On Best Proximity Point Results for Some Type of Mappings

Vahid Parvaneh , Mohammad Reza Haddadi, and Hassen Aydi 4,5 Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Correspondence should be addressed to Hassen Aydi; hassen.aydi@isima.rnu.tn Received 10 February 2020; Revised 17 May 2020; Accepted 18 May 2020; Published 28 May 2020 Academic Editor: Calogero Vetro Copyright © 2020 Vahid Parvaneh 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. In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghtyand Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.


Introduction and Preliminaries
The Banach contraction principle (BCP) in metric spaces has been generalized and extended in various ways. As a generalization of the BCP, Geraghty [1] proved the following.
Theorem 1 [1]. Let ðX, dÞ be a complete metric space and let T : X ⟶ X be an operator. Suppose that there exists β : ½0, +∞Þ ⟶ ½0, 1Þ satisfying the condition β t n ð Þ ⟶ 1 implies t n ⟶ 0, as n ⟶ +∞: If T satisfies the following inequality then T has a unique fixed point.
The following are two examples of Geraghty functions β.
One of the important extensions of the BCP was given by Caristi [2]. Theorem 2 [2]. Let Γ be a self-mapping on a complete metric space ðX, dÞ. Assume that there is a bounded below and lower semicontinuous function ψ : X ⟶ ℝ so that On the other hand, Kirk et al. [3] in 2003 introduced the notion of a cyclic representation. Definition 3 [3]. Let A and B be nonempty subsets of a metric space ðX, dÞ and T : A ∪ B ⟶ A ∪ B. Then, T is called a cyclic map if TðAÞ ⊆ B and TðBÞ ⊆ A.
The following interesting theorem for a cyclic map was given in [3]. Theorem 4 [3]. Let A and B be nonempty closed subsets of a complete metric space ðX, dÞ. Suppose that T : for all x ∈ A and y ∈ B, where k ∈ ½0, 1Þ is a constant. Then, T has a unique fixed point u and u ∈ A ∩ B.
Let M and N be nonempty sets of a metric space ðX, dÞ. Given a map, Γ : The set of best proximity points of Γ is denoted by P Γ ðM, N Þ. The research of best proximity points is meaningful in optimization. The problem of existence of best proximity points in uniformly convex Banach spaces and in metric spaces as well as the convergence of sequences to such points has been focused on and successfully solved in some classic pioneering works (see [4]).
Definition 5 [5]. Let ðX, dÞ be a metric space and A and B be subsets of X. A map T : A ∪ B ⟶ A ∪ B is said to be a cyclic contractive map if it satisfies (i) dðTx, TyÞ ≤ kdðx, yÞ + ð1 − kÞdistðA, BÞ, for all x ∈ A and y ∈ B.
Eldred and Veeramani [5] extended Theorem 4 to include the case A ∩ B = ∅, by the following existence result of a best proximity point.
Theorem 6 [5]. Let A and B be nonempty closed subsets of a metric space X and let T : A ∪ B ⟶ A ∪ B be a cyclic contraction map. If either A or B is boundedly compact, then there exists x ∈ A ∪ B such that dðx, TxÞ = distðA, BÞ: A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations. The various related performed researches include the cases of cyclic ϕ-contractions [6,7], cyclic Meir-Keeler contractions [8], weak cyclic Bianchini contractions [9], weak cyclic Kannan contractions [10], p-cyclic summing iterated contractions [11], and MF-cyclic contractions with Property UC [12]. Some contractive conditions and related properties under general contractive conditions including some proximal rational types have been also investigated [13]. In this paper, we ensure the existence of best proximity points for Geraghty and Caristi type contraction mappings.

A Best Proximity Point Result for Geraghty-Type Contractions
In this section, we introduce cyclic Geraghty contraction maps and give new conditions for existence and uniqueness of a best proximity point.
Definition 7. Let X be a complete metric space and M and N be subsets of X.
We give the following theorem (comparable to Theorem 3.1 of [1]).

Theorem 8.
Let M and N be closed subsets of a complete metric space X such that diamðMÞ, diamðN Þ < dðM, N Þ. Suppose Γ : M ∪ N ⟶ M ∪ N is a cyclic Geraghty contraction map. Then, P Γ ðM, N Þ ≠ 0. Further, if x 0 ∈ M and x n+1 = Γ x n , then fx 2n g converges to a best proximity point.

A Best Proximity Point Result for Caristi-Type Mappings
Recently, Du [14] established a direct proof of Caristi's fixed point theorem without using Zorn's lemma. In this section, we introduce a generalization of Caristi's fixed point theorem and provide a proof without using Zorn's lemma. We start with the following definition. Our related result is as follows.
Theorem 11. Let ðX, dÞ be a complete metric space and ðM, N Þ be a pair of nonempty closed subsets of X such that M is boundedly compact. Also, let Γ : M ⟶ N a semicontraction map and is a bounded below and lower semicontinuous func- Then, there is u 0 ∈ M so that χðu 0 Þ = inf x∈M χðxÞ.
Proof. Assume that inf x∈M χðxÞ < χðyÞ for every y ∈ M. Given μ 0 ∈ M. Then, inf x∈M χðxÞ < χðμ 0 Þ. We have Γμ 0 ∈ N . Therefore, there is μ 1 ∈ M such that Let us define inductively a sequence fμ n g ⊆ S n , where so that Therefore, Since fμ n g ⊆ M and M is boundedly compact, fμ n g has a convergent subsequence to x ∈ M. Suppose μ n k ⟶ x, dðΓ μ n k , xÞ ≤ χðμ n k Þ − χðxÞ. Since Γx ∈ N , there is z ∈ M so that dðΓx, zÞ > dðM, N Þ and dðΓx, zÞ ≤ χðxÞ − χðzÞ. Therefore, We find that z ∈ S n k . By (30), we get Thus, It is a contradiction, so there is u 0 ∈ M such that χðu 0 Þ = inf x∈M χðxÞ.