#### Abstract

Let and be nonempty subsets of a metric space with the distance function d, and is a given non-self-mapping. The purpose of this paper is to solve the nonlinear programming problem that consists in minimizing the real-valued function , where belongs to a new class of contractive mappings. We provide also an iterative algorithm to find a solution of such optimization problems.

#### 1. Introduction

Let and be nonempty subsets of a metric space . Because the functional equation (), where is a given non-self-mapping, does not necessarily have a solution, it is desirable in this case to find an optimal approximate solution to the equation when the equation has no solution. In view of the fact that is a lower bound for , where an approximate solution to the equation produces the least possible error if . Such a solution is called a best proximity point of the mapping . Due to the fact that for all , a best proximity point provides the global minimum of the nonlinear programming problem . The results that provide sufficient conditions that ensure the existence of a best proximity point are known as best proximity point theorems. Best proximity point theorems for various classes of non-self-mapppings have been established in .

This work focuses on best proximity point theorems for a new family of non-self-mapppings known as MK-proximal contractions. An iterative algorithm is presented to compute an optimal approximate solution to some fixed point equations. The presented theorems extend and generalize several existing results in the literature including the well-known result of Meir and Keeler .

#### 2. Preliminaries

We present in this section some notations and notions that will be used later.

Let be a metric space; and are two nonempty subsets of . We consider the following notations:

Definition 2.1 (see ). is said to be approximatively compact with respect to if every sequence of satisfying the condition that for some in has a convergent subsequence.

Definition 2.2. An element is said to be a best proximity point of the non-self-mappping if it satisfies the condition that
Because of the fact that for all , the global minimum of the mapping is attained at a best proximity point. Moreover, if the underlying mapping is a self-mapping, then it can be observed that a best proximity point is essentially a fixed point.

Definition 2.3. One says that is an isometry if for any , one has

Definition 2.4 (see ). Given a mapping and an isometry , the mapping is said to preserve the isometric distance with respect to if for any , one has

Definition 2.5 (see ). Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the -property if and only if where and .
We introduce the concept of the weakly -property as follows.

Definition 2.6. Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the weakly -property if and only if where and .
Note that if the pair has the -property, then it has the weakly -property. The following example shows that the converse is not true in general.

Example 2.7. Consider the Euclidean space endowed with the Euclidean metric . Let us define the sets Clearly, we have . On the other hand, we have but . Thusdoes not satisfy the -property. Now, suppose that for some . This implies immediately that , that is, . Similarly, if for some , we get that . This implies that has the weakly -property.

Definition 2.8. A self-mapping is said to be an MK-contraction of the first kind if, for all , there exists such that
The class of MK-contractions of the first kind was introduced by Meir and Keeler in . It is easy to show that every contraction is an MK-contraction of the first kind.

Definition 2.9. A self-mapping is said to be an MK-contraction of the second kind if, for all , there exists such that
Clearly every MK-contraction of the first kind is an MK-contraction of the second kind.

Definition 2.10. A non-self-mappping is said to be an MK-proximal contraction of the first kind if, for all , there exists such that
Clearly, if , an MK-proximal contraction of the first kind is an MK-contraction of the first kind.

Lemma 2.11. Let be an MK-proximal contraction of the first kind. Suppose that the pair has the weakly -property. Then,

Proof. Let such that and . Let . Since , we have . If , since has the weakly -property, we get that , that is, .

Definition 2.12. A non-self-mappping is said to be an MK-proximal contraction of the second kind if, for all , there exists such that
If , an MK-proximal contraction of the second kind is an MK-contraction of the second kind.
Similarly, we have the following.

Lemma 2.13. Let be an MK-proximal contraction of the second kind. Suppose that the pair has the weakly -property. Then,

#### 3. Main Results

We have the following best proximity point result for MK-proximal contractions.

Theorem 3.1. Let and be closed subsets of a complete metric space such that is nonvoid and the pair satisfies the weakly -property. Suppose that the mappings and satisfy the following conditions: (a)is an MK-proximal contraction of the first and second kinds; (b); (c) is an isometry; (d); (e) preserves the isometric distance with respect to .
Then, there exists a unique element such that Further, for any fixed element , the iterative sequence , defined by converges to .

Proof. Let (such a point exists since ). From conditions (b) and (d), there exists such that Again, from conditions (b) and (d), there exists such that Continuing this process, we can construct a sequence such that Since is an MK-proximal contraction of the first kind and is an isometry, it follows from (3.5) and Lemma 2.11 that Further, since is an MK-proximal contraction of the second kind and preserves isometric distance with respect to , it follows from Lemma 2.13 that
Claim  1. We claim that is a Cauchy sequence.
Let . From (3.6), is a nonnegative, bounded below and decreasing sequence of real numbers and hence converges to some nonnegative real number (). Let us suppose that . This implies that there exists such that . Since is an MK-proximal contraction of the first kind and is an isometry, this implies that , which is a contradiction. Thus we have , that is, Fix . Without restriction of the generality, we can suppose that . Then, there exists such that Let us denote by the subset of defined by We will prove that Let such that for some . We distinguish two cases.
Case  1. If , we have
Case  2. If , we have Thus, in all cases, we have , and (3.11) is proved.
Now, we shall prove that Clearly, for , (3.14) is satisfied. Moreover, from (3.9), (3.14) is satisfied for . Now, from (3.5), we have . Since we have also , from (3.11), we get that . Continuing this process, by induction, we get (3.14).
Finally, for all , from (3.14), we have which implies that is a Cauchy sequence. Thus, Claim 1 is proved.
Claim  2. We claim that is a Cauchy sequence.
Let . From (3.7), is a nonnegative, bounded below and decreasing sequence of real numbers and hence converges to some nonnegative real number . Let us suppose that . This implies that there exists such that . Since is an MK-proximal contraction of the second kind and preserves the isometric distance with resect to , we get which is a contradiction. Thus we have , that is, Fix . Without restriction of the generality, we can suppose that . Then, there exists such that Let us denote by the subset of defined by We shall prove that Let such that for some . We distinguish two cases.
Case  1. If , we have
Case  2. If , we have Thus, in all cases, we have , and (3.20) is proved.
Now, we shall prove that Clearly, for , (3.23) is satisfied. Moreover, from (3.18), (3.23) is satisfied for . Since and , from (3.20), we have . Continuing this process, by induction, we get (3.23).
Finally, for all , from (3.23), we have which implies that is a Cauchy sequence. Thus, Claim 2 is proved.
Now, since is complete and is closed, there exists such that as . Similarly, since is closed, there exists such that as . Therefore, we have This implies that . From condition (d), there exists such that . On the other hand, since is an isometry, we get that , which implies that . Now, from condition (b), we have ; that is, there exists such that . Since is an MK-proximal contraction of the first kind, using Lemma 2.11, we get that Letting , it follows that that . Thus, it can be concluded that To assert the uniqueness, let us assume that is another element in such that . Due to the fact that is an MK-proximal contraction of the first kind and is an isometry, using Lemma 2.11, we get that which is a contradiction. Then and are identical. This completes the proof.

If is the identity mapping, then Theorem 3.1 yields the following result.

Corollary 3.2. Let and be closed subsets of a complete metric space such that is nonvoid and the pair satisfies the weakly -property. Suppose that the mapping satisfies the following conditions: (a) is an MK-proximal contraction of the first and second kinds; (b).
Then, there exists a unique element such that Further, for any fixed element , the iterative sequence , defined by converges to .

Example 3.3. We endow with the standard metric: for all . Consider the sets defined by Then and are nonempty closed subsets of with . It is easy to show that in this case we have and . On the other hand, for all , we have Clearly, for all , , Thus the pair satisfies the weakly -property.
Let be the mapping defined by

Claim 1. is an MK-proximal contraction of the first kind.
Let be fixed and . Let such that We consider three cases.

Case 1. There exist such that In this case, from (3.36), we have

Case 2. There exists such that In this case, from (3.36), we have

Case 3. There exists such that In this case, from (3.36), we have Thus, in all cases we have . Then Claim 1 holds.

Claim 2. is an MK-proximal contraction of the second kind.
Let be fixed and . Let such that We consider also three cases.

Case 1. There exist such that In this case, from (3.43), we have

Case 2. There exists such that In this case, from (3.43), we have

Case 3. There exists such that In this case, from (3.43), we have Thus, in all cases we have . Then Claim 2 holds.
Finally, from Corollary 3.2, there exists a unique such that . In this example, we have .

The preceding best proximity point result (see Corollary 3.2) gives rise to the following fixed point theorem, due to Meir and Keeler , which in turn extends the famous Banach contraction principle .

Corollary 3.4 (Meir-Keeler ). Let be a complete metric space and be an MK-contraction of the first kind. Then has a unique fixed point , and for each , .

The following result furnishes another best proximity point theorem for MK-proximal contractions.

Theorem 3.5. Let and be closed subsets of a complete metric space such that is nonvoid, the pair satisfies the weakly -property, and is approximatively compact with respect to . Suppose that the mappings and satisfy the following conditions: (a) is an MK-proximal contraction of the first kind; (b); (c) is an isometry; (d).
Then, there exists a unique element such that Further, for any fixed element , the iterative sequence , defined by converges to .

Proof. Proceeding as in Theorem 3.1, it can be shown that there is a sequence of elements in such that and any sequence satisfying the above condition must converge to some element . On the other hand, we have Letting in the above inequality, we get as . Since is approximatively compact with respect to , it follows that the sequence has a subsequence converging to some element . Thus we have The rest part of the proof follows as in Theorem 3.1.

If is the identity mapping, the preceding best proximity point theorem yields the following special case.

Corollary 3.6. Let and be closed subsets of a complete metric space such that is nonvoid, the pair satisfies the weakly -property, and is approximatively compact with respect to . Suppose that the mapping satisfies the following conditions: (a) is an MK-proximal contraction of the first kind; (b).
Then, there exists a unique element such that Further, for any fixed element , the iterative sequence , defined by converges to .

#### Acknowledgment

The first and third authors are supported by the Research Center, College of Science, King Saud University.