Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

Abstract

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.

1. Introduction and Preliminaries

A self-mapping , defined on a metric space , is said to be a contraction if there exists a constant such that the inequality holds for all . Moreover, a self-mapping is called a contractive mapping if holds for all with .

The celebrated Banach contraction principle says that if is complete, then every contraction has a unique fixed point. In fact, the fixed point of a contraction mapping is obtained as a limit of repeated iteration of the mapping for any (initial) point of . Let be the class of continuous, nondecreasing mapping such that is positive on and . A function is called an altering distance function.

A mapping is called a weak- contraction if there exists a such that for each . The notion of weak- contraction was defined by Alber and Guerre-Delabriere [1] to generalize the well-known Banach contraction principle in the setting of Hilbert spaces. Later, Rhoades [2] noticed that most of the results of Alber and Guerre-Delabriere [1] are valid for any Banach space. Rhoades also proved the following generalization of the Banach contraction principle (see also [3–7]).

Theorem 1. Let be a nonempty complete metric space and let be a weak- contraction on ; then has a unique fixed point.

Recently, Dutta and Choudhury [8] proved the following generalization of Theorem 1 by using -weak contraction map.

Theorem 2. Let be a nonempty complete metric space and let be a self-mapping satisfying the inequality for all , where of all function. Then has a unique fixed point.

Let be the class of all function;    is a lower semicontinuous with if and only if . In [9] Dorić proved the following generalization of Theorem 2 by using generalized -weak contractions which contains the -weak contractions as a subclass.

Theorem 3. Let be a nonempty complete metric space and let be a generalized -weak contraction map; that is, satisfies the following inequality: where  ,  , and for all . Then has a unique fixed point.

One of the aims of this paper is to extend Theorem 3 via best proximity point. For this purpose, we recollect the basic definitions and fundamentals results as follows.

1.1. Best Proximity Point Theorems

We first recall the notion of best proximity point for nonself-mappings.

Definition 4. Let be a pair of two nonempty subsets of a metric space . An element is said to be a best proximity point of the nonself-mappings if it satisfies the condition that where and are nonempty subsets of a metric space.

Best proximity point theorems have been studied to find necessary condition such that the minimization problem has at least one solution.

Existence and convergence of best proximity point is an interesting topic of optimization theory which recently attracted the attention of many authors [10–15]. A best proximity point theorem for nonself-proximal contractions has been investigated in [16–18].

In this paper, let us consider the mappings , where and are nonempty subsets of a metric space with generalized proximal weak contraction on which ensure the existence of a unique point which satisfies . When the map is considered to be self-map, then our result reduces to Theorem 3.

Given nonempty subsets and of a metric space , the following notions are used subsequently: In [13], the authors discussed sufficient conditions which guarantee the nonemptiness of and . Also, in [14], the authors proved that is contained in the boundary of .

Definition 5. The set is said to be approximatively compact with respect to if every sequence in satisfying the condition that for some in has a convergent subsequence.

Note that every set is approximatively compact with respect to itself and that every compact set is approximatively compact. Further, and are nonempty if is compact and is approximatively compact with respect to .

Let us define the notion of generalized proximal weak contraction maps as follows. For this goal, we first introduce the following class of the mapping. Let be the set of all function;   is nondecreasing with the following property: if and only if .

Definition 6. A mapping is said to be a generalized proximal weak contraction on if there exists functions satisfying the following condition: where for all in , where and .

Remark 7. Definition 6 guarantees that if a mapping has a best proximity point then it should be unique. Indeed, we can prove our claim easily. Let be a best proximity point of . Suppose, on the contrary, that there is another element such that . Since is a generalized proximal weak contraction on , we have where .
From (5), we obtain , which implies , and by our assumption about , we get , or equivalently, .

Definition 8. A mapping is said to be a generalized proximal weak contraction on if there exist functions satisfying the following condition: where , for all in , where and .

For self-mappings, it is clear that every generalized proximal weak contraction on is a generalized proximal weak contraction on . An operator is said to be a generalized proximal weak contraction if it is both generalized proximal weak contraction on and generalized proximal weak contraction on .

2. Main Results

We start this section with our main result.

Theorem 9. Let be a pair of two nonempty closed subsets of a complete metric space such that is nonempty. Let be a map satisfying the following conditions:(i)is a generalized proximal weak contraction,(ii). Then, there exists a unique such that . Further, for any fixed element , the sequence , defined by , converges to the element .

Proof. We prove the theorem in several steps.
Step 1. Let . Since , there exists such that . Due to the fact that , there exists such that . Recursively, we find a sequence in such that If there exists such that , then ; that is, is a best proximity point of . Thus, the proof is finished. Hence, we suppose that for all . Since is a generalized proximal weak contraction on , it follows that Using the monotone property of the -function, we get .
Now from the triangle inequality for , we have If , then . From (8), we obtain , which is a contradiction. So, we have Hence, the sequence is monotone nonincreasing and bounded. Thus, there exists such that Suppose that . Then the inequality implies that But as and is nondecreasing function, and this gives us which contradicts to (13). Hence,
Step 2. We will show that is a Cauchy sequence. Suppose, on the contrary, that is not a Cauchy sequence. Thus, there exists for which we can find subsequences and of such that is the smallest index for which .
This means that Letting and by using (15), we conclude that Again, Therefore, Letting and by using (17) together with (15), it follows that Similarly, we derive that Then, we have Using the fact that is generalized proximal weak contraction on for and , we obtain By using (22) and the continuity of in the above inequality, we find that But from , we can find such that for any and consequently, Therefore, and this contradicts to (24). Thus, is a Cauchy sequence in and hence converges to some element in . Analogously, by using the fact that is generalized proximal weak contraction on , we conclude that is a Cauchy sequence in . Hence, converges to some element in .
Step 3. Let us now prove that is best proximity point for .
Recall that in and in . Therefore, from (7), we get and hence is a member of . Since , we get ; hence there exists such that Since is a generalized proximal weak contraction on , we obtain where .
By using the fact that , we get Regarding (29) and continuity of in (28), we can obtain But from we find such that for any and consequently, since is nondecreasing, we get By using (30) in the inequality above, we get and from the property of , we obtain or equivalently, . Hence, from (27), we have .

Example 10. Consider the complete space with usual metric.

Suppose that and . Then and are nonempty closed subsets of and and . Note that . Let be defined as Suppose that , , ,  are elements in such that Then, and become the members of . Consequently, we have By assuming that , such that and , we get

Therefore, is both generalized proximal weak contraction on and generalized proximal weak contraction on . Hence is generalized proximal weak contraction such that . So, all the hypotheses of Theorem 9 are satisfied. Further, it is easy to see that is the unique element satisfying the conclusion of Theorem 9.

It is easy to see that a self-mapping that is a generalized proximal weak contraction reduces to a generalized -weak contraction. Hence the above Theorem 9 gives rise to the following fixed point theorem, due to Dorić [9], which in turn extends the famous contraction principle.

Corollary 11. Let be a nonempty complete metric space and let be a generalized proximal weak contraction map. Then has a unique fixed point.

Theorem 12. Let be a pair of two nonempty closed subsets of a complete metric space such that is approximatively compact with respect to and is nonempty. Let be a map satisfying the following conditions:(i) is a generalized proximal weak contraction on ,(ii). Then, there exists a unique such that . Further, for any fixed element , the sequence , defined by , converges to the element .

Proof. Following the proof of Theorem 9, there exists a sequence in satisfying the following conditions: and converges to in . Let us now prove that is best proximity point for .
Note that from (37), we have Since , we get . Since is approximatively compact with respect to the set , it follows that the sequence has a subsequence converging to some in . Now arguing like Step 3 and Step 4 of Theorem 9, we get the required result.

In what follows we prove that Theorem 12 is still valid for not necessarily approximatively compact with respect to , assuming that is closed.

Theorem 13. Assume that “ is closed” instead of “the statement is approximatively compact with respect to A” in Theorem 12.

Proof. Following the proof of Theorem 9, there exists a sequence in satisfying the following condition: and converges to in . Note that the sequence in and is closed. Therefore, . Since , we get . Since , there exists such that . Since is a generalized proximal weak contraction on , we have where .
Now arguing like Step 3 and Step 4 of Theorem 9, we get the required result.

Theorem 14. Assume that “ is continuous” instead of the statement “ is approximatively compact with respect to ” in the Theorem 12.

Proof. Following the proof of Theorem 9, there exists a sequence in satisfying the following condition: and converges to in . Since are continuous, we have Uniqueness follows the same as in Theorem 9.

Conflict of Interests

The authors declare that there is no 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.