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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 730825, 7 pages
http://dx.doi.org/10.1155/2013/730825
Research Article

Fixed Point Results for --Contractions on Gauge Spaces and Applications

1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 21 July 2013; Accepted 25 August 2013

Academic Editor: Calogero Vetro

Copyright © 2013 Mohamed Jleli 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 extend the concept of α-ψ-contractive mappings introduced recently by Samet et al. (2012) to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

1. Introduction and Preliminaries

Fixed point theory plays an important role in nonlinear analysis. This is because many practical problems in applied science, economics, physics, and engineering can be reformulated as a problem of finding fixed points of nonlinear mappings. The Banach contraction principle [1] is one of the fundamental results in fixed point theory. It guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to approximate those fixed points.

Theorem 1 (see [1]). Let be a complete metric space. Let be a contraction self-mapping on ; that is, there exists a constant such that for all . Then the mapadmits a unique fixed point. Moreover, for any , the sequence converges to this fixed point.

During the last few decades, several extensions of this famous principle have been established. In 1961, Edelstein [2] established the following result.

Theorem 2 (see [2]). Let be complete and -chainable for some . Let be such that where is a constant. Thenhas a unique fixed point.

Kirk et al. [3] introduced the concept of cyclic mappings and proved the following fixed point theorem.

Theorem 3 (see [3]). Let and be two nonempty closed subsets of a complete metric space . Let be a self-mapping such that where is a constant. Suppose also that and . Then has a unique fixed point in .

Ran and Reurings [4] extended the Banach contraction principle to a metric space endowed with a partial order. They established the following result.

Theorem 4 (see [4]). Let be a complete metric space endowed with a partial order . Let be a continuous mapping such that where is a constant. Suppose also that there exists such that . Then has a fixed point.

Many extensions of the previous result exist in the literature; for more details, we refer the reader to [511] and the references therein.

Observe that all the contractive conditions (2), (3), and (4) can be written as where is a subset of . In Theorem 2, we have In Theorem 3, we have In Theorem 4, we have The contractive condition (5) is said to be a partial contraction, that is, a contraction satisfied only on a subset of .

Very recently, Samet et al. [12] observed that a partial contraction can be considered as a total contraction, that is, a contraction satisfied for every pair . More precisely, if we define the function by we show that (5) is equivalent to In [12], the authors considered a more general inequality; that is, where is a function satisfying some conditions. The above inequality is called an --contraction. In [12], some fixed point results were established under this contractive condition. For other works in this direction, we refer the reader to [1316].

The aim of this work is to extend, generalize, and improve the obtained results in [12]. More precisely, the concept of --contractive mappings is extended to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

Through this paper, will denote a gauge space endowed with a separating gauge structure , where is a directed set.

A sequence is said to be convergent if there exists an such that for every and , there is an with , for all .

A sequence is said to be Cauchy, if for every and , there is an with , for all and .

A gauge space is called complete if any Cauchy sequence is convergent.

A subset ofis said to be closed if it contains the limit of any convergent sequence of its elements.

For more details on guage spaces, we refer the reader to Dugundji [17].

We denote by the set of functions satisfying the following conditions:(C1) is nondecreasing;(C2), for all , where is the th iterate of ;(C3), for all .It is easy to show that under the conditions (C1) and (C2), we have , for all . Moreover, under the conditions (C1) and (C3), we have for all and .

Example 5. Let be the function defined by Then .

Definition 6. Let be a given function. Let and . We say that is an --path fromtoif

We denote Let and . For and , and let

2. Fixed Point Results for --Contractions

Definition 7. Let be a given self-mapping. Let be a given function, and let . We say that is an --contraction if for all and .

Definition 8. Let be a given self-mapping. Let be a given function. We say that is -admissible if

The following lemma will be useful to establish our fixed point results.

Lemma 9. Let be a self-mapping. Suppose that there exist and such that the following conditions hold:(i) is an --contraction;(ii) is -admissible.Let and . Then, for every , , and , one has(I) with(II)for all , one has with

Proof. Let , , and . Let be an --path from tosuch that Since , we have Since , we have Recursively from to , since , we have On the other hand, sinceis-admissible, is an --path from to . So, we have Thus, we proved (I).
Again, since , we have Since , we have Recursively from to , since , we have On the other hand, since is -admissible, is an --path from to . So, we have Continuing this process, by induction, we get (II).

Definition 10. Let be a self-mapping, and let be a given function. For , we say that a sequence is an --Picard trajectory from if for all . We denote by , the set of all --Picard trajectories from.

Definition 11. Let be a self-mapping, and let be a given function. For , we say that is --Picard continuous from if the limit of any convergent sequence is a fixed point of .

We have the following fixed point result.

Theorem 12. Let be a self-mapping on the complete gauge space . Let and be a given function. Suppose that the following conditions hold:(i) is an -contraction;(ii) is -admissible;(iii)there exist and such that ;(iv) is --Picard continuous from .Then has a fixed point.

Proof. Let and . From condition (iii) and Lemma 9, we have and Again, from Lemma 9, we have and Continuing this process, by induction, for , we have and Thus, and, for , From condition (C2), we have which implies that is a Cauchy sequence in the complete gauge space . Since is --Picard continuous from , the limit of is a fixed point of .

Corollary 13. Let be a self-mapping on the complete gauge space . Let and be a given function. Suppose that the following conditions hold:(i) is an --contraction;(ii) is -admissible;(iii)there exist and such that ;(iv) is continuous.Then has a fixed point.

Proof. Let be such that . Since is continuous, we have . Since is endowed with a separating gauge structure, we have . The conclusion follows from Theorem 12.

Corollary 14. Let be a self-mapping on the complete gauge space . Let and be a given function. Suppose that the following conditions hold:(i) is an contraction;(ii) is -admissible;(iii)there exist and such that ;(iv)for every such that , there exist a subsequence of  and   such that for . Then has a fixed point.

Proof. Let be such that . From condition (iv), there exist a subsequence of and such that for . Since is an --contraction, for all and , we have Letting in the above inequality, we obtain that Since is endowed with a separating gauge structure, we have . The conclusion follows from Theorem 12.

For , we denote by the set of fixed points of ; that is, The next result gives us a sufficient condition that ensures the uniqueness of the fixed point.

Theorem 15. Suppose that all the conditions of Theorem 12 are satisfied. Moreover, suppose that (v)for every with , there exists such that.Then has a unique fixed point.

Proof. From Theorem 12, The mapping has at least one fixed point. Suppose that are two fixed points of with . From the condition (v), there exists such that . Let and . From Lemma 9, we have for all . Letting in the above inequality, we obtain that for all , which is a contradiction with (since we have a separating gauge structure). We deduce that .

The following result follows immediately from Theorems 12 and 15 with and for every .

Corollary 16. Let be a self-mapping on the complete gauge space . Let . Suppose that for all , for all , one has Then has a unique fixed point.

Corollary 17. Let be a self-mapping on the complete gauge space . Let and be a given function. Suppose that the following conditions hold:(i) is an --contraction;(ii) is -admissible;(iii)there exists such that ;(iv) is continuous(or) for any sequence such that , and for , there exist a subsequence of and such that for .
Then has a fixed point. Moreover, if (v)for every with , there exists such that and ,one has uniqueness of the fixed point.

Proof. The existence follows from Theorem 12 with . The uniqueness follows from Theorem 15 with .

Corollary 18. Let be a partial order on the complete gauge space . Let be a self-mapping and . Suppose that the following conditions hold:(i)for all , for all such that and are comparable, one has (ii) and are comparable and are comparable;(iii)there exists such that and are comparable;(iv) is continuous,(or) for any sequence such that , , and are comparable for , there exist a subsequence of and such that and are comparable for .
Then has a fixed point. Moreover, if (v) for every with , there exists such that and are comparable, and are comparable,one has uniqueness of the fixed point.

Proof. It follows from Corollary 17 with

3. Applications

In this section, we are interested in the study of the existence of solutions to the nonlinear integral equation on the real axis where and . Here, is a Banach space with respect to a given norm .

Let and the family of pseudonorms defined by For every , define now Then is a separating gauge structure on.

We have the following existence result.

Theorem 19. Suppose that the following conditions hold:(i)there exist a nonempty set and a constant such that for all , ;(ii) there exists a nonempty set such that (iii) for all , one has for all ;(iv) there exists such that for all ;(v) if is a sequence such that   for and (with respect to ), then there exist a subsequence of and such that for all , .
Then (42) has at least one solution in .

Proof. Consider the mapping defined by for all . We have to prove that has at least a fixed point.
Define the function by We claim that for all , for all , where for all , for all . Clearly, since , . If , (52) holds immediately. So, suppose that ; that is, . Let . From conditions (i) and (ii), for all , we have which implies that Thus, we proved (52).
We will prove thatis-admissible. Let such that ; that is, . From condition (iii), we have for all , which implies from condition (ii) that ; that is, . So, is -admissible.
From conditions (iv) and (ii), we have , which is equivalent to say that .
Finally, condition (v) implies that for every such that , there exist a subsequence of and such that for .
Now, All the hypotheses of Corollary 14 are satisfied; we deduce that has at least a fixed point, which is a solution to (52).

Theorem 20. In addition to the assumptions of Theorem 19, suppose that (vi) for all , there exists such that and .Then (42) has one and only one solution in .

Proof. It follows immediately from Theorem 15.

Conflict of Interests

The authors declare that there is no competing/conflict of interests regarding the publication of this paper.

Authors’ Contribution

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

Acknowledgment

This work is supported by the Research Center, College of Science, King Saud University.

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