Abstract and Applied Analysis

Volume 2012 (2012), Article ID 193085, 14 pages

doi:10.1155/2012/193085

Research Article

Best Proximity Point Results for MK-Proximal Contractions

Mohamed Jleli,¹ Erdal Karapınar,² and Bessem Samet¹

¹Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

²Department of Mathematics, Atilim University, İncek, 06836, Ankara, Turkey

Received 10 October 2012; Accepted 15 December 2012

Academic Editor: Qamrul Hasan Ansari

Copyright © 2012 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.

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 [1–32].

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 [33].

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 [3]). 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 [6]). 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 [27]). 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 [33]. 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 [33], which in turn extends the famous Banach contraction principle [34].

Corollary 3.4 (Meir-Keeler [33]). 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.

References

M. A. Al-Thagafi and N. Shahzad, “Best proximity sets and equilibrium pairs for a finite family of multimaps,” Fixed Point Theory and Applications, Article ID 457069, 10 pages, 2008.

J. Anuradha and P. Veeramani, “Proximal pointwise contraction,” Topology and its Applications, vol. 156, no. 18, pp. 2942–2948, 2009.

S. S. Basha, “Extensions of Banach's contraction principle,” Numerical Functional Analysis and Optimization, vol. 31, no. 5, pp. 569–576, 2010.

S. S. Basha, “Best proximity points: global optimal approximate solutions,” Journal of Global Optimization, vol. 49, no. 1, pp. 15–21, 2011.

S. S. Basha, N. Shahzad, and R. Jeyaraj, “Common best proximity points: global optimization of multi-objective functions,” Applied Mathematics Letters, vol. 24, no. 6, pp. 883–886, 2011.

S. Sadiq Basha, “Best proximity point theorems: an exploration of a common solution to approximation and optimization problems,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9773–9780, 2012.

H. H. Cuenya and A. G. Bonifacio, “Best proximity pairs in uniformly convex spaces,” Bulletin of the Institute of Mathematics, vol. 3, no. 3, pp. 391–398, 2008.

M. De la Sen, “Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces,” Fixed Point Theory and Applications, vol. 2012, Article ID 510974, 12 pages, 2010.

C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 69, no. 11, pp. 3790–3794, 2008.

A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.

A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, no. 3, pp. 283–293, 2005.

A. Fernández-León, “Existence and uniqueness of best proximity points in geodesic metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 4, pp. 915–921, 2010.

E. Karapınar, “Best proximity points of Kannan type cylic weak phi-contractions in ordered metric spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta. In press

E. Karapınar, “Best proximity points of cyclic mappings,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1761–1766, 2012.

E. Karapınar, G. Petruþel, and K. Taþ, “Best Proximity Point Theorems for KT-Types Cyclic Orbital Contraction Mappings,” Fixed Point Theory, vol. 13, no. 2, pp. 537–545, 2012.

E. Karapınar and I. M. Erhan, “Cyclic contractions and fixed point theorems,” Filomat, vol. 26, pp. 777–782, 2012.

E. Karapınar, İ. M. Erhan, and A. Y. Ulus, “Fixed point theorem for cyclic maps on partial metric spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 2, pp. 239–244, 2012.

E. Karapınar, “Fixed point theory for cyclic -weak contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011.

M. Jleli, E. Karapınar, and B. Samet, “Fixed point results for almost generalized cyclic -weak contractive type mappings with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 917831, 17 pages, 2012.

E. Karapinar, S. Romaguera, and K. Tas, “Fixed points for cyclic orbital generalized contractions on complete metric spaces,” Central European Journal of Mathematics, vol. 11, no. 3, pp. 552–560, 2013.

E. Karapinar and K. Sadarangani, “Fixed point theory for cyclic -contractions,” Fixed Point Theory and Applications, vol. 2011, article 69, 2011.

E. Karapınar and I. S. Yuce, “Fixed point theory for cyclic generalized weak -contraction on partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 491542, 12 pages, 2012.

E. Karapınar and M. Erhan, “Best proximity point on different type contractions,” Applied Mathematics & Information Sciences, vol. 5, no. 3, pp. 558–569, 2011.

S. Karpagam and S. Agrawal, “Existence of best proximity points of -cyclic contractions,” Fixed Point Theory, vol. 13, no. 1, pp. 99–105, 2012.

W. K. Kim, S. Kum, and K. H. Lee, “On general best proximity pairs and equilibrium pairs in free abstract economies,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 68, no. 8, pp. 2216–2227, 2008.

W. A. Kirk, S. Reich, and P. Veeramani, “Proximinal retracts and best proximity pair theorems,” Numerical Functional Analysis and Optimization, vol. 24, no. 7-8, pp. 851–862, 2003.

V. S. Raj, “A best proximity point theorem for weakly contractive non-self-mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 14, pp. 4804–4808, 2011.

N. Shahzad, S. Sadiq Basha, and R. Jeyaraj, “Common best proximity points: global optimal solutions,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 69–78, 2011.

P. S. Srinivasan, “Best proximity pair theorems,” Acta Scientiarum Mathematicarum, vol. 67, no. 1-2, pp. 421–429, 2001.

A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, no. 3, pp. 283–293, 2005.

C. Vetro, “Best proximity points: convergence and existence theorems for -cyclic mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 7, pp. 2283–2291, 2010.

K. Włodarczyk, R. Plebaniak, and C. Obczyński, “Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 2, pp. 794–805, 2010.

A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 326–329, 1969.

S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.