#### 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 where is a non-self mapping.

#### 1. Introduction

Fixed point theorems delve into the existence of a solution to the equations of the form where is a self-mapping. However, when is a nonself-mapping, the equation 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 is a nonempty convex compact subset of a Hausdorff topological vector space and is a continuous non-self mapping from to , then there exists an element in such that . 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 , they may not render an approximate solution that is optimal. On the contrary, best proximity point theorems are intended to furnish an approximate solution that is optimal in the sense that the error is minimum. Indeed, in light of the fact that is at least , a best proximity point theorem guarantees the global minimization of by the requirement that an approximate solution satisfies the condition . Such optimal approximate solutions are called best proximity points of the mapping .

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 where 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 and is said to form a *K-Cyclic* mapping between and if there exists a nonnegative real number such that
for all and .

*Definition 2.2. *A pair of mappings and is said to form a *C-Cyclic* mapping between and if there exists a non-negative real number such that
for all and .

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

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 and be two non-empty subsets of a metric space. Suppose that the mappings and form a K-Cyclic map between and . For a fixed element in , let and . Then, .*

*Proof. *As and form a K-Cyclic map,
So, it follows that .

Similarly, it can be seen that
Hence, it follows by induction that
Therefore, because of the fact that .

Lemma 3.2. *Let and be non-empty closed subsets of a metric space. Let the mappings and form a K-Cyclic map between and . For a fixed element in , let and . Then, the sequence is bounded.*

*Proof. *It follows from Lemma 3.1 that is convergent and hence it is bounded. Further, since and form a K-cyclic mapping, it follows that
Therefore, the subsequence is bounded. Similarly, it can be shown that is also bounded.

Lemma 3.3. *Let and be non-empty closed subsets of a metric space. Let the mappings and form a K-Cyclic map between and . For a fixed element in , let and . Suppose that the sequence has a subsequence converging to some element in . Then, is a best proximity point of . *

*Proof. *Suppose that a subsequence converges to in . It follows from Lemma 3.1 that converges to . As and form a K-cyclic mapping, it follows that
Therefore, .

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 and be two non-empty and closed subsets of a metric space. Let the mappings and form a K-Cyclic map between and . If is boundedly compact, then 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 be a non-empty, closed, and convex subset and be a non-empty and closed subset of a uniformly convex Banach space. Suppose that and are sequences in and is a sequence in satisfying the following conditions:*(a)*,
* (b)* for every , , ** for sufficiently large values of and .**Then, for every , for sufficiently large values of and .*

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

Theorem 3.6. *Let and be non-empty, closed, and convex subsets of a uniformly convex Banach space. If the mappings and form a K-Cyclic map between and , then there exist a unique element and a unique element such that
**
Further, if is any fixed element in , and , then the sequences and converge to the best proximity points and , respectively.*

*Proof. *It follows from Lemma 3.1 that
Therefore, for every ,
for sufficiently large values of and . As and form a K-cyclic mapping,
for sufficiently large values of and . Thus, is a Cauchy sequence by Lemma 3.5. Since the space is complete, converges to some element , which becomes a best proximity point of the mapping by Lemma 3.3. Similarly, converges to some element , which is a best proximity point of the mapping . Further, . Therefore, . By strict convexity of the space, and should be identical, and and should be identical. Consequently, . To prove the uniqueness, let us suppose that there exists another element such that
Then, . Consequently, . By strict convexity of the space, . Moreover,
Therefore, . By strict convexity of the space, and 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 with the norm .

Let
Then, and for all in and in . Let and be defined as
For any positive number ,
So, the mappings and form a K-cyclic mapping. Further, it can be observed that every element of is a best proximity point of the mapping .

#### 4. C-Cyclic Mappings

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

Lemma 4.1. *Let and be two non-empty subsets of a metric space. Suppose that the mappings and form a C-cyclic mapping between and . For a fixed element in , let and . Then, .*

*Proof. *Since and form a C-cyclic mapping,
So, it follows that .

Similarly, .

It can be shown by induction that
Therefore, because of the fact that .

Lemma 4.2. *Let and be non-empty closed subsets of a metric space. Let the mappings and form a C-cyclic map between and . For a fixed element in , let and . Suppose that the sequence has a subsequence converging to some element in . Then, is a best proximity point of . *

*Proof. *Suppose that a subsequence converges to in . Then, it follows from Lemma 4.1 that . Further, we have
So, it follows that
Letting , . This completes the proof of the Lemma.

Lemma 4.3. *Let and be non-empty closed subsets of a metric space. Let the mappings and form a C-cyclic map between and . For a fixed element in , let and . Then, the sequence is bounded.*

*Proof. *By Lemma 4.1, is convergent and hence it is bounded. Further, we have
Therefore, .

Therefore, the subsequence is bounded. Similarly, it can be shown that 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 and be two non-empty and closed subsets of a metric space. Let the mappings and form a C-cyclic map between and . If is boundedly compact, then 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 and be non-empty, closed, and convex subsets of a uniformly convex Banach space. Let the mappings and form a C-Cyclic map between and . If is any fixed element in , and , then the sequence converges to a best proximity of and the sequence converges to a best proximity point of such that .*

*Proof. *It follows from Lemma 4.1 that
Therefore, for every ,
for sufficiently large values of and . As and form a K-cyclic mapping,
Thus, it follows that
Therefore, it can be concluded that
for sufficiently large values of and . Thus, is a Cauchy sequence by Lemma 3.5. Since the space is complete, converges to some element , which becomes a best proximity point of the mapping by Lemma 4.2. Similarly, converges to some element , which is a best proximity point of the mapping . Further, . However, by Lemma 4.1, . Consequently, . 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.