Research

Existence of a solution of integral equations via fixed point theorem

Selma Gülyaz1, Erdal Karapınar2*, Vladimir Rakocević3 and Peyman Salimi4

Author Affiliations

1 Department of Mathematics, Cumhuriyet University, Sivas, Turkey

2 Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey

3 Faculty of Sciences and Mathematics, University of Nis, Visegradska 33, Nis, 18000, Serbia

4 Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

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Journal of Inequalities and Applications 2013, 2013:529  doi:10.1186/1029-242X-2013-529

 Received: 14 May 2013 Accepted: 18 October 2013 Published: 11 November 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we establish a solution to the following integral equation:

(1)

where , and are continuous functions. For this purpose, we also obtain some auxiliary fixed point results which generalize, improve and unify some fixed point theorems in the literature.

MSC: 47H10, 54H25.

Keywords:
cyclic representation; fixed point

1 Introduction and preliminaries

Fixed point theory is one of the most efficient tools in nonlinear functional analysis to solve the nonlinear differential and integral equations. The existence/uniqueness of a solution of differential/integral equations turns into the existence/uniqueness of a (common) fixed point of the operators which are obtained after suitable substitutions and elementary calculations; see, e.g., [1-14].

In this paper, we first obtain some fixed point theorems to solve the integral equation mentioned above. For the sake of completeness, we recollect some basic definitions and elementary results. Let X be a nonempty set and T be a self-mapping on X. Then, the set of all fixed points of T on X is denoted by . Let Ψ be the set of all functions satisfying the following conditions:

(1) ψ is continuous,

(2) if and only if ,

(3) .

Cyclic mapping and cyclic contraction were introduced by Kirk-Srinavasan-Veeramani to improve the well-known Banach fixed point theorem. Later, various types of cyclic contraction have been investigated by a number of authors; see, e.g., [6,15-17].

Definition 1.1[18]

Suppose that is a metric space and T is a self-mapping on X. Let m be a natural number and , , be nonempty sets. Then is called a cyclic representation of X with respect to T if

where .

Definition 1.2[17]

Let , and be two functions. We say that T is r--admissible if

(i) for some implies ,

(ii) for some implies .

Definition 1.3 Let be a metric space and be a self-mapping, where is a cyclic representation of Y with respect to T. Let be two functions. An operator is called:

• a cyclic weak r--C-contractive mapping of the first kind if

(2)

holds for all and , where .

• a cyclic weak r--C-contractive mapping of the second kind if

(3)

such that holds for all and , where .

2 Auxiliary fixed point results

We state the main result of this section as follows.

Theorem 2.1Letbe a complete metric space, , be nonempty closed subsets ofand. Suppose thatis a cyclic weakr--C-contractive mapping of the first kind such that

(ii) there existssuch thatand;

(iii) ifis a sequence inYsuch thatandfor allandas, thenand.

ThenThas a fixed point. Moreover, if, , , for all, thenThas a unique fixed point.

Proof Let there exist such that and . Since T is r--admissible, then and . Again, since T is r--admissible, then and . By continuing this process, we get

(4)

On the other hand, since , there exists some such that . Now implies that . Thus there exists in such that . Similarly, , where . Hence, for , there exists such that and . In case for some  , then it is clear that is a fixed point of T. Now assume that for all n. Hence, we have for all n. Set . We shall show that the sequence is non-increasing. Due to (2) with and , we get

which implies

(5)

and so for all . Then there exist such that . Suppose, on the contrary, that . Also, taking limit as in (5), we deduce

that is,

(6)

Taking limit as in (5) and using (6), we get

Consequently, we have , which yields . Hence

(7)

We shall show that is a Cauchy sequence. To reach this goal, first we prove the following claim:

(K) For every , there exists such that if with , then .

Suppose, to the contrary, that there exists such that for any , we can find with satisfying

(8)

Now, we take . Then, corresponding to , one can choose in such a way that it is the smallest integer with satisfying and . Therefore, . By using the triangular inequality,

Letting in the last inequality, keeping (7) in mind, we derive that

(9)

Again,

Taking (7) and (9) into account, we get

(10)

as in (9).

Also we have the following inequalities:

(11)

and

(12)

Letting in (11) and (12), we derive that

(13)

Again, we have

(14)

and

(15)

Letting in (14) and (15), we conclude that

(16)

Since and lie in different adjacently labeled sets and for certain , using the fact that T is a cyclic weak r--C-contractive mapping of the first kind, we have

(17)

which implies

Letting in the inequality above and keeping the expressions (7), (9), (10), (13), (16) in mind, we conclude that

Thus, we have , which yields that . Hence, (K) is satisfied.

We shall show that the sequence is Cauchy. Fix . By the claim, we find such that if with , then

(18)

Since , we also find such that

(19)

for any . Suppose that and . Then, there exists such that . Therefore, for . So, we have, for ,

By (18) and (19) and from the last inequality, we get

This proves that is a Cauchy sequence. Since Y is closed in , then is also complete, there exists such that in . In what follows, we prove that x is a fixed point of T. In fact, since and, as is a cyclic representation of Y with respect to T, the sequence has infinite terms in each for . Suppose that , and we take a subsequence of with . Now from (iii) we have and . By using the contractive condition, we can obtain

(20)

which implies

Passing to the limit as in the last inequality, we get

which implies , i.e., . Finally, to prove the uniqueness of the fixed point, suppose that such that , , , , where . The cyclic character of T and the fact that are fixed points of T imply that . Suppose that . That is, . Using the contractive condition, we obtain

which implies

Then and so , i.e., , which is a contradiction. This finishes the proof. □

Example 2.2 Let with the metric for all . Suppose and and . Define and by

Also, define by . Clearly, , and and . Let , then . On the other hand, for all , i.e., . Similarly, implies . Therefore, T is an r--admissible mapping. Let be a sequence in X such that , and as . Then . So, , i.e., and .

Let and . Now, if or , then . Also, if and , then . That is, for all and all . Hence,

for all and . Then T is a cyclic weak r--C-contractive mapping of the first kind. Therefore all the conditions of Theorem 2.1 hold and T has a fixed point in . Here, is a fixed point of T.

Theorem 2.3Letbe a complete metric space, , be nonempty closed subsets ofand. Suppose thatis a cyclic weakr--C-contractive mapping of the second kind such that

(ii) there existssuch thatand;

(iii) ifis a sequence inYsuch thatandfor allandas, thenand.

ThenThas a fixed point. Moreover, if, , , for all, thenThas a unique fixed point.

Proof By a similar method as in the proof of Theorem 2.1, we have

(21)

We shall show that the sequence is non-increasing. Due to (3) with and , we get

which implies

(22)

and so for all . Then there exists such that . We shall show that by the method of reductio ad absurdum. Suppose that . By letting in (22), we deduce

that is,

(23)

Taking limit as in (22) and using (23), we get

Thus, we have and hence , which is a contradiction. Consequently, we have

(24)

We shall show that is a Cauchy sequence. To reach this goal, first we prove the following claim:

(K) For every , there exists such that if with , then .

Suppose, to the contrary, that there exists such that for any we can find with satisfying

(25)

Following the related lines in Theorem 2.1, we deduce

(26)

(27)

(28)

and

(29)

Since and lie in different adjacently labeled sets and for certain , using the fact that a cyclic weak r--C-contractive mapping of the second kind, we have

which implies

Letting in the inequality above and by applying (24) (26), (27), (28), (29), we deduce that

Consequently, we have , and hence . As a result, we conclude that (K) is satisfied. We assert that the sequence is Cauchy. Fix . By the claim, we find such that if with , then

(30)

Since , we also find such that

(31)

for any . Suppose that and . Then there exists such that . Therefore, for . So, we have, for , ,

By (30) and (31) and from the last inequality, we get

This proves that is a Cauchy sequence. Since Y is closed in , then is also complete, there exists such that in . In what follows, we prove that x is a fixed point of T. In fact, since and, as is a cyclic representation of Y with respect to T, the sequence has infinite terms in each for . Suppose that , and we take a subsequence of with . Now from (iii) we have and . By using the contractive condition, we can obtain

(32)

which implies

Passing to the limit as in the last inequality, we get

which implies , i.e., . Finally, to prove the uniqueness of the fixed point, suppose that such that , , , , where . The cyclic character of T and the fact that are fixed points of T imply that . Suppose that . That is, . Using the contractive condition, we obtain

which implies

Hence, we obtain , which implies , that is, a contradiction. □

3 Existence of solutions of an integral equation

For , we denote by the set of real continuous functions on . We endow X with the metric

It is evident that is a complete metric space.

Consider the integral equation

(33)

(1) and are continuous functions.

(2) Let , such that

(34)

Assume that for all , we have

(35)

and

(36)

Let for all , be a decreasing function, that is,

(37)

Let . There exist and such that if and with ( and ) or ( and ), then for every , we have

(38)

(3) Assume that

(39)

for all , where . Suppose that

(40)

(4) If is a sequence in such that for all and as , then .

(5) There exists such that .

Theorem 3.1Under assumptions (1)-(5), integral equation (33) has a solution in.

Proof Define the closed subsets of X, and by

and

Also define the mapping by

Let us prove that

(41)

Suppose , that is,

Applying condition (37), since for all , we obtain that

The above inequality with condition (35) imply that

for all . Then we have .

Similarly, let , that is,

Using condition (37), since for all , we obtain that

The above inequality with condition (36) imply that

for all . Then we have . Also, we deduce that (41) holds.

Now, let , that is, for all ,

This implies from condition (34) that for all ,

Now, by conditions (39) and (38), we have, for all ,

which implies

Define by and . Further, .

Hence,

for all . By a similar method, we can show that the above inequality holds if . Now, all the conditions of Theorem 2.1 hold and T has a fixed point in

That is, is the solution to (33). □

Example 3.2 In this example, we denote by the set of real continuous functions on . We endow X with the metric

Consider the following continuous functions:

and

Let and . Then, for , we have

and

Also, . Define by

Clearly, . Also, if , then . On the other hand,

for all . That is, . Hence, implies .

Assume and with ( and ) or ( and ). Thus, and , which implies . That is,

for all , where . Further,

and so

Assume that is a sequence in X such that for all and as . Then . So, . That is, .

Therefore, all of the conditions of Theorem 3.1 are satisfied. Then the integral equation

has a solution in . Here, is a solution.

But if we chose and , then and . That is,

Also,

and so

That is, Theorem 3.1 of [6] cannot be applied to this example.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper. The third author (V Rakocević) is supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

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