Best Proximity Point Results for MK-Proximal Contractions

Abstract

Let 𝐴 and 𝐵 be nonempty subsets of a metric space with the distance function d, and 𝑇∶𝐴→𝐵 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 𝑥↦𝑑(𝑥,𝑇𝑥), where 𝑇 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 𝐴 and 𝐵 be nonempty subsets of a metric space (𝑋,𝑑). Because the functional equation 𝑇𝑥=𝑥 (𝑥∈𝐴), where 𝑇∶𝐴→𝐵 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 𝑇𝑥=𝑥 when the equation has no solution. In view of the fact that 𝑑(𝐴,𝐵) is a lower bound for 𝑑(𝑥,𝑇𝑥), where 𝑑(𝐴,𝐵)=inf{𝑑(𝑎,𝑏)∶𝑎∈𝐴,𝑏∈𝐵}, (1.1) an approximate solution 𝑥^∗∈𝐴 to the equation 𝑇𝑥=𝑥 produces the least possible error if 𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝐴,𝐵). Such a solution is called a best proximity point of the mapping 𝑇∶𝐴→𝐵. Due to the fact that 𝑑(𝑥,𝑇𝑥)≥𝑑(𝐴,𝐵) for all 𝑥∈𝐴, a best proximity point provides the global minimum of the nonlinear programming problem min_𝑥∈𝐴𝑑(𝑥,𝑇𝑥). 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 (𝑋,𝑑) be a metric space; 𝐴 and 𝐵 are two nonempty subsets of 𝑋. We consider the following notations: 𝑑(){𝑑()};(){𝑑()};𝐴𝐴,𝐵∶=inf𝑎,𝑏∶𝑎∈𝐴,𝑏∈𝐵𝑎∈𝐴,𝑑𝑎,𝐵∶=inf𝑎,𝑏∶𝑏∈𝐵₀{()()};𝐵∶=𝑎∈𝐴∶𝑑𝑎,𝑏=𝑑𝐴,𝐵forsome𝑏∈𝐵₀∶={𝑏∈𝐵∶𝑑(𝑎,𝑏)=𝑑(𝐴,𝐵)forsome𝑎∈𝐴}. (2.1)

Definition 2.1 (see [3]). 𝐵is said to be approximatively compact with respect to 𝐴 if every sequence {𝑦_𝑛} of 𝐵 satisfying the condition that 𝑑(𝑥,𝑦_𝑛)→𝑑(𝑥,𝐵) for some 𝑥 in 𝐴 has a convergent subsequence.

Definition 2.2. An element 𝑥^∗∈𝐴 is said to be a best proximity point of the non-self-mappping 𝑇∶𝐴→𝐵 if it satisfies the condition that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (2.2)
Because of the fact that 𝑑(𝑥,𝑇𝑥)≥𝑑(𝐴,𝐵) for all 𝑥∈𝐴, the global minimum of the mapping 𝑥↦𝑑(𝑥,𝑇𝑥) 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 𝑔∶𝐴→𝐴 is an isometry if for any 𝑥,𝑦∈𝐴, one has 𝑑𝑔𝑥,𝑔𝑦=𝑑𝑥,𝑦. (2.3)

Definition 2.4 (see [6]). Given a mapping 𝑇∶𝐴→𝐵 and an isometry 𝑔∶𝐴→𝐴, the mapping 𝑇 is said to preserve the isometric distance with respect to 𝑔 if for any 𝑥,𝑦∈𝐴, one has 𝑑𝑇𝑔𝑥,𝑇𝑔𝑦=𝑑𝑇𝑥,𝑇𝑦. (2.4)

Definition 2.5 (see [27]). Let (𝐴,𝐵) be a pair of nonempty subsets of a metric space (𝑋,𝑑) with 𝐴₀≠∅. Then the pair (𝐴,𝐵) is said to have the 𝑃-property if and only if 𝑑𝑥₁,𝑦₁()𝑑𝑥=𝑑𝐴,𝐵₂,𝑦₂()𝑥=𝑑𝐴,𝐵⇒𝑑₁,𝑥₂=𝑑𝑦₁,𝑦₂, (2.5) where 𝑥₁,𝑥₂∈𝐴 and 𝑦₁,𝑦₂∈𝐵.
We introduce the concept of the weakly 𝑃-property as follows.

Definition 2.6. Let (𝐴,𝐵) be a pair of nonempty subsets of a metric space (𝑋,𝑑) with 𝐴₀≠∅. Then the pair (𝐴,𝐵) is said to have the weakly 𝑃-property if and only if 𝑑𝑥₁()𝑑𝑥,y=𝑑𝐴,𝐵₂,y=𝑑(𝐴,𝐵)⇒𝑥₁=𝑥₂, (2.6) where 𝑥₁,𝑥₂∈𝐴 and 𝑦∈𝐵.
Note that if the pair (𝐴,𝐵) has the 𝑃-property, then it has the weakly 𝑃-property. The following example shows that the converse is not true in general.

Example 2.7. Consider the Euclidean space ℝ² endowed with the Euclidean metric 𝑑. Let us define the sets 𝐴∶={(𝑎,0)∶𝑎≥0},𝐵∶={(𝑏,1)∶𝑏≥0}∪{(𝑏,−1)∶𝑏≥0}. (2.7) Clearly, we have 𝑑(𝐴,𝐵)=1. On the other hand, we have 𝑑((0,0),(0,1))=𝑑((2,0),(2,−1))=𝑑(𝐴,𝐵) but 𝑑((0,0),(2,0))=2≠4=𝑑((0,1),(2,−1)). Thus(𝐴,𝐵)does not satisfy the 𝑃-property. Now, suppose that 𝑑((𝑎,0),(𝑏,1))=1=𝑑((𝑎^′,0),(𝑏,1)) for some 𝑎,𝑏,𝑎^′≥0. This implies immediately that 𝑎=𝑎^′, that is, (𝑎,0)=(𝑎^′,0). Similarly, if 𝑑((𝑎,0),(𝑏,−1))=1=𝑑((𝑎^′,0),(𝑏,−1)) for some 𝑎,𝑏,𝑎^′≥0, we get that (𝑎,0)=(𝑎^′,0). This implies that (𝐴,𝐵) has the weakly 𝑃-property.

Definition 2.8. A self-mapping 𝑇∶𝐴→𝐴 is said to be an MK-contraction of the first kind if, for all 𝜀>0, there exists 𝛿(𝜀)>0 such that 𝑥,𝑦∈𝐴,𝜀≤𝑑𝑥,𝑦<𝜀+𝛿(𝜀)⇒𝑑𝑇𝑥,𝑇𝑦<𝜀. (2.8)
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 𝑇∶𝐴→𝐴 is said to be an MK-contraction of the second kind if, for all 𝜀>0, there exists 𝛾(𝜀)>0 such that 𝑥,𝑦∈𝐴,𝜀≤𝑑𝑇𝑥,𝑇𝑦<𝜀+𝛾(𝜀)⇒𝑑𝑇²𝑥,𝑇²𝑦<𝜀. (2.9)
Clearly every MK-contraction of the first kind is an MK-contraction of the second kind.

Definition 2.10. A non-self-mappping 𝑇∶𝐴→𝐵 is said to be an MK-proximal contraction of the first kind if, for all 𝜀>0, there exists 𝛿(𝜀)>0 such that ()(),(𝜀)()𝑥,𝑦,𝑢,𝑣∈𝐴,𝑑𝑢,𝑇𝑥=𝑑𝑣,𝑇𝑦=𝑑𝐴,𝐵𝜀≤𝑑𝑥,𝑦<𝜀+𝛿⇒𝑑𝑢,𝑣<𝜀. (2.10)
Clearly, if 𝐵=𝐴, an MK-proximal contraction of the first kind is an MK-contraction of the first kind.

Lemma 2.11. Let 𝑇∶𝐴→𝐵 be an MK-proximal contraction of the first kind. Suppose that the pair (𝐴,𝐵) has the weakly 𝑃-property. Then, 𝑥,𝑦,𝑢,𝑣∈𝐴,𝑑(𝑢,𝑇𝑥)=𝑑𝑣,𝑇𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑢,𝑣)≤𝑑𝑥,𝑦. (2.11)

Proof. Let 𝑥,𝑦,𝑢,𝑣∈𝐴 such that 𝑑(𝑢,𝑇𝑥)=𝑑(𝑣,𝑇𝑦)=𝑑(𝐴,𝐵) and 𝑥≠𝑦. Let 𝜀=𝑑(𝑥,𝑦)>0. Since 𝜀≤𝑑(𝑥,𝑦)<𝜀+𝛿(𝜀), we have 𝑑(𝑢,𝑣)<𝜀=𝑑(𝑥,𝑦). If 𝑥=𝑦, since (𝐴,𝐵) has the weakly 𝑃-property, we get that 𝑢=𝑣, that is, 𝑑(𝑇𝑥,𝑇𝑦)=𝑑(𝑢,𝑣).

Definition 2.12. A non-self-mappping 𝑇∶𝐴→𝐵 is said to be an MK-proximal contraction of the second kind if, for all 𝜀>0, there exists 𝛾(𝜀)>0 such that ()(),(𝜀)()𝑥,𝑦,𝑢,𝑣∈𝐴,𝑑𝑢,𝑇𝑥=𝑑𝑣,𝑇𝑦=𝑑𝐴,𝐵𝜀≤𝑑𝑇𝑥,𝑇𝑦<𝜀+𝛾⇒𝑑𝑇𝑢,𝑇𝑣<𝜀. (2.12)
If 𝐵=𝐴, 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 𝑇∶𝐴→𝐵 be an MK-proximal contraction of the second kind. Suppose that the pair (𝐴,𝐵) has the weakly 𝑃-property. Then, 𝑥,𝑦,𝑢,𝑣∈𝐴,𝑑(𝑢,𝑇𝑥)=𝑑𝑣,𝑇𝑦=𝑑(𝐴,𝐵)⇒𝑑(𝑇𝑢,𝑇𝑣)≤𝑑𝑇𝑥,𝑇𝑦. (2.13)

3. Main Results

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

Theorem 3.1. Let 𝐴 and 𝐵 be closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonvoid and the pair (𝐴,𝐵) satisfies the weakly 𝑃-property. Suppose that the mappings 𝑔∶𝐴→𝐴 and 𝑇∶𝐴→𝐵 satisfy the following conditions: (a)𝑇is an MK-proximal contraction of the first and second kinds; (b)𝑇(𝐴₀)⊆𝐵₀; (c)𝑔 is an isometry; (d)𝐴₀⊆𝑔(𝐴₀); (e)𝑇 preserves the isometric distance with respect to 𝑔.
Then, there exists a unique element 𝑥^∗∈𝐴 such that 𝑑𝑔𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (3.1) Further, for any fixed element 𝑥₀∈𝐴₀, the iterative sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵), (3.2) converges to 𝑥^∗.

Proof. Let 𝑥₀∈𝐴₀ (such a point exists since 𝐴₀≠∅). From conditions (b) and (d), there exists 𝑥₁∈𝐴₀ such that 𝑑𝑔𝑥₁,𝑇𝑥₀=𝑑(𝐴,𝐵). (3.3) Again, from conditions (b) and (d), there exists 𝑥₂∈𝐴₀ such that 𝑑𝑔𝑥₂,𝑇𝑥₁=𝑑(𝐴,𝐵). (3.4) Continuing this process, we can construct a sequence {𝑥_𝑛}⊂𝐴₀ such that 𝑑𝑔𝑥_𝑛,𝑇𝑥_𝑛−1=𝑑(𝐴,𝐵),∀𝑛∈ℕ. (3.5) Since 𝑇 is an MK-proximal contraction of the first kind and 𝑔 is an isometry, it follows from (3.5) and Lemma 2.11 that 𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2≤𝑑𝑥_𝑛,𝑥_𝑛+1,∀𝑛∈ℕ∪{0}. (3.6) Further, since 𝑇 is an MK-proximal contraction of the second kind and preserves isometric distance with respect to 𝑔, it follows from Lemma 2.13 that 𝑑𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+2=𝑑𝑇𝑔𝑥_𝑛+1,𝑇𝑔𝑥_𝑛+2≤𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,∀𝑛∈ℕ∪{0}. (3.7)
Claim  1. We claim that {𝑥_𝑛} is a Cauchy sequence.
Let 𝑡_𝑛=𝑑(𝑥_𝑛,𝑥_𝑛+1). From (3.6), {𝑡_𝑛} is a nonnegative, bounded below and decreasing sequence of real numbers and hence converges to some nonnegative real number 𝑡 (𝑡_𝑛→𝑡⁺). Let us suppose that 𝑡>0. This implies that there exists 𝑝∈ℕ such that 𝑡≤𝑑(𝑥_𝑝−1,𝑥_𝑝)<𝑡+𝛿(𝑡). Since 𝑇 is an MK-proximal contraction of the first kind and 𝑔 is an isometry, this implies that 𝑑(𝑥_𝑝,𝑥_𝑝+1)=𝑑(𝑔𝑥_𝑝,𝑔𝑥_𝑝+1)<𝑡, which is a contradiction. Thus we have 𝑡=0, that is, lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (3.8) Fix 𝜀>0. Without restriction of the generality, we can suppose that 𝛿(𝜀)≤𝜀. Then, there exists 𝑁∈ℕ such that 𝑑𝑥_𝑁,𝑥_𝑁+1<𝛿(𝜀). (3.9) Let us denote by 𝐴(𝑥_𝑁,𝜀) the subset of 𝐴 defined by 𝐴𝑥_𝑁,𝜀∶=𝑥∈𝐴∶𝑑𝑥_𝑁,𝑥<𝜀+𝛿(𝜀). (3.10) We will prove that 𝑥∈𝐴𝑥_𝑁,𝜀,𝑑𝑔𝑢,𝑇𝑥=𝑑(𝐴,𝐵),𝑢∈𝐴⇒𝑢∈𝐴𝑥_𝑁,𝜀. (3.11) Let 𝑥∈𝐴(𝑥_𝑁,𝜀) such that 𝑑(𝑔𝑢,𝑇𝑥)=𝑑(𝐴,𝐵) for some 𝑢∈𝐴. We distinguish two cases.
Case  1. If 𝑑(𝑥_𝑁,𝑥)≤𝜀, we have 𝑑𝑢,𝑥_𝑁≤𝑑𝑢,𝑥_𝑁+1+𝑑𝑥_𝑁+1,𝑥_𝑁,since𝑔isanisometry=𝑑𝑔𝑢,𝑔𝑥_𝑁+1+𝑑𝑥_𝑁+1,𝑥_𝑁,(from(3.9))<𝑑𝑔𝑢,𝑔𝑥_𝑁+1(𝜀),()+𝛿fromLemma2.11≤𝑑𝑥,𝑥_𝑁(𝜀)(𝜀).+𝛿≤𝜀+𝛿 (3.12)
Case  2. If 𝜀<𝑑(𝑥_𝑁,𝑥)<𝜀+𝛿(𝜀), we have 𝑑𝑢,𝑥_𝑁≤𝑑𝑢,𝑥_𝑁+1+𝑑𝑥_𝑁+1,𝑥_𝑁,since𝑔isanisometry=𝑑𝑔𝑢,𝑔𝑥_𝑁+1+𝑑𝑥_𝑁+1,𝑥_𝑁,(from(3.9))<𝑑𝑔𝑢,𝑔𝑥_𝑁+1(𝜀),(𝜀).+𝛿since𝑇isanMK-proximalcontractionofthefirstkind<𝜀+𝛿 (3.13) Thus, in all cases, we have 𝑢∈𝐴(𝑥_𝑁,𝜀), and (3.11) is proved.
Now, we shall prove that 𝑥_𝑛∈𝐴𝑥_𝑁,𝜀,∀𝑛≥𝑁. (3.14) Clearly, for 𝑛=𝑁, (3.14) is satisfied. Moreover, from (3.9), (3.14) is satisfied for 𝑛=𝑁+1. Now, from (3.5), we have 𝑑(𝑔𝑥_𝑁+2,𝑇𝑥_𝑁+1)=𝑑(𝐴,𝐵). Since we have also 𝑥_𝑁+1∈𝐴(𝑥_𝑁,𝜀), from (3.11), we get that 𝑥_𝑁+2∈𝐴(𝑥_𝑁,𝜀). Continuing this process, by induction, we get (3.14).
Finally, for all 𝑛,𝑚≥𝑁, from (3.14), we have 𝑑𝑥_𝑛,𝑥_𝑚≤𝑑𝑥_𝑛,𝑥_𝑁+𝑑𝑥_𝑚,𝑥_𝑁<2(𝜀+𝛿(𝜀))≤4𝜀, (3.15) which implies that {𝑥_𝑛} is a Cauchy sequence. Thus, Claim 1 is proved.
Claim  2. We claim that {𝑇𝑥_𝑛} is a Cauchy sequence.
Let 𝑠_𝑛=𝑑(𝑇𝑥_𝑛,𝑇𝑥_𝑛+1). From (3.7), {𝑠_𝑛} is a nonnegative, bounded below and decreasing sequence of real numbers and hence converges to some nonnegative real number 𝑠. Let us suppose that 𝑠>0. This implies that there exists 𝑝∈ℕ such that 𝑠≤𝑑(𝑇𝑥_𝑝−1,𝑇𝑥_𝑝)<𝑠+𝛾(𝑠). Since 𝑇 is an MK-proximal contraction of the second kind and 𝑇 preserves the isometric distance with resect to 𝑔, we get 𝑑𝑇𝑥_𝑝,𝑇𝑥_𝑝+1=𝑑𝑇𝑔𝑥_𝑝,𝑇𝑔𝑥_𝑝+1<𝑠, (3.16) which is a contradiction. Thus we have 𝑠=0, that is, lim_𝑛→∞𝑠_𝑛=lim_𝑛→∞𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛+1=0. (3.17) Fix 𝜀>0. Without restriction of the generality, we can suppose that 𝛾(𝜀)≤𝜀. Then, there exists 𝑁∈ℕ such that 𝑑𝑇𝑥_𝑁,𝑇𝑥_𝑁+1<𝛾(𝜀). (3.18) Let us denote by 𝐴(𝑥_𝑁,𝜀) the subset of 𝐴 defined by 𝐴𝑥_𝑁,𝜀∶=𝑥∈𝐴∶𝑑𝑇𝑥_𝑁,𝑇𝑥<𝜀+𝛾(𝜀). (3.19) We shall prove that 𝑥∈𝐴𝑥_𝑁,𝜀,𝑑𝑔𝑢,𝑇𝑥=𝑑(𝐴,𝐵),𝑢∈𝐴⇒𝑢∈𝐴𝑥_𝑁,𝜀. (3.20) Let 𝑥∈𝐴(𝑥_𝑁,𝜀) such that 𝑑(𝑔𝑢,𝑇𝑥)=𝑑(𝐴,𝐵) for some 𝑢∈𝐴. We distinguish two cases.
Case  1. If 𝑑(𝑇𝑥_𝑁,𝑇𝑥)≤𝜀, we have 𝑑𝑇𝑢,𝑇𝑥_𝑁≤𝑑𝑇𝑢,𝑇𝑥_𝑁+1+𝑑𝑇𝑥_𝑁+1,𝑇𝑥_𝑁,(from(e))=𝑑𝑇𝑔𝑢,𝑇𝑔𝑥_𝑁+1+𝑑𝑇𝑥_𝑁+1,𝑇𝑥_𝑁,(from(3.18))<𝑑𝑇𝑔𝑢,𝑇𝑔𝑥_𝑁+1(𝜀),()+𝛾fromLemma2.13≤𝑑𝑇𝑥,𝑇𝑥_𝑁(𝜀)(𝜀).+𝛾≤𝜀+𝛾 (3.21)
Case  2. If 𝜀<𝑑(𝑇𝑥_𝑁,𝑇𝑥)<𝜀+𝛾(𝜀), we have 𝑑𝑇𝑢,𝑇𝑥_𝑁≤𝑑𝑇𝑢,𝑇𝑥_𝑁+1+𝑑𝑇𝑥_𝑁+1,𝑇𝑥_𝑁,(from(e))=𝑑𝑇𝑔𝑢,𝑇𝑔𝑥_𝑁+1+𝑑𝑥_𝑁+1,𝑥_𝑁,(from(3.18))<𝑑𝑇𝑔𝑢,𝑇𝑔𝑥_𝑁+1(𝜀),(𝜀).+𝛾since𝑇isanMK-proximalcontractionofthesecondkind<𝜀+𝛾 (3.22) Thus, in all cases, we have 𝑢∈𝐴(𝑥_𝑁,𝜀), and (3.20) is proved.
Now, we shall prove that 𝑥_𝑛∈𝐴𝑥_𝑁,𝜀,∀𝑛≥𝑁. (3.23) Clearly, for 𝑛=𝑁, (3.23) is satisfied. Moreover, from (3.18), (3.23) is satisfied for 𝑛=𝑁+1. Since 𝑑(𝑔𝑥_𝑁+2,𝑇𝑥_𝑁+1)=𝑑(𝐴,𝐵) and 𝑥_𝑁+1∈𝐴(𝑥_𝑁,𝜀), from (3.20), we have 𝑥_𝑁+2∈𝐴(𝑥_𝑁,𝜀). Continuing this process, by induction, we get (3.23).
Finally, for all 𝑛,𝑚≥𝑁, from (3.23), we have 𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑚≤𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑁+𝑑𝑇𝑥_𝑚,𝑇𝑥_𝑁<2𝛾(𝜀)+𝜀≤4𝜀, (3.24) which implies that {𝑇𝑥_𝑛} is a Cauchy sequence. Thus, Claim 2 is proved.
Now, since (𝑋,𝑑) is complete and 𝐴 is closed, there exists 𝑥^∗∈𝐴 such that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Similarly, since 𝐵 is closed, there exists 𝑦^∗∈𝐵 such that 𝑇𝑥_𝑛→𝑦^∗ as 𝑛→∞. Therefore, we have 𝑑𝑔𝑥^∗,𝑦^∗=lim_𝑛→∞𝑑𝑔𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵). (3.25) This implies that 𝑔𝑥^∗∈𝐴₀. From condition (d), there exists 𝑥_∗∈𝐴₀ such that 𝑔𝑥^∗=𝑔𝑥_∗. On the other hand, since 𝑔 is an isometry, we get that 𝑑(𝑥^∗,𝑥_∗)=𝑑(𝑔𝑥^∗,𝑔𝑥_∗)=0, which implies that 𝑥^∗=𝑥_∗∈𝐴₀. Now, from condition (b), we have 𝑇𝑥^∗∈𝐵₀; that is, there exists 𝑥∈𝐴₀ such that 𝑑(𝑥,𝑇𝑥^∗)=𝑑(𝐴,𝐵). Since 𝑇 is an MK-proximal contraction of the first kind, using Lemma 2.11, we get that 𝑑𝑥,𝑔𝑥_𝑛+1≤𝑑𝑥^∗,𝑥_𝑛. (3.26) Letting 𝑛→∞, it follows that that 𝑥=𝑔𝑥^∗. Thus, it can be concluded that 𝑑𝑔𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (3.27) To assert the uniqueness, let us assume that 𝑧^∗ is another element in 𝐴 such that 𝑑(𝑔𝑧^∗,𝑇𝑧^∗)=𝑑(𝐴,𝐵). Due to the fact that 𝑇 is an MK-proximal contraction of the first kind and 𝑔 is an isometry, using Lemma 2.11, we get that 𝑑𝑥^∗,𝑧^∗=𝑑𝑔𝑥^∗,𝑔𝑧^∗<𝑑𝑥^∗,𝑧^∗, (3.28) which is a contradiction. Then 𝑥^∗ and 𝑧^∗ are identical. This completes the proof.

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

Corollary 3.2. Let 𝐴 and 𝐵 be closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonvoid and the pair (𝐴,𝐵) satisfies the weakly 𝑃-property. Suppose that the mapping 𝑇∶𝐴→𝐵 satisfies the following conditions: (a)𝑇 is an MK-proximal contraction of the first and second kinds; (b)𝑇(𝐴₀)⊆𝐵₀.
Then, there exists a unique element 𝑥^∗∈𝐴 such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (3.29) Further, for any fixed element 𝑥₀∈𝐴₀, the iterative sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵), (3.30) converges to 𝑥^∗.

Example 3.3. We endow 𝑋=ℝ⁴ with the standard metric: 𝑑𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑦₁,𝑦₂,𝑦₃,𝑦₄=𝑥₁−𝑦₁+𝑥₂−𝑦₂+𝑥₃−𝑦₃+𝑥₄−𝑥₄, (3.31) for all (𝑥₁,𝑥₂,𝑥₃,𝑥₄),(𝑦₁,𝑦₂,𝑦₃,𝑦₄)∈ℝ⁴. Consider the sets 𝐴,𝐵⊂𝑋 defined by 1𝐴∶=0,0,𝑛,2𝑛,1∶𝑛∈ℕ{(0,0,0,0)}𝐵∶=1,2,𝑛,2𝑛∶𝑛∈ℕ{(1,2,0,0)}. (3.32) Then 𝐴 and 𝐵 are nonempty closed subsets of ℝ⁴ with 𝑑(𝐴,𝐵)=3. It is easy to show that in this case we have 𝐴₀=𝐴 and 𝐵₀=𝐵. On the other hand, for all (𝑥,𝑦)∈𝐴×𝐵, we have 𝑑⟺∈1𝑥,𝑦=3𝑥,𝑦0,0,𝑛,2𝑛,11,2,𝑛,2𝑛,((0,0,0,0),(1,2,0,0))∶𝑛∈ℕ∶=Ω. (3.33) Clearly, for all 𝑥₁,𝑥₂∈𝐴, 𝑦∈𝐵, 𝑥₁,𝑦,𝑥₂,𝑦∈Ω⇒𝑥₁=𝑥₂. (3.34) Thus the pair (𝐴,𝐵) satisfies the weakly 𝑃-property.
Let 𝑇∶𝐴→𝐵 be the mapping defined by 𝑇10,0,𝑛,2𝑛=11,2,,23𝑛𝑇()=().3𝑛,∀𝑛∈ℕ,0,0,0,01,2,0,0 (3.35)

Claim 1. 𝑇 is an MK-proximal contraction of the first kind.
Let 𝜀>0 be fixed and 𝛿(𝜀)=2𝜀. Let 𝑥,𝑦,𝑢,𝑣∈𝐴 such that 𝑑(𝑢,𝑇𝑥)=𝑑𝑣,𝑇𝑦=𝑑(𝐴,𝐵)=3,𝜀≤𝑑𝑥,𝑦<𝜀+𝛿(𝜀)=3𝜀. (3.36) We consider three cases.

Case 1. There exist 𝑝,𝑘∈ℕ such that 1𝑢=0,0,,23𝑝13𝑝,𝑥=0,0,𝑝,2𝑝,1𝑣=0,0,,23𝑘13𝑘,𝑦=0,0,𝑘,2𝑘. (3.37) In this case, from (3.36), we have 𝑑()=1𝑢,𝑣𝑝−1𝑘<𝜀. (3.38)

Case 2. There exists 𝑝∈ℕ such that 1𝑢=0,0,,23𝑝13𝑝,𝑥=0,0,𝑝,2𝑝,𝑣=𝑦=(0,0,0,0). (3.39) In this case, from (3.36), we have 𝑑()=1𝑢,𝑣𝑝<𝜀. (3.40)

Case 3. There exists 𝑘∈ℕ such that ()1𝑢=0,0,0,0=𝑥,𝑣=0,0,,23𝑘13𝑘,𝑦=0,0,𝑘,2𝑘. (3.41) In this case, from (3.36), we have 𝑑()=1𝑢,𝑣𝑘<𝜀. (3.42) Thus, in all cases we have 𝑑(𝑢,𝑣)<𝜀. Then Claim 1 holds.

Claim 2. 𝑇 is an MK-proximal contraction of the second kind.
Let 𝜀>0 be fixed and 𝛾(𝜀)=2𝜀. Let 𝑥,𝑦,𝑢,𝑣∈𝐴 such that 𝑑(𝑢,𝑇𝑥)=𝑑𝑣,𝑇𝑦=𝑑(𝐴,𝐵)=3,𝜀≤𝑑𝑇𝑥,𝑇𝑦<𝜀+𝛾(𝜀)=3𝜀. (3.43) We consider also three cases.

Case 1. There exist 𝑝,𝑘∈ℕ such that 1𝑢=0,0,,23𝑝13𝑝,𝑥=0,0,𝑝,2𝑝,1𝑣=0,0,,23𝑘13𝑘,𝑦=0,0,𝑘,2𝑘. (3.44) In this case, from (3.43), we have 𝑑()=3𝑇𝑢,𝑇𝑣91𝑝−1𝑘<𝜀. (3.45)

Case 2. There exists 𝑝∈ℕ such that 1𝑢=0,0,,23𝑝13𝑝,𝑥=0,0,𝑝,2𝑝,𝑣=𝑦=(0,0,0,0). (3.46) In this case, from (3.43), we have 𝑑()=1𝑇𝑢,𝑇𝑣3𝑝<𝜀. (3.47)

Case 3. There exists 𝑘∈ℕ such that ()1𝑢=0,0,0,0=𝑥,𝑣=0,0,,23𝑘13𝑘,𝑦=0,0,𝑘,2𝑘. (3.48) In this case, from (3.43), we have 𝑑()=1𝑇𝑢,𝑇𝑣3𝑘<𝜀. (3.49) Thus, in all cases we have 𝑑(𝑇𝑢,𝑇𝑣)<𝜀. Then Claim 2 holds.
Finally, from Corollary 3.2, there exists a unique 𝑥^∗∈𝐴 such that 𝑑(𝑥^∗,𝑇𝑥^∗)=𝑑(𝐴,𝐵)=3. In this example, we have 𝑥^∗=(0,0,0,0).

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 (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be an MK-contraction of the first kind. Then 𝑇 has a unique fixed point 𝑥^∗∈𝑋, and for each 𝑥∈𝑋, lim_𝑛→∞𝑇^𝑛𝑥=𝑥^∗.

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

Theorem 3.5. Let 𝐴 and 𝐵 be closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonvoid, the pair (𝐴,𝐵) satisfies the weakly 𝑃-property, and 𝐵 is approximatively compact with respect to 𝐴. Suppose that the mappings 𝑔∶𝐴→𝐴 and 𝑇∶𝐴→𝐵 satisfy the following conditions: (a)𝑇 is an MK-proximal contraction of the first kind; (b)𝑇(𝐴₀)⊆𝐵₀; (c)𝑔 is an isometry; (d)𝐴₀⊆𝑔(𝐴₀).
Then, there exists a unique element 𝑥^∗∈𝐴 such that 𝑑𝑔𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (3.50) Further, for any fixed element 𝑥₀∈𝐴₀, the iterative sequence {𝑥_𝑛}, defined by 𝑑𝑔𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵), (3.51) converges to 𝑥^∗.

Proof. Proceeding as in Theorem 3.1, it can be shown that there is a sequence {𝑥_𝑛} of elements in 𝐴₀ such that 𝑑𝑔𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵), (3.52) and any sequence satisfying the above condition must converge to some element 𝑥^∗∈𝐴. On the other hand, we have 𝑑𝑔𝑥^∗,𝐵≤𝑑𝑔𝑥^∗,𝑇𝑥_𝑛≤𝑑𝑔𝑥^∗,𝑔𝑥_𝑛+1+𝑑𝑔𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑𝑥^∗,𝑥_𝑛+1()𝑥+𝑑𝐴,𝐵≤𝑑^∗,𝑥_𝑛+1+𝑑𝑔𝑥^∗,𝐵. (3.53) Letting 𝑛→∞ in the above inequality, we get 𝑑(𝑔𝑥^∗,𝑇𝑥_𝑛)→𝑑(𝑔𝑥^∗,𝐵) as 𝑛→∞. Since 𝐵 is approximatively compact with respect to 𝐴, it follows that the sequence {𝑇𝑥_𝑛} has a subsequence {𝑇𝑥_𝑛(𝑘)} converging to some element 𝑦^∗∈𝐵. Thus we have 𝑑𝑔𝑥^∗,𝑦^∗=lim_𝑘→∞𝑑𝑔𝑥_𝑛(𝑘)+1,𝑇𝑥_𝑛(𝑘)=𝑑(𝐴,𝐵). (3.54) The rest part of the proof follows as in Theorem 3.1.

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

Corollary 3.6. Let 𝐴 and 𝐵 be closed subsets of a complete metric space (𝑋,𝑑) such that 𝐴₀ is nonvoid, the pair (𝐴,𝐵) satisfies the weakly 𝑃-property, and 𝐵 is approximatively compact with respect to 𝐴. Suppose that the mapping 𝑇∶𝐴→𝐵 satisfies the following conditions: (a)𝑇 is an MK-proximal contraction of the first kind; (b)𝑇(𝐴₀)⊆𝐵₀.
Then, there exists a unique element 𝑥^∗∈𝐴 such that 𝑑𝑥^∗,𝑇𝑥^∗=𝑑(𝐴,𝐵). (3.55) Further, for any fixed element 𝑥₀∈𝐴₀, the iterative sequence {𝑥_𝑛}, defined by 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑(𝐴,𝐵), (3.56) converges to 𝑥^∗.

Acknowledgment

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