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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 386253, 15 pages
http://dx.doi.org/10.1155/2012/386253
Research Article

Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations

1Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Chhattisgarh, Raipur 492101, India
2Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Beograd, Serbia
3Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Received 28 July 2012; Accepted 5 September 2012

Academic Editor: Sehie Park

Copyright © 2012 Hemant Kumar Nashine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

1. Introduction and Preliminaries

It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in classical functional analysis. The Banach contraction principle is a very popular tool which is used to solve existence problems in many branches of mathematical analysis and its applications. It is no surprise that there is a great number of generalizations of this fundamental theorem. They go in several directions modifying the basic contractive condition or changing the ambient space. This celebrated theorem can be stated as follows.

Theorem 1.1 (see [1]). Let be a complete metric space and let be a mapping of into itself satisfying: where is a constant in . Then, has a unique fixed point .

There is in the literature a great number of generalizations of the Banach contraction principle (see, e.g., [2] and references cited therein).

Inequality (1.1) implies continuity of . A natural question is whether we can find contractive conditions which will imply existence of a fixed point in a complete metric space but will not imply continuity.

On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al. [3].

Definition 1.2 (see [3, 4]). Let be a metric space. Let be a positive integer and let be nonempty subsets of . Then is said to be a cyclic representation of with respect to if
(i) , are nonempty closed sets, and
(ii) , .

Kirk et al. [3] proved the following result.

Theorem 1.3 (see [3]). Let be a metric metric space and let be a cyclic representation of with respect to . If holds for all ,   where , and , then has a unique fixed point and .

Notice that, while contractions are always continuous, cyclic contractions might not be.

Following [3], a number of fixed point theorems on cyclic representations of with respect to a self-mapping have appeared (see, e.g., [412]).

In this paper, we introduce a new class of cyclic contractive mappings satisfying an implicit relation in the framework of metric spaces and then derive the existence and uniqueness of fixed points for such mappings. Suitable examples are provided to demonstrate the validity of our results. Our main result generalizes and improves many existing theorems in the literature. We also give an application of the presented results in the area of integral equations and prove an existence theorem for solutions of a system of integral equations in the last section.

2. Notation and Definitions

First, we introduce some further notations and definitions that will be used later.

2.1. Implicit Relation and Related Concepts

In recent years, Popa [13] used implicit functions rather than contraction conditions to prove fixed point theorems in metric spaces whose strength lies in its unifying power. Namely, an implicit function can cover several contraction conditions which include known as well as some new conditions. This fact is evident from examples furnished in Popa [13]. Implicit relations on metric spaces have been used in many articles (for details see [1419] and references cited therein).

In this section, we define a suitable implicit function involving six real nonnegative arguments to prove our results, that was given in [20].

Let denote the nonnegative real numbers and let be the set of all continuous functions satisfying the following conditions: : is non-increasing in variables ; : there exists a right continuous function , , for , such that for , or implies ; : , , for all .

Example 2.1. , where .

Example 2.2. , where .

Example 2.3. , where is right continuous and , for .

Example 2.4. , where , , and .

We need the following lemma for the proof of our theorems.

Lemma 2.5 (see [21]). Let be a right continuous function such that for every . Then , where denotes the times repeated composition of with itself.

Next, we introduce a new notion of cyclic contractive mapping and establish a new results for such mappings.

Definition 2.6. Let be a metric space. Let be a positive integer, let be nonempty subsets of , and . An operator is called an implicit relation type cyclic contractive mapping if
(*) is a cyclic representation of with respect to ;
(**) for any ,   (with ), for some .

Using Example 2.2, we present an example of an implicit relation type cyclic contractive mapping.

Example 2.7. Let with the usual metric. Suppose , , and ; note that . Define such that Clearly, and are closed subsets of . Moreover, for , so that is a cyclic representation of with respect to . Furthermore, if is given by then . We will show that implicit relation type cyclic contractive conditions are verified. We will distinguish the following cases:(1), .(i)When and , we deduce and inequality (2.3) is trivially satisfied.(ii)When and , we deduce and then . Inequality (2.3) holds as it reduces to .(2), .(i)When and , we deduce and inequality (2.3) is trivially satisfied.(ii)When and , we deduce and Then . Inequality (2.3) holds as it reduces to .
Hence, is an implicit relation type cyclic contractive mapping.

3. Main Result

Our main result is the following.

Theorem 3.1. Let be a complete metric space, , nonempty closed subsets of , and . Suppose is an implicit relation type cyclic contractive mapping, for some . Then has a unique fixed point. Moreover, the fixed point of belongs to .

Proof. Let (such a point exists since ). Define the sequence in by We will prove that If for some , we have , then (3.2) follows immediately. So, we can suppose that for all . From the condition , we observe that for all , there exists such that . Then, from the condition , we have and so Now using , we have and from , there exists a right continuous function , , , for , such that for all , If we continue this procedure, we can have and so from Lemma 2.5,
Next we show that is a Cauchy sequence. Suppose it is not true. Then we can find a and two sequences of integers , , with We may also assume by choosing to be the smallest number exceeding for which (3.9) holds. Now (3.7), (3.9), and (3.10) imply and so
On the other hand, for all , there exists such that . Then (for large enough, ) and lie in different adjacently labelled sets and for certain . Using the triangle inequality, we get which, by (3.12), implies that Using (3.2), we have Again, using the triangle inequality, we get Passing to the limit as in the above inequality and using (3.16) and (3.14), we get Similarly, we have Passing to the limit as and using (3.2) and (3.14), we obtain Similarly, we have
Using the condition (2.3) for and , we have and so Now letting and using (3.12), (3.14), and (3.18)–(3.21), we have, by continuity of , that a contradiction with since we have supposed that . Thus, is a Cauchy sequence in . Since is complete, there exists such that We will prove that From condition , and since , we have . Since is closed, from (3.25), we get that . Again, from the condition , we have . Since is closed, from (3.25), we get that . Continuing this process, we obtain (3.26).
Now, we will prove that is a fixed point of . Indeed, from (3.26), for all , there exists such that . Applying with and , we obtain and so letting from the last inequality, we also have which is a contradiction to . Thus, and so ; that is, is a fixed point of .
Finally, we prove that is the unique fixed point of . Assume that is another fixed point of , that is, . By the condition , this implies that . Then we can apply for and . Hence, we obtain Since and are fixed points of , we can show easily that . If , we get which is a contradiction to . Then we have , that is, . Thus, we have proved the uniqueness of the fixed point.

In what follows, we deduce some fixed point theorems from our main result given by Theorem 3.1.

If we take and in Theorem 3.1, then we get immediately the following fixed point theorem.

Corollary 3.2. Let be a complete metric space and let satisfy the following condition: there exists such that for all . Then has a unique fixed point.

Corollary 3.3. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that
(*)' is a cyclic representation of with respect to ;
(**)' for any , with , where . Then has a unique fixed point. Moreover, the fixed point of belongs to .

Remark 3.4. Corollary 3.3 is an extension to Theorem 2.1 in [3, 4].

Corollary 3.5. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that
(*)' is a cyclic representation of with respect to ;
(**)' for any , with , where is right continuous and for . Then has a unique fixed point. Moreover, the fixed point of belongs to .

Remark 3.6. Taking in Corollary 3.5, with , we obtain a generalized version of Theorem 3 in [3, 8].

Corollary 3.7. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that(*)' is a cyclic representation of with respect to ;(**)' for any , with , where , , . Then has a unique fixed point. Moreover, the fixed point of belongs to .

The following example demonstrates the validity of Theorem 3.1.

Example 3.8. Let with the usual metric. Suppose , , and . Define by , for all . Clearly, are closed subsets of . Moreover, for so that is a cyclic representation of with respect to . Moreover, mapping is implicit relation type cyclic contractive, with defined by Indeed, to see this fact we examine the following cases.
Inequality (2.3) reduces to (I) For , :(i)suppose and . Then inequality (2.3) holds as it reduces to ;(ii)suppose and . Then inequality (2.3) holds as it reduces to ;(iii)suppose and . Then inequality (2.3) holds as it reduces to ;(iv)suppose and . Then inequality (2.3) holds as it reduces to ;(v)suppose and . Then inequality (2.3) holds as it reduces to .(II) For , :(i)suppose and . Then inequality (2.3) holds as it reduces to ;(ii)suppose and . Then inequality (2.3) holds as it reduces to ;(iii)suppose and . Then inequality (2.3) holds as it reduces to .(III) For , , inequality (2.3) trivially holds.
Similarly other cases can be verified. Hence, is an implicit relation type cyclic contractive mapping. Therefore, all conditions of Theorem 3.1 are satisfied and so has a fixed point (which is ).
We illustrate Theorem 3.1 by another example which is obtained by modifying the one from [22].

Example 3.9. Let and we define by and let , , and be three subsets of .
Define by Let the function be defined by where , , , , , and , for all . Then is an implicit type cyclic contractive mapping for for . Therefore, all conditions of Theorem 3.1 are satisfied and so has a fixed point (which is ).

4. An Application to Integral Equations

In this section, we apply Theorem 3.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.

We consider the following nonlinear integral equation, where , and are continuous functions.

Let be the set of real continuous functions on . We endow with the standard metric It is well known that is a complete metric space. Define the mapping by

Let , such that We suppose that for all , we have We suppose that for all , is a decreasing function, that is, We suppose that Finally, we suppose that for all , for all with and or and , where .

Now, define the set We have the following result.

Theorem 4.1. Under the assumptions (4.4)–(4.9), Problem (4.1) has one and only one solution .

Proof. Define the closed subsets of , , and by We will prove that Let , that is, Using condition (4.7), since for all , we obtain that The above inequality with condition (4.5) imply that for all . Then we have .
Similarly, let , that is, Using condition (4.7), since for all , we obtain that The above inequality with condition (4.6) imply that for all . Then we have . Finally, we deduce that (4.12) holds.
Now, let , that is, for all , This implies from condition (4.4) that for all , Now, using conditions (4.8) and (4.9), we can write that for all , we have This implies that Using the same technique, we can show that the above inequality holds also if we take .
Now, all the conditions of Corollary 3.3 are satisfied (with ) and we deduce that has a unique fixed point ; that is, is the unique solution to (4.1).

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC Grant no. 55000613). Moreover, the second author is grateful to the Ministry of Science and Technological Development of Serbia and the third author gratefully acknowledges the support provided visitor project by the Department of Mathematic and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta, Canada’ as a visitor for the short-term research.

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