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In a recent paper ?Filomat, 35:5 (2021), 1555-1564? the authors proved some best proximity point theorems for classes of contractive pair of mappings as well as cyclic ?-contractions under some sufficient conditions. In this article, we present some counterexamples of the existence results of contractive pair of mappings and present the corrected version of these results. We also show that the best proximity point theorem for cyclic ?-contractions is a straightforward consequence of Boyd-Wong fixed point theorem by dropping an additional assumption.

This work intends to treat the existence of mild solutions for the Hilfer fractional hybrid differential equation (HFHDE) with linear perturbation of first and second type in partially ordered Banach spaces. First, we establish the results concerning the actuality of fixed point of sum and product of operators via the concepts of measure of noncompactness and simulation functions in partially ordered spaces. Then combining these fixed point theorems with the concepts in fractional calculus, new existence results for mild solutions of HFHDE’s are established. Furthermore, the presented fixed point results and existence results improve and extend the present state-of-art in the literature. Competent examples in support of theory are illustrated for better understanding.

We establish the existence of best proximity points (pairs) for a new class of cyclic (noncyclic) condensing operators by using the concept of measure of noncompactness. Our conclusions extend and improve the main results of [Indagationes Math. 29 (2018), 895-906]. By applying our results, we prove a coupled best proximity point theorem and investigate the existence of a solution for a system of differential equations.

We propose a new concept of condensing operators by using a notion of measure of non-compactness in the setting of Banach spaces and establish a new generalization of Darbo’s fixed point theorem. We also show the applicability of our results to integral equations. A concrete example will be presented to support the application part.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In this article, we survey the existence of best proximity pairs for noncyclic contractions with respect to orbits which are defined on a non-convex and weakly compact pair of subsets of a strictly...

In the current paper, we discuss sufficient and necessary conditions for the existence of best proximity points for cyclic f- contractive mappings in the setting of strictly convex Banach spaces. E...

We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces.

Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

Related to earlier work on existence results of best proximity points (pairs) for cyclic (noncyclic) condensing operators of integral type in the setting of reflexive Busemann convex spaces, in this paper, we introduce another class of cyclic (noncyclic) condensing operators and study the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in such spaces. Then, we present an application of our main existence result to study the existence of an optimal solution for a system of differential equations.

Let A and B be nonempty subsets of a Banach space X and T : A → B be a non-self mapping. An approximate sequence of best proximity points for the mapping T is a sequence {xn} in A such that limn→∞ ...

Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and $$(E,F)\subseteq (A,B)$$ be a nonempty and proximinal pair in X such that $$\mathrm{dist}(E,F)=\mathrm{dist}(A,B)$$. We prove that the pair $$(\overline{\mathrm{con}}(E),\overline{\mathrm{con}}(F))$$ is also proximinal, where $$\overline{\mathrm{con}}(E)$$ denotes the closed convex hull of the set E. Moreover, we introduce a new notion of cyclic (noncyclic) mappings involving measure of noncompactness and obtain some new existence results of best proximity points (pairs). As an application of our main conclusions, we study the existence of an optimal solution for a system of integrodifferential equations under new sufficient conditions.

In this chapter, we present some best proximity point theorems for Kannan cyclic mappings in the setting of Busemann convex spaces which are reflexive. To this end, we recall some results obtained in the framework of the fixed point theory for Kannan self mappings and generalize them to cyclic mappings in order to study the existence of best proximity points. We do it from two different approaches. The first one is based on a geometric property defined on a nonempty and convex pair in a geodesic space, called proximal normal structure, and the other one will be done by considering some sufficient conditions on the cyclic mappings. We also study the structure of minimal sets for Kannan cyclic nonexpansive mappings.

Given A and B two nonempty subsets of a metric space, a mapping T : A ∪ B → A ∪ B is noncyclic provided that T ( A ) ⊆ A and T ( B ) ⊆ B . A point ( p , q ) ∈ A × B is called a best proximity pair for the noncyclic mapping T if p = T p , q = T q and d ( p , q ) = dist ( A , B ) . In this article, we survey the convergence of Picard’s iteration to a best proximity pair for noncyclic contractions using a projection algorithm in uniformly convex Banach spaces, where the initial point is in the proximal set of A . We also provide some sufficient conditions to ensure the existence of a common best proximity pairs for a pair of noncyclic mappings.

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.

In this article, we consider the class of noncyclic Meir-Keeler contractions and study the existence and convergence of best proximity pairs for such mappings in the setting of complete CAT(0) spaces. We also discuss asymptotic pointwise noncyclic Meir-Keeler contractions in the framework of uniformly convex Banach spaces and generalize a main result of Chen [Chen C. M., A note on asymptotic pointwise weaker Meir-Keeler type contractions, Appl. Math. Lett., 2012, 25, 1267-1269]. Examples are given to support our main results.

New classes of mappings, called cyclic (noncyclic) condensing operators, are introduced and used to investigate the existence of best proximity points (best proximity pairs) with the help of a suitable measure of noncompactness. In this way, we obtain some real generalizations of Schauder and Darbo’s fixed point theorems. In the last section, we apply such results to study the existence of optimum solutions to a system of differential equations.

The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings,” Demonstr. Math. 53 (2020), 38–43.

We establish the existence of a solution for the following system of differential equations \begin{equation*} \label{system}\begin{cases}x'(t) = f (t, x(t)), & x(t_0) = x^*,\\ y'(t) = g (t, y(t)), & y(t_0) = x^{**}, \end{cases} \end{equation*} in the space of all bounded and continuous real functions on $[0,+\infty[$. We use best proximity point methods and measure of noncompactness theory under suitable assumptions on $f$ and $g$. Some new best proximity point theorems play a key role in the above result.

We establish a best proximity pair theorem for noncyclic φ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic φ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be concluded. As an application of our results, we study the existence of a global optimal solution for a system of ordinary differential equations.

In this article, we introduce a new class of noncyclic mappings called generalized pointwise noncyclic contractions, and we prove a best proximity pair theorem for this class of noncyclic mappings in the setting of strictly convex Banach spaces. Our conclusions generalize a result due to Kirk and Royalty. We also study convergence of iterates of noncyclic contraction mappings in uniformly convex Banach spaces.

Let T,S : A U B ? A U B be mappings such that T(A) ? B,T(B)? A and S(A) ? A,S(B)?B. Then the pair (T,S) of mappings defined on A[B is called cyclic-noncyclic pair, where A and B are two nonempty subsets of a metric space (X,d). A coincidence best proximity point p ? A U B for such a pair of mappings (T,S) is a point such that d(Sp,Tp) = dist(A,B). In this paper, we study the existence and convergence of coincidence best proximity points in the setting of convex metric spaces. We also present an application of one of our results to an integral equation.

Consider a self-mapping T defined on a union of two subsets A and B of a Banach space such that T(A) ⊆ B and T(B) ⊆ A. In this work we survey the existence of an optimal approximate solution, known as a best proximity point for a class of cyclic mappings, called cyclic Kannan nonexpansive mappings, in Banach spaces under appropriate sufficient conditions. In this order, the notion of T-uniformly semi-normal structure is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we conclude an old fixed point problem in Banach spaces which are not reflexive necessarily. Examples are given to support the usability of our main conclusions.Keywords: Best proximity point, fixed point, cyclic Kannan nonexpansive mapping, T-uniformly semi-normal structure, uniformly convex Banach space

In this article, we survey the existence, uniqueness and convergence of a common best proximity point for a cyclic pair of mappings, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Finally, we provide an extension of Edelstein’s fixed point theorem in strictly convex Banach spaces. Examples are given to illustrate our main conclusions.

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.