Optimal Approximate Solutions of Fixed Point Equations

Abstract

The main objective of this paper is to present some best proximity point theorems for K-cyclic mappings and C-cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form 𝑇𝑥=𝑥 where 𝑇 is a non-self mapping.

1. Introduction

Fixed point theorems delve into the existence of a solution to the equations of the form 𝑇𝑥=𝑥 where 𝑇 is a self-mapping. However, when 𝑇 is a nonself-mapping, the equation 𝑇𝑥=𝑥 does not necessarily have a solution, in which case best approximation theorems explore the existence of an approximate solution whereas best proximity point theorems analyze the existence of an approximate solution that is optimal. Indeed, a classical and well-known best approximation theorem, due to Fan [1], contends that if 𝐾 is a nonempty convex compact subset of a Hausdorff topological vector space 𝐸 and 𝑇 is a continuous non-self mapping from 𝐾 to 𝐸, then there exists an element 𝑥 in 𝐾 such that 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵). Subsequently, many authors, including Prolla [2], Reich [3], and Sehgal and Singh [4, 5], accomplished several appealing extensions and variants of the preceding best approximation theorem. Further, Vetrivel et al. [6] elicited a more generalized result that unifies and subsumes many such results. Despite the fact that best approximation theorems produce an approximate solution to the equation 𝑇𝑥=𝑥, they may not render an approximate solution that is optimal. On the contrary, best proximity point theorems are intended to furnish an approximate solution 𝑥 that is optimal in the sense that the error 𝑑(𝑥,𝑇𝑥) is minimum. Indeed, in light of the fact that 𝑑(𝑥,𝑇𝑥) is at least 𝑑(𝐴,𝐵), a best proximity point theorem guarantees the global minimization of 𝑑(𝑥,𝑇𝑥) by the requirement that an approximate solution 𝑥 satisfies the condition 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵). Such optimal approximate solutions are called best proximity points of the mapping 𝑇.

Eldred et al. [7] have established interesting best proximity point theorems for relatively nonexpansive mappings. A Best proximity point theorem for contractive mapping has been explored in [8]. Best proximity point theorems for various types of contractions have been obtained in [9–13]. Best proximity point theorems for several types of set valued mappings have been derived in [14–25]. Moreover, common best proximity point theorems for pairs of contractions and for pairs of contractive mappings have been elicited in [26].

The main objective of this article is to prove some best proximity point theorems for K-cyclic mappings and C-cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form 𝑇𝑥=𝑥 where 𝑇 is a non-self-K-cyclic mapping or a non-self-C-cyclic mapping.

2. Preliminaries

The following notions will be used in the sequel.

Definition 2.1. A pair of mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 is said to form a K-Cyclic mapping between 𝐴 and 𝐵 if there exists a nonnegative real number 𝑘<1/2 such that 𝑑𝑇𝑥,𝑆𝑦≤𝑘𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑆𝑦+(1−2𝑘)𝑑(𝐴,𝐵), (2.1) for all 𝑥∈𝐴 and 𝑦∈𝐵.

Definition 2.2. A pair of mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 is said to form a C-Cyclic mapping between 𝐴 and 𝐵 if there exists a non-negative real number 𝑘<1/2 such that 𝑑𝑇𝑥,𝑆𝑦≤𝑘𝑑𝑥,𝑆𝑦+𝑑𝑦,𝑇𝑥+(1−2𝑘)𝑑(𝐴,𝐵), (2.2) for all 𝑥∈𝐴 and 𝑦∈𝐵.

Definition 2.3. A subset 𝐶 of a metric space is said to be boundedly compact if every bounded sequence in 𝐶 has a subsequence converging to some element in 𝐶.

It is evident that every compact set is boundedly compact but the converse is not true.

3. K-Cyclic Mappings

This section is concerned with best proximity point theorems for K-cyclic non-self mappings.

Lemma 3.1. Let 𝐴 and 𝐵 be two non-empty subsets of a metric space. Suppose that the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a K-Cyclic map between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Then, 𝑑(𝑥_𝑛,𝑥_𝑛+1)→𝑑(𝐴,𝐵).

Proof. As 𝑇 and 𝑆 form a K-Cyclic map, 𝑑𝑥₁,𝑥₂=𝑑𝑇𝑥₀,𝑆𝑥₁≤𝑘𝑑𝑥₀,𝑇𝑥₀+𝑑𝑥₁,𝑆𝑥₁+()𝑑()𝑑𝑥1−2𝑘𝐴,𝐵=𝑘₀,𝑥₁+𝑑𝑥₁,𝑥₂+(1−2𝑘)𝑑(𝐴,𝐵). (3.1) So, it follows that 𝑑(𝑥₁,𝑥₂)≤(𝑘/(1−𝑘))𝑑(𝑥₀,𝑥₁)+[1−(𝑘/(1−𝑘))]𝑑(𝐴,𝐵).
Similarly, it can be seen that 𝑑𝑥₂,𝑥₃≤𝑘1−𝑘²𝑑𝑥₀,𝑥₁+𝑘1−1−𝑘²𝑑(𝐴,𝐵). (3.2) Hence, it follows by induction that 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑘1−𝑘^𝑛𝑑𝑥₀,𝑥₁+𝑘1−1−𝑘^𝑛𝑑(𝐴,𝐵). (3.3) Therefore, 𝑑(𝑥_𝑛,𝑥_𝑛+1)→𝑑(𝐴,𝐵) because of the fact that 𝑘<1/2.

Lemma 3.2. Let 𝐴 and 𝐵 be non-empty closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a K-Cyclic map between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Then, the sequence {𝑥_𝑛} is bounded.

Proof. It follows from Lemma 3.1 that 𝑑(𝑥_2𝑛−1,𝑥_2𝑛) is convergent and hence it is bounded. Further, since 𝑆 and 𝑇 form a K-cyclic mapping, it follows that 𝑑𝑥_2𝑛,𝑇𝑥₀≤𝑘𝑑𝑥_2𝑛−1,𝑥_2𝑛+𝑑𝑥₀,𝑇𝑥₀+(1−2𝑘)𝑑(𝐴,𝐵). (3.4) Therefore, the subsequence {𝑥_2𝑛} is bounded. Similarly, it can be shown that {𝑥_2𝑛+1} is also bounded.

Lemma 3.3. Let 𝐴 and 𝐵 be non-empty closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a K-Cyclic map between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Suppose that the sequence {𝑥_2𝑛} has a subsequence converging to some element 𝑥 in 𝐴. Then, 𝑥 is a best proximity point of 𝑇.

Proof. Suppose that a subsequence {𝑥_2𝑛𝑘} converges to 𝑥 in 𝐴. It follows from Lemma 3.1 that 𝑑(𝑥_{2𝑛𝑘−1},𝑥_2𝑛𝑘) converges to 𝑑(𝐴,𝐵). As 𝑆 and 𝑇 form a K-cyclic mapping, it follows that 𝑑(𝐴,𝐵)≤𝑑𝑥_2𝑛𝑘,𝑇𝑥≤𝑘𝑑𝑥_{2𝑛𝑘−1},𝑥_2𝑛𝑘+𝑑(𝑥,𝑇𝑥)+(1−2𝑘)𝑑(𝐴,𝐵). (3.5) Therefore, 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵).

The preceding two lemmas yield the following best proximity point theorem for K-cyclic mappings in the setting of metric spaces.

Corollary 3.4. Let 𝐴 and 𝐵 be two non-empty and closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a K-Cyclic map between 𝐴 and 𝐵. If 𝐴 is boundedly compact, then 𝑇 has a best proximity point.

The following lemma, due to Eldred and Veeramani [10], will be required subsequently to establish the next best proximity point theorem of this section.

Lemma 3.5. Let 𝐴 be a non-empty, closed, and convex subset and 𝐵 be a non-empty and closed subset of a uniformly convex Banach space. Suppose that {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝐴 and {𝑧_𝑛} is a sequence in 𝐵 satisfying the following conditions:(a)‖𝑦_𝑛−𝑧_𝑛‖→𝑑(𝐴,𝐵), (b) for every 𝜖>0, ‖𝑥_𝑚−𝑧_𝑛‖≤𝑑(𝐴,𝐵)+𝜖, for sufficiently large values of 𝑚 and 𝑛.
Then, for every 𝜖>0, ‖𝑥_𝑚−𝑦_𝑛‖≤𝜖 for sufficiently large values of 𝑚 and 𝑛.

The following best proximity point theorem is for K-cyclic mappings in the setting of uniformly convex Banach spaces.

Theorem 3.6. Let 𝐴 and 𝐵 be non-empty, closed, and convex subsets of a uniformly convex Banach space. If the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a K-Cyclic map between 𝐴 and 𝐵, then there exist a unique element 𝑥∈𝐴 and a unique element 𝑦∈𝐵 such that 𝑑()(),𝑑(),𝑑().𝑥,𝑇𝑥=𝑑𝐴,𝐵𝑦,𝑆𝑦=𝑑𝐴,𝐵𝑥,𝑦=𝑑𝐴,𝐵 (3.6) Further, if 𝑥₀ is any fixed element in 𝐴, 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1, then the sequences {𝑥_2𝑛} and {𝑥_2𝑛+1} converge to the best proximity points 𝑥 and 𝑦, respectively.

Proof. It follows from Lemma 3.1 that 𝑥_2𝑚−1−𝑥_2𝑚(),𝑥⟶𝑑𝐴,𝐵_2𝑛−𝑥_2𝑛+1⟶𝑑(𝐴,𝐵). (3.7) Therefore, for every 𝜖>0, ‖𝑥_2𝑚−1−𝑥_2𝑚()+𝜖‖<𝑑𝐴,𝐵,2𝑘‖𝑥_2𝑛−𝑥_2𝑛+1()+𝜖‖<𝑑𝐴,𝐵,2𝑘 (3.8) for sufficiently large values of 𝑚 and 𝑛. As 𝑇 and 𝑆 form a K-cyclic mapping, ‖𝑥_2𝑚−𝑥_2𝑛+1‖≤𝑘‖𝑥_2𝑚−1−𝑥_2𝑚‖+‖𝑥_2𝑛−𝑥_2𝑛+1‖+()𝑑()()1−2𝑘𝐴,𝐵<𝑑𝐴,𝐵+𝜖, (3.9) for sufficiently large values of 𝑚 and 𝑛. Thus, {𝑥_2𝑛} is a Cauchy sequence by Lemma 3.5. Since the space is complete, {𝑥_2𝑛} converges to some element 𝑥∈𝐴, which becomes a best proximity point of the mapping 𝑇 by Lemma 3.3. Similarly, {𝑥_2𝑛+1} converges to some element 𝑦∈𝐵, which is a best proximity point of the mapping 𝑆. Further, 𝑑(𝑇𝑥,𝑆𝑦)≤𝑘[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑆𝑦)]+(1−2𝑘)𝑑(𝐴,𝐵)=𝑑(𝐴,𝐵). Therefore, 𝑑(𝑇𝑥,𝑆𝑦)=𝑑(𝐴,𝐵). By strict convexity of the space, 𝑇𝑥 and 𝑦 should be identical, and 𝑆𝑦 and 𝑥 should be identical. Consequently, 𝑑(𝑥,𝑦)=𝑑(𝐴,𝐵). To prove the uniqueness, let us suppose that there exists another element 𝑥^∗ such that ‖𝑥^∗−𝑇𝑥^∗‖=𝑑(𝐴,𝐵). (3.10) Then, ‖𝑇𝑥^∗−𝑆𝑇𝑥^∗‖≤𝑘[‖𝑥^∗−𝑇𝑥^∗‖+‖𝑇𝑥^∗−𝑆𝑇𝑥^∗‖]+(1−2𝑘)𝑑(𝐴,𝐵). Consequently, ‖𝑇𝑥^∗−𝑆𝑇𝑥^∗‖=𝑑(𝐴,𝐵). By strict convexity of the space, 𝑆𝑇𝑥^∗=𝑥^∗. Moreover, ‖𝑇𝑥−𝑥^∗‖=‖𝑇𝑥−𝑆𝑇𝑥^∗‖≤𝑘‖𝑥−𝑇𝑥‖+‖𝑇𝑥^∗−𝑆𝑇𝑥^∗‖+()𝑑()().1−2𝑘𝐴,𝐵=𝑑𝐴,𝐵 (3.11) Therefore, ‖𝑇𝑥−𝑥^∗‖=𝑑(𝐴,𝐵). By strict convexity of the space, 𝑥 and 𝑥^∗ are identical. This completes the proof of the theorem.

The following example illustrates Lemma 3.3. Further, it shows that uniqueness of best proximity point is not feasible.

Example 3.7. Consider the nonuniformly convex Banach space 𝑅² with the norm ‖(𝑥,𝑦)‖=max{|𝑥|,|𝑦|}.
Let )},)}.𝐴∶={(𝑥,0∶0≤𝑥≤1𝐵∶={(𝑥,1∶0≤𝑥≤1 (3.12) Then, 𝑑(𝐴,𝐵)=1 and 𝑑(𝑢,𝑣)=1 for all 𝑢 in 𝐴 and 𝑣 in 𝐵. Let 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 be defined as 𝑇=(),𝑆=().((𝑥,0))𝑥,1((𝑥,1))𝑥,0 (3.13) For any positive number 𝑘, 𝑇𝑥₁,0−𝑆𝑥₂‖𝑥,1=1=𝑘₁,0−𝑇𝑥₁,0‖+‖𝑥₂,1−𝑆𝑥₂,1‖+(1−2𝑘)𝑑(𝐴,𝐵). (3.14) So, the mappings 𝑆 and 𝑇 form a K-cyclic mapping. Further, it can be observed that every element of 𝐴 is a best proximity point of the mapping 𝑇.

4. C-Cyclic Mappings

This section is concerned with best proximity point theorems for C-cyclic non-self mappings.

Lemma 4.1. Let 𝐴 and 𝐵 be two non-empty subsets of a metric space. Suppose that the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a C-cyclic mapping between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Then, 𝑑(𝑥_𝑛,𝑥_𝑛+1)→𝑑(𝐴,𝐵).

Proof. Since 𝑇 and 𝑆 form a C-cyclic mapping, 𝑑𝑥₁,𝑥₂=𝑑𝑇𝑥₀,𝑆𝑥₁≤𝑘𝑑𝑥₁,𝑇𝑥₀+𝑑𝑥₀,𝑆𝑥₁+()𝑑()𝑥1−2𝑘𝐴,𝐵=𝑘𝑑₀,𝑥₂+()𝑑()𝑑𝑥1−2𝑘𝐴,𝐵≤𝑘₀,𝑥₁+𝑑𝑥₁,𝑥₂+(1−2𝑘)𝑑(𝐴,𝐵). (4.1) So, it follows that 𝑑(𝑥₁,𝑥₂)≤(𝑘/(1−𝑘))𝑑(𝑥₀,𝑥₁)+[1−(𝑘/(1−𝑘))]𝑑(𝐴,𝐵).
Similarly, 𝑑(𝑥₂,𝑥₃)≤(𝑘/(1−𝑘))²𝑑(𝑥₀,𝑥₁)+[1−(𝑘/(1−𝑘))²]𝑑(𝐴,𝐵).
It can be shown by induction that 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑘1−𝑘^𝑛𝑑𝑥₀,𝑥₁+𝑘1−1−𝑘^𝑛𝑑(𝐴,𝐵). (4.2) Therefore, 𝑑(𝑥_𝑛,𝑥_𝑛+1)→𝑑(𝐴,𝐵) because of the fact that 𝑘<1/2.

Lemma 4.2. Let 𝐴 and 𝐵 be non-empty closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a C-cyclic map between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Suppose that the sequence {𝑥_2𝑛} has a subsequence converging to some element 𝑥 in 𝐴. Then, 𝑥 is a best proximity point of 𝑇.

Proof. Suppose that a subsequence {𝑥_2𝑛𝑘} converges to 𝑥 in 𝐴. Then, it follows from Lemma 4.1 that 𝑑(𝑥_{2𝑛𝑘−1},𝑥_2𝑛𝑘)→𝑑(𝐴,𝐵). Further, we have 𝑑𝑥_2𝑛𝑘,𝑇𝑥=𝑑𝑆𝑥_{2𝑛𝑘−1}𝑑𝑥,𝑇𝑥≤𝑘_{2𝑛𝑘−1},𝑇𝑥+𝑑𝑥,𝑆𝑥_{2𝑛𝑘−1}+()𝑑()𝑑𝑥1−2𝑘𝐴,𝐵≤𝑘_{2𝑛𝑘−1},𝑥_2𝑛𝑘+𝑑𝑥_2𝑛𝑘,𝑇𝑥+𝑑𝑥,𝑥_2𝑛𝑘+(1−2𝑘)𝑑(𝐴,𝐵). (4.3) So, it follows that 𝑑(𝐴,𝐵)≤𝑑𝑥_2𝑛𝑘≤𝑘,𝑇𝑥𝑑𝑥1−𝑘^{2𝑛𝑘−1},𝑥_2𝑛𝑘+𝑑𝑥,𝑥_2𝑛𝑘+𝑘1−𝑑()1−𝑘𝐴,𝐵. (4.4) Letting 𝑘→∞, 𝑑(𝑥,𝑇𝑥)=𝑑(𝐴,𝐵). This completes the proof of the Lemma.

Lemma 4.3. Let 𝐴 and 𝐵 be non-empty closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a C-cyclic map between 𝐴 and 𝐵. For a fixed element 𝑥₀ in 𝐴, let 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1. Then, the sequence {𝑥_𝑛} is bounded.

Proof. By Lemma 4.1, 𝑑(𝑥_2𝑛−1,𝑥_2𝑛) is convergent and hence it is bounded. Further, we have 𝑑𝑥_2𝑛,𝑇𝑥₀=𝑑𝑆𝑥_2𝑛−1,𝑇𝑥₀≤𝑘𝑑𝑥_2𝑛−1,𝑇𝑥₀+𝑑𝑥₀,𝑆𝑥_2𝑛−1+()𝑑()𝑥1−2𝑘𝐴,𝐵≤𝑘[𝑑_2𝑛−1,𝑥_2𝑛+2𝑑𝑥_2𝑛,𝑇𝑥₀+𝑑𝑥₀,𝑇𝑥₀+(1−2𝑘)𝑑(𝐴,𝐵). (4.5) Therefore, 𝑑(𝑥_2𝑛,𝑇𝑥₀)≤(𝑘/(1−2𝑘))[𝑑(𝑥_2𝑛−1,𝑥_2𝑛)+𝑑(𝑥₀,𝑇𝑥₀)]+𝑑(𝐴,𝐵).
Therefore, the subsequence {𝑥_2𝑛} is bounded. Similarly, it can be shown that {𝑥_2𝑛+1} is also bounded.

The preceding two lemmas give rise to the following best proximity point theorem for C-cyclic mappings in the setting of metric spaces.

Corollary 4.4. Let 𝐴 and 𝐵 be two non-empty and closed subsets of a metric space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a C-cyclic map between 𝐴 and 𝐵. If 𝐴 is boundedly compact, then 𝑇 has a best proximity point.

The following best proximity point theorem is for C-cyclic mappings in the setting of uniformly convex Banach spaces.

Theorem 4.5. Let 𝐴 and 𝐵 be non-empty, closed, and convex subsets of a uniformly convex Banach space. Let the mappings 𝑇∶𝐴→𝐵 and 𝑆∶𝐵→𝐴 form a C-Cyclic map between 𝐴 and 𝐵. If 𝑥₀ is any fixed element in 𝐴, 𝑥_2𝑛+1=𝑇𝑥_2𝑛 and 𝑥_2𝑛=𝑆𝑥_2𝑛−1, then the sequence {𝑥_2𝑛} converges to a best proximity 𝑥 of 𝑇 and the sequence {𝑥_2𝑛+1} converges to a best proximity point 𝑦 of 𝑆 such that 𝑑(𝑥,𝑦)=𝑑(𝐴,𝐵).

Proof. It follows from Lemma 4.1 that ‖𝑥_2𝑚−1−𝑥_2𝑚‖⟶𝑑(𝐴,𝐵),‖𝑥_2𝑛−𝑥_2𝑛+1‖⟶𝑑(𝐴,𝐵). (4.6) Therefore, for every 𝜖>0, ‖𝑥_2𝑚−1−𝑥_2𝑚()+𝜖()‖<𝑑𝐴,𝐵1−2𝑘,2𝑘‖𝑥_2𝑛−𝑥_2𝑛+1()+𝜖()‖<𝑑𝐴,𝐵1−2𝑘,2𝑘 (4.7) for sufficiently large values of 𝑚 and 𝑛. As 𝑇 and 𝑆 form a K-cyclic mapping, ‖𝑥_2𝑚−𝑥_2𝑛+1‖=‖𝑆𝑥_2𝑚−1−𝑇𝑥_2𝑛‖≤𝑘‖𝑥_2𝑚−1−𝑇𝑥_2𝑛‖+‖𝑥_2𝑛−𝑆𝑥_2𝑚−1‖+(1−2𝑘)𝑑(𝐴,𝐵)≤𝑘‖𝑥_2𝑚−1−𝑥_2𝑚‖+‖𝑥_2𝑚−𝑥_2𝑛+1‖+‖𝑥_2𝑛−𝑥_2𝑛+1‖+‖𝑥_2𝑛+1−𝑥_2𝑚‖+(1−2𝑘)𝑑(𝐴,𝐵). (4.8) Thus, it follows that ‖𝑥_2𝑚−𝑥_2𝑛+1𝑘‖≤1−2𝑘‖𝑥_2𝑚−1−𝑥_2𝑚‖+‖𝑥_2𝑛−𝑥_2𝑛+1‖+𝑑(𝐴,𝐵). (4.9) Therefore, it can be concluded that ‖𝑥_2𝑚−𝑥_2𝑛+1‖<𝑑(𝐴,𝐵)+𝜖, (4.10) for sufficiently large values of 𝑚 and 𝑛. Thus, {𝑥_2𝑛} is a Cauchy sequence by Lemma 3.5. Since the space is complete, {𝑥_2𝑛} converges to some element 𝑥∈𝐴, which becomes a best proximity point of the mapping 𝑇 by Lemma 4.2. Similarly, {𝑥_2𝑛+1} converges to some element 𝑦∈𝐵, which is a best proximity point of the mapping 𝑆. Further, 𝑑(𝑥_2𝑛,𝑥_2𝑛+1)→𝑑(𝑥,𝑦). However, by Lemma 4.1, 𝑑(𝑥_2𝑛,𝑥_2𝑛+1)→𝑑(𝐴,𝐵). Consequently, 𝑑(𝑥,𝑦)=𝑑(𝐴,𝐵). This completes the proof of the theorem.

Acknowledgment

The research of the second author (N. Shahzad) was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Project no. 3-021/430.