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In this paper, we derive a best proximity point theorem for non-self-mappings satisfied proximal cyclic contraction in PM-spaces and this shows the existence of optimal approximate solutions of certain simultaneous fixed point equations in the event that there is no solution. As an application we consider a nonlinear programming problem. Our results extend and improve the recent results of (Sadiq Basha in Nonlinear Anal. 74(17):5844-5850, 2011). [ABSTRACT FROM AUTHOR]

Let us consider the mappings $${S : A \to A}$$ and $${T : A \to B}$$ , where A and B are two nonempty subsets of a metric space ( X, d). The purpose of this paper is to provide sufficient conditions to ensure the existence and uniqueness of an element $${p \in A}$$ such that $${d(Sp, Tp) = {\rm dist}(A, B) := {\rm inf}\{d(x, y) : (x, y) \in A \times B\}}$$ . In particular, the existence and uniqueness of a best proximity point for non-self-mappings are obtained. Examples are given to show usability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of best proximity point for mappings satisfying generalized proximal weak contraction in a complete ordered metric spaces. Further, our result provides an extension of a result due to Stojan Radenović and Zoran Kadelburg to the case of non-self-mappings. [ABSTRACT FROM AUTHOR]

In this paper the existence and uniqueness of best proximity point for Reich-type contraction on b-metric spaces endowed with a graph is established. These results are significant, since we replace the condition of continuity of mapping with the condition of G-continuity of mapping and we consider b-metric spaces endowed with a graph instead of metric spaces, under which can be unified some theorems of the existing literature. [ABSTRACT FROM AUTHOR]

T. Zamfirescu, [Fixed point theorems in metric spaces, Arch. Math. (Basel), 23 (1972), 292-298] obtained a very interesting fixed point theorem on complete metric spaces, by combining results of Banach, Kannan and Chatterjea. In this paper, we introduce the concept of Zamfirescu- Perov type cyclic contraction and obtain best proximity point theorems for such mapping in the frame work of regular cone metric spaces. Examples are given to support our results. Our results extend and generalize several comparable existing results in literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the coupled fixed point problem and the coupled best proximity problem for single-valued and multi-valued contraction type operators defined on cyclic representations of the space. The approach is based on fixed point results for appropriate operators generated by the initial problems. [ABSTRACT FROM AUTHOR]

This article studies new classes of contractions called the p-cyclic Reich contraction and p-cyclic Reich contraction pair and develops certain best proximity point results for such contractions in the setting of partial metric spaces. Furthermore, the best proximity point results for p-proximal cyclic Reich contractions of the first and second types are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of an α3-proximal admissible mappings and establish the existence of best proximity point theorems for implicit relation type modified α3-proximal contractions. As applications of our theorems, we derive some new best proximity point results for implicit relation type contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some best proximity point results in the literature. Several interesting consequences of our theorems are also provided. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study certain new concepts of α-ψ-proximal contractions in an intuitionistic fuzzy metric space. Then we establish certain best proximity point theorems for such proximal contractions in intuitionistic fuzzy metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered intuitionistic fuzzy metric spaces. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain some recent fixed point theorems as special cases. Moreover, we discuss some illustrative examples to highlight the realized improvements. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the novel concept of generalized distance denoted as Jθ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance Jθ(u,H★), a generalized Hausdorff type distance and a PJθ-property of a pair (H★,K★) of nonempty subsets of extended b-metric space (U★,ρθ). Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples.

In this paper we present some best proximity point theorems for a combination of weak Kannan, Chatterjea cyclic condition and the MT - functions through rational expression in metric spaces (X, d). We provide an example to illustrate and support our main result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new idea of best proximity point of F-contraction on a closed ball and obtain new theorems in a complete metric space. That is why this outcome becomes useful for contraction of a mapping on a closed ball instead of the whole space. At the same time, some comparative examples are constructed which establish the superiority of our results. Our results that have come into being give a proof of extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature. [ABSTRACT FROM AUTHOR]

In this paper, first we introduce the notion of 1/2-modular metric spaces and weak (α,Θ)-ω-contractions in this spaces and we establish some results of best proximity points. Finally, as consequences of these theorems, we derive best proximity point theorems in modular metric spaces endowed with a graph and in partially ordered metric spaces. We present an example to illustrate the usability of these theorems. [ABSTRACT FROM AUTHOR]

Fixed point (briefly FP ) theory is a potent tool for resolving several actual problems since many problems may be simplified to the FP problem. The idea of Banach contraction mapping is a foundational theorem in FP theory. This idea has wide applications in several fields; hence, it has been developed in numerous ways. Nevertheless, all of these results are reliant on the existence and uniqueness of a FP on some suitable space. Because the FP problem could not have a solution in the case of non self-mappings,the idea of the best proximity point (briefly Bpp) is offered to approach the best solution. This paper investigates the existence and uniqueness of the Bpp of non self-mappings in fuzzy normed space(brieflyFNspace) to arrive at the best solution. Following the introduction of the definition of the Bpp, the existence, and uniqueness of the Bpp are shown in a FNs pace for diverse fuzzy proximal contractions such as 𝔅 ̃𝜓- fuzzy proximal contractive mapping and 𝔅h𝔍h- fuzzy proximal contractive mapping.

In this work, we prove the existence of the best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide suitable examples to demonstrate the validity of our results. [ABSTRACT FROM AUTHOR]

The symmetry of fuzzy metric spaces has benefits for flexibility, ambiguity tolerance, resilience, compatibility, and applicability. They provide a more comprehensive description of similarity and offer a solid framework for working with ambiguous and imprecise data. We give fuzzy versions of some celebrated iterative mappings. Further, we provide different concrete conditions on the real valued functions J , S : (0 , 1 ] → R for the existence of the best proximity point of generalized fuzzy (J , S) -iterative mappings in the setting of fuzzy metric space. Furthermore, we utilize fuzzy versions of J , S -proximal contraction, J , S -interpolative Reich–Rus–Ciric-type proximal contractions, J , S -Kannan type proximal contraction and J , S -interpolative Hardy Roger's type proximal contraction to examine the common best proximity points in fuzzy metric space. Also, we establish several non-trivial examples and an application to support our results. [ABSTRACT FROM AUTHOR]

In this work, we introduce some new concepts such as n-cyclic Fisher quasi-contraction mappings, full-n-noncyclic and regular-n-noncyclic Fisher quasi-contraction mappings in metric spaces. We then generalize the results by Safari-Hafshejani, Amini-Harandi and Fakhar. Meanwhile, we answer the question "under what conditions does a full-n-noncyclic Fisher quasi-contraction mapping have n (n − 1) / 2 unique optimal pairs of fixed points?". Further, to support the main results, we highlight all of the new concepts via non-trivial examples. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concepts of $ d $- and $ d^{-1} $-proximal Ćirić contraction mappings. Also, we present new definitions and notations by taking into account the lack of symmetry property of quasi-metric spaces. Moreover, we give some examples to support our definitions and notations. Then, we prove some right and left best proximity point results for these mappings on best orbitally complete quasi-metric spaces. Hence, we obtain some generalizations of famous results in the literature.

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations.

In this paper, we introduce the new concept of proximal mapping, namely proximal weak contractions and proximal Berinde nonexpansive mappings. We prove the existence of best proximity points for proximal weak contractions in metric spaces, and for proximal Berinde nonexpansive mappings on starshape sets in Banach spaces. Examples supporting our main results are also given. Our main results extend and generalize some of well-known best proximity point theorems of proximal nonexpansive mappings in the literatures. 1 [ABSTRACT FROM AUTHOR]

In this paper, using the concept of C-class fuction, we prove the existence of common fixed point for generalized Zam rescu-type mappings and generalized weakly Zam rescu-type mappings. Our results generalize so many results in the literature. [ABSTRACT FROM AUTHOR]

In this paper the concept of set-valued cyclic Meir-Keeler contraction map is introduced. The existence of best proximity point for such maps on a metric space with the UC property is presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a pair of generalized proximal contraction mappings and prove the existence of a unique best proximity point for such mappings in a complete metric space. We provide examples to illustrate our result. Our result extends some of the results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness theorems of the best proximity points for Meir-Keeler contraction type of non-self quasi-triangular admissible mappings in the setting of proximity condition and give nontrivial examples to validate the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a split best proximity point and equilibrium problem, and find a solution of the best proximity point problem such that its image under a given bounded linear operator is a solution of the equilibrium problem. We construct an iterative algorithm to solve such problem in real Hilbert spaces and obtain a weak convergence theorem. Finally, we also give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

Inspired by the work of Jachymski, we slightly extend some fixed point theorems with a graph and show that some best proximity point theorems for α-ψ-contraction mappings of Jleli and Samet can be deduced by our results. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and uniqueness of the best proximity points of nonself-mappings in the context of non-Archimedean fuzzy metric spaces. The existence of different proximal quasi-contractive mappings allowed us to generalize some results concerning the existence and uniqueness of the best proximity points in the existing literature. Moreover, an application in computer science, particularly in the domain of words has been provided.

The aim of this paper is to present some best proximity results for multivalued cyclic operators satisfying a Ćirićtype condition. Our results extend to the multivalued case some recent results in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve some global optimization problems for Geraghty type proximal contractions in the setting of partially ordered sets with a metric by using a w-distance and an algorithm for determining such an optimal approximate solution, also, we give some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Best approximation results provide an approximate solution to the fixed point equation Tx = x, when the non-self mapping T has no fixed point. In particular a well- known best approximation theorem, due to Fan [6], asserts that if K is a nonempty compact convex subset of a Hausdor locally convex topological vector space E and T : K → E is a continuous mapping, then there exists an element x satisfying the condition d(x; Tx) = inf{d(y; Tx) : y ϵ K} where d is a metric on E. Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, Article ID 837943) introduced proximal contractive mappings and established certain best proximity point results for these mappings in G-metric spaces. The aim of this paper is to introduce certain new classes of auxiliary functions and proximal contraction mappings and establish best proximity point theorems for such kind of mappings in G-metric spaces. As consequences of these results, we deduce certain new best proximity and fixed point results in G-metric spaces. Moreover, we present certain examples to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper we define a generalized proximal G-contraction on a metric space having the additional structure of a directed graph. We obtain a best proximity point result for such contractions which is with a view to obtaining minimum distance between the domain and range sets. An example illustrating the main theorem is also discussed. The work is in the line of research on mathematical analysis as well as optimization in metric spaces with a graph. [ABSTRACT FROM AUTHOR]

Existence of fixed point for C-class functions was first proved by Ansari in 2014. Then, many authors gave interesting results using C-class functions. In this paper, we prove the existence of strong coupled proximity point for generalised cyclic coupled proximal maps. Our result generalises the results of Kadwin and Marudai. [ABSTRACT FROM AUTHOR]

In the present work, Banach and Kannan integral type contractions in metric spaces endowed with a graph are considered and the existence and uniqueness of best proximity points for mappings satisfying in these contractions are proved. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new notion, so-called K W -type F-contraction mapping, inspired by the ideas of Wardowski and Klim–Wardowski. Then, we investigate the existence of a best proximity point for such mappings by considering a new family, which is larger than the family of functions that is often used in fixed-point results for multivalued mappings. To demonstrate the effectiveness of our result, we also give a comparative example to which similar results in the literature cannot be applied. Moreover, we present an application of our main result to homotopic mappings. [ABSTRACT FROM AUTHOR]

We introduce a new type of interpolative proximal contractive condition that ensures the existence of the best proximity points of fuzzy mappings in the complete non-archimedean fuzzy metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing fuzzy proximal interpolative contractions. The obtained results improve and generalize the best proximity point theorems published in Fuzzy Information and Engineering, 5 (2013), 417--429. Moreover, we provide many nontrivial examples to validate our best proximity point theorem. [ABSTRACT FROM AUTHOR]

First, we use a new technique to obtain an existence result for the newly introduced S - cyclic mappings. Afterwards, we study the structure of minimal sets of tricyclic relatively nonexpansive mappings in the setting of Kohlenbach hyperbolic spaces. This way, we obtain a best proximity point theorem for such mappings. [ABSTRACT FROM AUTHOR]

We introduce the concept of q-ordered proximal nonunique contraction for the non self mappings and then obtain some proximity point results for these mappings. We also furnish examples to support our claims. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions to ensure the existence of the best proximity point of monotone relatively nonexpansive mappings defined on partially ordered Banach spaces. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution ω that is optimal in the sense that the error σ (ω , J ω) assumes the global minimum value σ (θ , ϑ) . The aim of this paper is to define the notion of Suzuki α - Θ -proximal multivalued contraction and prove the existence of best proximity points ω satisfying σ (ω , J ω) = σ (θ , ϑ) , where J is assumed to be continuous or the space M is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems. [ABSTRACT FROM AUTHOR]