904)|) 119889119904

= 120572max119905isin119868

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

(46)

Using the Holder inequality in the last integral we get

int

119879

0

119866 (119905 119904) 120601 (|119906 (119904) minus V (119904)|) 119889119904

le (int

119879

0

119866(119905 119904)119901119889119904)

1119901

(int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

(47)

where 119902 gt 0 with 1119901 + 1119902 = 1 The first integral gives us

int

119879

0

119866(119905 119904)119901119889119904 = int

119905

0

119866(119905 119904)119901119889119904 + int

119879

119905

119866(119905 119904)119901119889119904

= int

119905

0

119890119901120582(119879+119904minus119905)

(119890120582119879minus 1)119901119889119904 + int

119879

119905

119890119901120582(119904minus119905)

(119890120582119879minus 1)119901119889119904

=

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901

(48)

As the function 120601(119909) is increasing and 120572 le (119901120582(119890120582119879 minus 1)119901119879119901minus1(119890119901120582119879

minus 1))1119901 from (47) we obtain

119889 (119865119906 119865V) le 120572max119905isin119868

[(int

119879

0

119866(119905 119904)119901119889119904)

1119901

times (int

119879

0

120601(|119906 (119904) minus V (119904)|)119902119889119904)1119902

]

le 120572(

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

(int

119879

0

120601(119906 minus V)119902119889119904)1119902

= 120572 sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119906 minus V)

le (

119901120582(119890120582119879minus 1)

119901

119879119901minus1(119890119901120582119879

minus 1)

)

1119901

sdot (

119890119901120582119879

minus 1

119901120582(119890120582119879minus 1)119901)

1119901

sdot 1198791119902sdot 120601 (119889 (119906 V))

6 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

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 Abstract and Applied Analysis

= 120601 (119889 (119906 V)) =120601 (119889 (119906 V))119889 (119906 V)

sdot 119889 (119906 V)

= 120573 (119889 (119906 V)) 119889 (119906 V) (49)

FromCorollary 13 we deduce that119865 has a unique fixed point

FromTheorem 18 we obtain the following result whenwetake 119901 = 2

Corollary 19 Consider problem (1) with 119891 continuous andsuppose that there exist 120582 120572 gt 0 with

120572 le (

2120582 (119890120582119879minus 1)

119879 (119890120582119879+ 1)

)

12

(50)

such that for 119909 119910 isin R1003816100381610038161003816119891 (119905 119910) + 120582119910 minus [119891 (119905 119909) + 120582119909]

1003816100381610038161003816le 120572120601 (

1003816100381610038161003816119910 minus 119909

1003816100381610038161003816) (51)

where 120601 isin A Then there exists a unique solution for (1)

Remark 20 From [6 Remark 33] we see that if 120601 isin A then120593(119909) = radic119909120601(119909) isin A The condition (8) of Theorem 3 impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)ThusCorollary 19 is an improvement of Theorem 3

Remark 21 We easily check that 120601(119909) = radicln(1199092 + 1) and120601(119909) = radic119909

2(1199092+ 1) are inA Therefore the condition of [8

Theorem 31] [11 Theorem 25] or [7 Theorem 32] impliesthe condition (51) of Corollary 19 In addition Corollary 19need not assume the existence of a lower solution for (1)Thus Corollary 19 is an improvement of [8 Theorem 31] [11Theorem 25] and [7 Theorem 32]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by Scientific Studies ofHigher Education Institution of InnerMongolia (NJZZ13019)and in part by Program of Higher-level talents of InnerMongolia University (30105-125150 30105-135117)

References

[1] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinary dif-ferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[2] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications to ordi-nary differential equationsrdquo Acta Mathematica Sinica vol 23no 12 pp 2205ndash2212 2007

[3] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysis Theory Methods amp Applications vol 71 no 7-8 pp 3403ndash3410 2009

[4] A Amini-Harandi and H Emami ldquoA fixed point theorem forcontraction type maps in partially ordered metric spaces andapplication to ordinary differential equationsrdquo Nonlinear Ana-lysis Theory Methods amp Applications vol 72 no 5 pp 2238ndash2242 2010

[5] M E Gordji and M Ramezani ldquoA generalization of MizoguchiandTakahashirsquos theorem for single-valuedmappings in partiallyordered metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 13 pp 4544ndash4549 2011

[6] J Caballero J Harjani and K Sadarangani ldquoContractive-likemapping principles in ordered metric spaces and application toordinary differential equationsrdquo Fixed Point Theory and Appli-cations vol 2010 Article ID 916064 9 pages 2010

[7] M Eshaghi Gordji H Baghani and G H Kim ldquoA fixed pointtheorem for contraction type maps in partially ordered met-ric spaces and application to ordinary differential equationsrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID981517 8 pages 2012

[8] F Yan Y Su andQ Feng ldquoAnew contractionmapping principlein partially ordered metric spaces and applications to ordinarydifferential equationsrdquo Fixed PointTheory and Applications vol2012 article 152 2012

[9] S Moradi E Karapınar and H Aydi ldquoExistence of solutionsfor a periodic boundary value problem via generalized weaklycontractionsrdquo Abstract and Applied Analysis vol 2013 ArticleID 704160 7 pages 2013

[10] N Hussain V Parvaneh and J Roshan ldquoFixed point results forG-120572-contractivemapswith application to boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[11] H H Alsulami S Gulyaz E Karapınar and I Erhan ldquoFixedpoint theorems for a class of 120572-admissible contractions andapplications to boundary value problemrdquo Abstract and AppliedAnalysis vol 2014 Article ID 187031 10 pages 2014

[12] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

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