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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 329451, 4 pages
http://dx.doi.org/10.1155/2013/329451
Research Article

A Best Proximity Point Result in Modular Spaces with the Fatou Property

1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 15 June 2013; Accepted 9 September 2013

Academic Editor: Salvador Hernandez

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

Consider a nonself-mapping , where is a pair of nonempty subsets of a modular space . A best proximity point of is a point satisfying the condition: . In this paper, we introduce the class of proximal quasicontraction nonself-mappings in modular spaces with the Fatou property. For such mappings, we provide sufficient conditions assuring the existence and uniqueness of best proximity points.

1. Introduction and Preliminaries

Through this paper, we denote by the set of positive integers including zero. Let be a vector space over . We denote by its zero vector. According to Orlicz [1], a functional is said to be modular, if, for any pair , the following conditions are satisfied:(i) if and only if ; (ii); (iii) whenever and .

If is a modular in , then the set called a modular space, is a vector space.

As a classical example of modulars, we may give the Orlicz modular defined for every measurable real function by where is the Lebesgue measure in and is a function satisfying some conditions. The modular space induced by the Orlicz modular is called the Orlicz space. For more examples of modular spaces, we refer the reader to [24].

Definition 1. Let be a modular space.(1)The sequence is said to be -convergent to if , as . (2)The sequence is said to be -Cauchy if , as . (3)A subset of is called -closed if the -limit of a -convergent sequence of always belongs to . (4)A subset of is called -complete if any -Cauchy sequence in is -convergent and its -limit belongs to .

Definition 2. The modular has the Fatou property if whenever -converges to .

Recently, the existence and uniqueness of best proximity points in metric spaces were investigated by many authors; see [2, 514] and references therein. In this paper, we introduce the family of proximal quasicontraction nonself-mappings on modular spaces with the Fatou property. Our main result is a best proximity point theorem providing sufficient conditions assuring the existence and uniqueness of best proximity points for such mappings.

Let be a pair of nonempty closed subsets of a modular space . Through this paper, we will use the following notations:

Definition 3. Let be a given nonself-mapping. We say that is a best proximity point of if

Clearly, from condition (i), if , a best proximity point of will be a fixed point of .

Definition 4. A nonself-mapping is said to be a proximal quasicontraction if there exists a number such that where .

Lemma 5. Let be a nonself-mapping. Suppose that (i); (ii). Then, for any , there exists a sequence such that

Proof. Let . From (ii), we have . By definition of the set , there exists such that . Again, we have , which implies that there exists such that . Continuing this process, by induction, we obtain a sequence satisfying (6).

Definition 6. Under the assumptions of Lemma 5, any sequence satisfying (6) is called a proximal Picard sequence associated to . We denote by the set of all proximal sequences associated to .

Definition 7. Under the assumptions of Lemma 5, we say that is proximal -orbitally -complete if every -Cauchy sequence for some -converges to an element in .

Let and . For all , We denote Since , we have

2. A Best Proximity Point Theorem

The following lemmas will be useful later.

Lemma 8. Let be a modular space. Suppose that a nonself-mapping , where is a pair of subsets of , satisfies the following conditions:(i); (ii); (iii) is proximal quasi-contraction. Then, for any , one has for any and .

Proof. Let and . From the definition of , for all , we have which implies, since is a proximal quasi-contraction, that This implies immediately that for all . Hence, for any , we have Using the above inequality, for all and , we have

Lemma 9. Let be a pair of subsets of a modular space . Let be a given nonself-mapping. Suppose that (i) is proximal -orbitally -complete; (ii); (iii) such that ; (iv) is proximal quasi-contraction; (v) satisfies the Fatou property. Then, any sequence -converges to some such that for all . Moreover, there exists such that

Proof. Let . From Lemma 8, we know that is -Cauchy. Since is proximal -orbitally -complete, then there exists such that -converges to . Again, by Lemma 8, we have for any and . Letting in the above inequality and using the Fatou property, we obtain for all . Now, since , by the definition of , there exists some such that .

Now, we are ready to state and prove our main result.

Theorem 10. Suppose that the assumptions of the previous lemma are satisfied. Assume and . Then, the -limit of is a best proximity point of . Moreover, if is any best proximity point of such that , then one has .

Proof. By Lemma 9, we have On the other hand, from the definition of , we have Since is proximal quasi-contraction, we get that Using Lemmas 8 and 9, we obtain that Again, from the definition of , we have Since is proximal quasi-contraction, we get that Thus, we proved that Continuing this process, by induction, we get that for all . Therefore, we have Using the Fatou property, we get which implies, since , that ; that is, . Thus, from (19), we get that Hence, is a best proximity point of .
Suppose now that is a best proximity point of such that . Since is proximal quasi-contraction, we obtain that Since , we have , which implies that .

Consider now the case . In this case, a best proximity point of will be a fixed point of the self-mapping .

Definition 11. We say that is -orbitally -complete if is a -Cauchy for every , then it is -convergent to an element of .

Similarly to Ćirić [15] definition, Khamsi [16] introduced the concept of quasicontraction self-mappings in modular spaces.

Definition 12. The self-mapping is said to be a quasicontraction if there exists a constant such that for all .

From Theorem 10, we can deduce the following result, that is, a slight extension of the fixed point theorem established by Khamsi in [16].

Corollary 13. Consider a self-mapping , where is a nonempty subset of . Suppose that the following conditions hold:(i) is -orbitally -complete; (ii) such that ; (iii) satisfies the Fatou property; (iv) is quasi-contraction. Then, the sequence -converges to some . Moreover, if and , then is a fixed point of  . If is a fixed point of   with , then .

Acknowledgment

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

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