Best Proximity Point Theorems for Two Weak Cyclic Contractions on Metric-Like Spaces

In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic φ-contraction via theMT -function. We express some examples to indicate the validity of the presented results. Our results unify and generalize a number of best proximity point results on the topic in the corresponding recent literature.


Introduction and Preliminaries
Fixed point theory provided not only the traditional tools but also the most crucial tools to prove the existence of solutions for several distinct and interesting problems both in pure and applied mathematics.For a self-mapping F on a non-empty set S, the equation Fx = x is named a fixed point equation.If a fixed point equation possesses a solution, we say that F has a fixed point.For example, if F is linear operator, then the fixed point equation Fx = x has no solution, or infinite solutions, or a unique solution.In this paper, we shall focus on the case that the fixed point equation Fx = x has no solution, which is also to say that F is fixed point free.In the setting of a metric space (S, d), if F is fixed point free, then we have d(x, Fx) > 0 for all x ∈ S. It is quite natural to ask the following: If d(x, Fx) > 0 for all x ∈ S is there x * ∈ S such that d(x * , Fx * ) ≤ d(x, Fx) for all x ∈ S, that is, is there any point x * ∈ S such that d(x * , Fx * ) is the minimum throughout the domain of F? Roughly speaking, if we have no exact solution of the fixed point equation Fx = x, we look for the approximative solution of the fixed point equation Fx = x.If the answer is affirmative, the point x * ∈ S is named the best proximity point of the domain and range of the mapping F. In the last decades, this topic has been discussed densely by several authors, see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12].
In what follows, we simply describe the research backgrounds and preliminaries.As it is used commonly, we shall denote the set of all non-negative real numbers by R + 0 .Through the paper, instead of considering the whole metric space (S, d), we shall restrict ourselves to two nonempty subsets, A and B, of it.Further, instead of considering self-mapping, we shall consider non-self mapping F : A → B. We formalize our consideration with In the case of the existence of such x ∈ A we shall say that x is the best proximity point of F for the pair (A, B).Commonly, the pair (A, B) is not mentioned and we only say that x is the best proximity point of F. Hence, x is an approximative solution of the fixed point equation Fx = x.Note that in case of A ∩ B = ∅, the best proximity point coincides with fixed point.On the other hand, even if A ∩ B = ∅, the corresponding fixed point equation still may not possess a solution.
First, we recall the notion of cyclic contraction.
Definition 1. Ref. [13] A mapping F : where A, B are nonempty subsets of a metric space (S, d).In addition, if there exists k ∈ [0, 1) such that for all x ∈ A and y ∈ B, then a mapping F is called cyclic contraction.
Now, we shall mention the result of Eldred and Veeramani [13] in which one of the initial results in this direction was given.
Theorem 1. Ref. [13] Suppose that a mapping F : A ∪ B → A ∪ B is cyclic contraction, where A, B are nonempty, closed, and convex subsets of a metric space (S, d) and k ∈ [0, 1).If we set x n+1 = Fx n for each n ∈ N ∪ {0}, for an arbitrary x 0 ∈ A, then there exists a unique x ∈ A such that x 2n → x and d(x, Tx) = d(A, B).That is, x is the best proximity point of T.
Here, we use the couple (X, σ) to describe "a metric-like space".
Let {p n } be a sequence in a metric-like space (X, σ).Then (1) {p n } converges to p ∈ X if and only if lim n→∞ σ(p n , p) = σ(p, p).
(2) if lim n→∞ σ(p n , p m ) exists and is finite, then we say that {p n } is fundamental (or, Cauchy) (3) if each fundamental (Cauchy) sequence is convergent, then we say that (X, σ) is complete.
The characterization of a best proximity point of F in the setting of metric-like space (X, σ) is follows: Let F : A → B be a mapping where A and B be two nonempty subsets X.Consider the distance of the sets A and B:

The Best Proximity Point Results of Meir-Keeler-Kannan Type Cyclic Contractions
A mapping F : A ∪ B → A ∪ B is called Kannan type cyclic contraction, if there exists k ∈ (0, 1  2 ) such that for all p ∈ A, and q ∈ B where A, B are nonempty subsets of a metric-like space (X, σ).
We have the following: (1) If p 0 ∈ A and {p 2n } has a subsequence {p 2n k } which converges to x * ∈ A with σ(p * , p * ) = 0, then, A function ξ : R + 0 → R + 0 called Meir-Keeler type (see, [16]), if We shall use M to denote set of all Meir-Keeler type function ξ.Note that if ξ ∈ M then, for all t ∈ (0, ∞), we have By using the Kannan type cyclic contraction and Meir-Keeler function, we define the new notion of Meir-Keeler-Kannan type cyclic contraction, as follows: Definition 3. Let φ ∈ M and T : A ∪ B → A ∪ B be a cyclic mapping, where A and B be nonempty subsets of a metric-like space (X, σ).Then, the mapping T is said to be a Meir-Keeler-Kannan type cyclic contraction, if for all x ∈ A and all y ∈ B.
In this section, we establish the best proximity point results of Meir-Keeler-Kannan type cyclic contraction.Our results generalize and improve Theorem 2.
Lemma 1.Let T : A ∪ B → A ∪ B be a cyclic Meir-Keeler-Kannan type contraction, where A and B be nonempty closed subsets of a metric-like space (X, σ), and φ : R Since φ ∈ M, we find that Attendantly, we deduce, for each n ∈ N ∪ {0}, that In other words, {σ(x n+1 , x n ) − σ(A, B)} is bounded below and monotone (non-increasing).Accordingly, there exists ≥ 0 such that In what follows, we assert that = 0. Suppose, on the contrary, that > 0. Keeping, φ ∈ M, in mind, corresponding to , there exist an η and a positive integer k 0 such that a contradiction.Consequently, we get = 0, and we have By Lemma 1, we shall derive the following result in the framework of best proximity theory.
Theorem 3. Let T : A ∪ B → A ∪ B be a cyclic Meir-Keeler-Kannan type contraction, where A and B are nonempty closed subsets of a complete metric-like space (X, σ).
If we set x n+1 = Tx n for each n ∈ N ∪ {0}, for an arbitrary x 0 ∈ A ∪ B, then we have the following: (1) Proof.Suppose x 0 ∈ A. Due to the fact that T is cyclic, we have x 2n ∈ A and Since φ ∈ M is increasing, we have We claim that σ(x * , Tx * ) − σ(A, B) = 0.If not, we assume that Letting k → ∞, by Lemma 1, we find that which implies a contradiction.Thus, σ(x * , Tx * ) = σ(A, B), that is, x * is a best proximity point of T.
The proof of (2) is a verbatim of (1), thus we omit it.

The Best Proximity Point Results of a Generalized Ćirić Type Cyclic ϕ-Contraction via the MT -Function
A mapping T : A ∪ B → A ∪ B is said to be a cyclic Ćirić type contraction if there exists k ∈ (0, 1) such that σ(Tx, Ty) ≤ k max{σ(x, y), σ(x, Tx), σ(y, Ty)} + (1 − k)σ(A, B), for all x ∈ A, and y ∈ B, where A and B are nonempty subsets of a metric-like space (X, σ).
In 2016, Aydi and Felhi [15] established the following best proximity point result for the cyclic Ćirić type contraction.Theorem 4. Ref. [15] Let a mapping T : A ∪ B → A ∪ B be a cyclic Ćirić type contraction, where A and B are nonempty closed subsets of a complete metric-like space (X, σ).If we set x n+1 = Tx n for each n ∈ N ∪ {0} and for an arbitrary x 0 ∈ A ∪ B, then We have the following: (1) If x 0 ∈ A and {x 2n } has a subsequence {x 2n k } which converges to x * ∈ A with σ(x * , x * ) = 0, then, In what follows, we recall the notion of MT -function (or, called the Reich's function).Definition 4. A function ψ : R + → [0, 1) is said to be an MT -function, if In 2012, Du [17] proved the following theorem and remark.Theorem 5. Ref. [17] Let ψ : R + → [0, 1) be a function.Then the following two statements are equivalent.
(a) ψ is an MT -function.(b) For any non-increasing sequence { n } n∈N in R + , we have Remark 1. Ref. [17] It is obvious that if ψ : R + → [0, 1) is non-increasing or non-decreasing, then ψ is an MT -function.

Definition 5. A mapping T
for all x ∈ A and all y ∈ B, where A and B are nonempty subsets of a metric-like space (X, σ), and ψ is an MT -function.
In this section, we establish the best proximity point results of a generalized MT -Ćirić -function type cyclic ϕ-contraction.Our results generalize and improve Theorem 4.

Lemma 2.
Let A and B be nonempty closed subsets of a metric-like space (X, σ).Let T : If σ(x n+2 , x n+1 ) > σ(x n+1 , x n ) for some n, then by the conditions of the function ϕ we have that which implies a contraction.So, we conclude that σ(x n+2 , x n+1 ) ≤ σ(x n+1 , x n ) for all n ∈ N. On the other hand, If σ(x n+1 , x n+2 ) > σ(x n , x n+1 ) for some n, then by the conditions of the function ϕ we have that which implies a contraction.So, we conclude that σ(x n+1 , x n+2 ) ≤ σ(x n , x n+1 ) for all n ∈ N.
From above argument, the sequence {σ(x n+1 , x n )} n∈N is non-increasing and bounded below in R + 0 .Since ψ is an MT -function, by Theorem 5 we conclude that Since T : A ∪ B → A ∪ B is a generalized MT -Ćirić -function type cyclic ϕ-contraction, we obtain that Since λ < 1, lim n→∞ λ n = 0, and we also get that By Lemma 2, we obtain the following best proximity point result of a generalized MT -Ćirić -function type cyclic ϕ-contraction.Theorem 6.Let T : A ∪ B → A ∪ B be a generalized MT -Ćirić -function type cyclic ϕ-contraction, where A and B are nonempty closed subsets of a complete metric-like space (X, σ).If we construct a sequence x n+1 = Tx n for each n ∈ N ∪ {0} for an arbitrary x 0 ∈ A ∪ B, then we have the following: (1) If x 0 ∈ A and {x 2n } has a subsequence {x 2n k } which converges to x * ∈ A with σ(x * , x * ) = 0, then σ(x * , Tx * ) = σ(A, B). (2) If x 0 ∈ B and {x 2n−1 } has a subsequence {x 2n k −1 } which converges to x * ∈ B with σ(x * , x * ) = 0, then σ(x * , Tx * ) = σ(A, B).
Proof.Suppose that x 0 ∈ A. On account of the fact that T is cyclic, we get x 2n ∈ A and x 2n+1 ∈ B for all n ∈ N ∪ {0}.Here, if {x 2n } has a subsequence {x 2n k } which converges to x * ∈ A with σ(x * , x * ) = 0, then Since T is a generalized MT -Ćirić -function type cyclic ϕ-contraction, we have which implies a contradiction.Thus, σ(x * , Tx * ) = σ(A, B), that is, x * is the best proximity point of T.
The proof of (2) is similar to (1), we omit it.