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Abstract Very recently Jain et al. (J. Inequal. Appl. 2022:73, 2022) introduced the concept of multivalued F-contraction with altering distance function and investigated the existence of best proximity points for this class of mappings. In this article we prove that the existence of best proximity points for multivalued F-contraction non-self mappings can be obtained from the corresponding fixed point result for multivalued F-contraction self mappings, and so the main conclusion due to Jain et al. is not a real generalization of fixed point theory.

In the paper, we introduce a new class of noncyclic φ-contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp'ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic φ-contractions as a consequence of our main theorems. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of proximal (α , F) -contractions in F -metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial example. Furthermore, we demonstrate the applicability of these results by proving the existence of solutions for Volterra integral equations related to population growth models. This approach not only extends best proximity theory, but also paves the way for further research in applied mathematics and beyond. [ABSTRACT FROM AUTHOR]

This paper explores the concept of normal structure by introducing a generalized form known as P-proximal normal structure. Focusing on the framework of Hausdorff locally convex spaces, the study establishes optimal proximity outcomes for both cyclic and noncyclic relatively P-nonexpansive mappings. Furthermore, the paper presents in the realm of probabilistic normed spaces, considered as instances of Hausdorff locally convex spaces, some theorems that address the existence of best proximity points within this context. [ABSTRACT FROM AUTHOR]

In this paper, we aim to overcome the problem given by Abkar et al. [Abstr. Appl. Anal., 2013 (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include R-functions. Thus, we present a new type of R-contraction and weaken R-contractions that have often been studied recently. We also give a new definition of the P-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of R-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a study of α-η-type generalized F-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the novel concept of generalized distance denoted as J θ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance J θ (u , H ★) , a generalized Hausdorff type distance and a P J θ -property of a pair (H ★ , K ★) of nonempty subsets of extended b-metric space (U ★ , ρ θ). 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. [ABSTRACT FROM AUTHOR]

Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Chen (Symmetry 11:750, 2019) built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative p-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The concept of measure of noncompactness (MNC) allows us to select an important class of mappings which are more general than compact operators. The proposed work exploits the axiomatic definition of MNC and introduces the notion of relatively nonexpansive cyclic (noncyclic) SR-condensing operators along with the aid of SR-functions. The first phase of the paper concentrates on establishing the best proximity point (pair) theorems for such operators. The main results in this manuscript extend and generalize several state of art literature on Darbo type fixed point results. In the second phase, proposed results are applied to show the actuality of optimum solutions for system of second order differential equations with two initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new notion of generalized α-φ-Geraghty proximal contraction mappings and investigate the existence of the best proximity point for such mappings in complete metric spaces. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature. [ABSTRACT FROM AUTHOR]

The paper presents existence results of (φ, φ) best proximity points for operators that fulfill implicit type inequalities. Classes of mappings endowed with continuity, monotone or monotone-type properties, and which additionally satisfy some adequate inequalities are studied from this point of view. Applications of our results are given with regard to fixed point theory. [ABSTRACT FROM AUTHOR]

Let (A1, A2, A3) be a triple of nonempty convex subsets of a metric space Ω. In this paper, we determine optimal problems of the best proximity pair by proximal normal structure between two sets A1 and A2 with the help of a third set A3 and we find some necessary and sufficient conditions for existence these optimal problems. Also, we provide an example to illustrate the convergence behavior of our proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [7] for closed and bounded valued F -contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this sense, we first introduce a new family, which is larger than F* that has often been used to give a positive answer to the problem. Then, we prove some best proximity point results for the new kind of F -contractions on quasi metric spaces via the new family. Additionally, we show that the note given by Almeida et al. [4] is not valid for our results. Therefore, our results are real generalizations of fixed point results in the literature. Moreover, we give comparative examples to demonstrate that our results unify and generalize some well-known results in the literature. As an application, we show that each homotopic mapping to f satisfying all the hypotheses of our best proximity point result has also a best proximity point. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new class of noncyclic $\varphi$-contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp\'{i}nola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic $\varphi$-contractions as a consequence of our main theorems.

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α - θ - Geraghty–Pata–Suzuki-type proximal contraction; an α - θ - generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α - θ - modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. [ABSTRACT FROM AUTHOR]

In the year 2016, Pragadeeswarar et al. [V. Pragadeeswarar, M. Marudai, P. Kumam, J. Nonlinear Sci. Appl., 9 (2016), 1911-1921] considered a special class of multi-valued mappings and investigated the existence of best proximity points for such class of mappings which generalizes the fixed point result established by Choudhury et al. [B. S. Choudhury, N. Metiya, Arab J. Math. Sci., 17 (2011), 135-151] in the context of partially ordered metric spaces. In this note, we have showed that the best proximity point result is a direct consequence of the corresponding fixed point result. In the last part of this note, we applied our result in [V. Pragadeeswarar, M. Marudai, P. Kumam, J. Nonlinear Sci. Appl., 9 (2016), 1911-1921, Example 2.2] to validate our claim. [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. [ABSTRACT FROM AUTHOR]

Let A and B be nonempty subsets of a metric space X. Let S: A → B and g: A → A. In this paper, we prove the existence of best approximate solution x* in the space such that Sx* is as close to gx* as possible. That is, we find the global minimizer of the map x 7→ ρ(gx,Sx) where S is either proximal F*-weak contraction of the first kind or second kind or both, and g is an isometry, in complete metric spaces. Examples are provided to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we point out an error in proving famous Achari type nonunique fixed point results. Also, we prove some best proximity point results in b -metric spaces by introducing new concepts. Hence, we develop some results existing in the literature. Finally, we give a result for the existence of the solution of nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper we list some geometric properties of Banach spaces and some fixed point theory results which we use in the subsequent topic . Certain geometric properties of a Banach space X are helpful to study the existence of fixed points for non expansive mappings and asymptotically non expansive mappings on closed bounded convex subsets of X. In this paper first we give a brief introduction to Geometry of Banach spaces, a brief introduction to fixed point theory, a brief introduction to best proximity point theorems, and finally we give contents of the paper. [ABSTRACT FROM AUTHOR]

Let X be an arbitrary topological space and g : X x X → R be a real valued continuous function defined on X x X. In this article, we introduce two notions like topologically Berinde weak proximal contraction and topologically proximal weakly contractive mapping with respect to g. We explore sufficient conditions for the existence and uniqueness of best proximity points for these classes of mappings. Moreover, in the last part of the paper, we show that the best proximity point theorem for topologically proximal weakly contractive mapping can be deduced from some fixed point theorems in topological spaces. [ABSTRACT FROM AUTHOR]

This paper addresses a problem of global optimization in a non-Archimedean fuzzy metric space context without fuzzy P-property. Specifically, it concerns the determination of the fuzzy distance between two subsets of a non-Archimedean fuzzy metric space. Our approach to solving this problem is to find an optimal approximate solution to a fixed point equation. This approach has been well studied within a category of problems called proximity point problems. We explore some new types of (ψ − ϕ) -weak proximal contractions and investigate the existence of the unique best proximity point for such kinds of mappings. Subsequently, some fixed point results for corresponding contractions are proved, and some illustrative examples are presented to support the validity of the main results. Moreover, an interesting application in computer science, particularly in the domain of words has been provided. Our work is a fuzzy generalization of the proximity point problem by means of fuzzy fixed point method. [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. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of the best proximity points in binormed linear spaces. As a consequence, we obtain some fixed point results. We also provide some illustrations to support our claims. As applications, we obtain the existence of a solution to split feasible and variational inequality problems. [ABSTRACT FROM AUTHOR]

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In this paper we establish best proximity point results for monotone multivalued mappings in partially ordered metric spaces. We consider three notions of monotonicity of multivalued mappings. The main theorem is obtained by utilizing property UC and MT-functions. There is no requirement of continuity on the multivalued function which is illustrated with two supporting examples of the results established in this paper. There are two corollaries. Some existing results are extended to the domain of partially ordered metric spaces through one of the corollaries. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the so-called S-probabilistic metric spaces. In S-probabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang (Fixed Point Theory Appl. 2014:170, 2014). The second class is the so-called Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of the homeomorphism metric space and to prove the fixed point theorems and the best proximity point theorems for generalized contractions in such spaces. The multiplicative metric space is a special form of the homeomorphism metric space. The results of this paper improve and extend the previously known ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of cyclic p-contraction pair for single-valued mappings. Then we present some best proximity point results for such mappings defined on proximally complete pair of subsets of a metric space. Also, we provide some illustrative examples that compared our results with some earliest. Finally, by taking into account a fixed point consequence of our main result we give an existence and uniqueness result for a common solution of a system of second order boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new concepts of Feng-Liu type multivalued contraction mapping and cyclic Feng-Liu type multivalued contraction mapping. Then, we obtain some new best proximity point results for such mappings on partial metric spaces by considering Feng-Liu's technique. Finally, we provide examples to show the effectiveness of our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce new class of proximal contractions in non-Archimedean modular metric spaces and to prove some best proximity point theorems for such kind of mappings. As application we deduce best proximity results in fuzzy metric spaces. Consequently, some basic fixed point results in both modular and fuzzy metric spaces are obtained as corollaries of our work. Finally, an example is provided to illustrate the usability of our obtained results. [ABSTRACT FROM AUTHOR]

There are situations when we have to resort to the approximate optimal solution of equations of the type g(t) = t when g is not a self-map, because exact solution of that equation does not exist. The existence of such optimal solutions are ensured by best proximity point theorems. In this paper, we define multivalued Geraghty contraction (MVGC) in a complete metric space and establish the corresponding best proximity point (BPP) result. Our result extends the famous result due to Geraghty on fixed points. [ABSTRACT FROM AUTHOR]

In this paper, we extend the result of Romaguera [21] with the aid of best proximity point theory on partial metric spaces by considering the approach of Haghi et al. [9], and so celebrated Boyd-Wong fixed point theorem [7]. We first introduce two concepts called generalized proximal BW-contraction and generalized best BW-contraction. Then, we obtain some best proximity point theorems for such mappings. To illustrate the effectiveness of our results, we provide some nontrivial and interesting examples. Finally, unlike homotopy applications existing in the literature, we present for the first time an application of the best proximity result to the homotopy theory. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of best proximity points of certain mappings via simulation functions in the frame of complete metric-like spaces. Some consequences and examples are given of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper we utilize a generalized weakly contractive mapping to establish some best proximity point results which are global optimization results for finding the minimum distances between two sets. Amongst many approaches to this problem, we adopt the approach where the problem is treated as that of finding global optimal approximate solution of the fixed point equation for the generalized weak contraction mapping. We use three control functions to define such mappings. The results are obtained in metric spaces with a partial ordering defined therein. There is a blending of analytic and order theoretic approaches in the proofs. The uniqueness is obtained by imposing some order theoretic conditions additionally. There are several corollaries. An illustration of the main theorem through an example is given which also shows that the corollaries are properly contained in the main theorem. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of a cyclic generalized proximal contraction, and then prove some best proximity point theorems in metric spaces. We will provide examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper we establish some best proximity point results using generalized weak contractions with discontinuous control functions. The theorems are established in metric spaces with a partial order. We view the main problem in the paper as a problem of finding an optimal approximate solution of a fixed point equation. We also discuss several corollaries and give an illustrative example. We apply our result to obtain some coupled best proximity point results. [ABSTRACT FROM AUTHOR]

In this paper, in the setting of metric spaces we introduce the notions of generalized cyclic quasi-contractions and the ultrametric property as an applied geometric concept. Then we study the existence and uniqueness of best proximity points for such mappings by using this property and the UC property. Also, iterative algorithms are furnished to determine such best proximity points. As a result, we establish a fixed point result and a common fixed point theorem. The presented results extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce the concept of a Geraghty's cyclic contraction which is a generalized cyclic contractive mapping and provide best proximity point theorems for such a mapping in complete metric space and uniformly convex Banach space. Moreover, we also give some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a weak MT-cyclic Reich type contractions and obtain the existence theorems for best proximity point for self-mappings defined on the complete metric spaces. Our results improve and generalize some results in literature. Also, we give some applications of our results to solving some classes of non-linear integral and differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we aim to overcome the problem given by Abkar et al. [Abstr. Appl. Anal., 2013 (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include $ \mathcal{R} $-functions. Thus, we present a new type of $ \mathcal{R} $ -contraction and weaken $ \mathcal{R} $-contractions that have often been studied recently. We also give a new definition of the $ P $-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of $ \mathcal{R} $-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result.

In the current paper, we first introduce a new class of contractions via a new notion called p-cyclic contraction mapping by combining the ideas of cyclic contraction mapping and p-contraction mapping. Then, we give a new definition of a cyclically 0-complete pair to weaken the completeness condition on the partial metric spaces. Following that, we prove some best proximity point results for p-cyclic contraction mappings on D ∪ E where D , E is a cyclically 0-complete pair in the setting of partial metric spaces. Hence, we generalize and unify famous and well-known results in the literature of metric fixed point theory. Additionally, we present some nontrivial examples to compare our results with earlier. Finally, we investigate the sufficient conditions for the existence of a solution to nonlinear Fredholm integral equations by the results in the paper. [ABSTRACT FROM AUTHOR]

Recall from [2], that a mapping T : X 7→ X is called a Reich mapping if it satisfies for all x; yϵ X, d(Tx; Ty) ≤ ad(x; Tx) + bd(y; Ty) + cd(x; y), where a,b,c are nonnegative and satisfy a + b + c < 1. Alternatively, one could define a Reich mapping as follows: T : X 7→ X is called a Reich mapping if there exists a nonnegative constant k with k < 1 3 such that d(Tx; Ty) ≤ k[d(x; Tx)+d(y; Ty)+d(x; y)]. In the present paper, we address the following: How do we characterize Theorem 3 [2], when T is a non-self map? We show such a characterization is given by Theorem 3.1 or Corollary 3.2 in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the existence of best proximity points for multivalued Suzuki-Edelstein-Wardowski type α-proximal contractions in the setting of complete metric spaces and partially ordered metric spaces. We give examples to illustrate the main results. Our results improve and extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we define α--β--ψ--proximal contractive mappings in partially ordered metric spaces and prove the existence of best proximity points of these maps in partially ordered complete metric spaces. These results extend/generalize the results of Asgari and Badehian, J. Nonl. Sci. and Appl., 2015. We provide illustrative examples in support of our theorems. [ABSTRACT FROM AUTHOR]