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We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a,b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space. Keywords. Best proximity pair; bipartite graph; complete bipartite graph; proximinal set; semimetric space; ultrametric space., 1. Introduction and preliminaries Let X be a set. A semimetric on X is a function d: X * X [right arrow] [0, [infinity]) such that d(x,y) = d(y,x) and [...]

Abstract In this paper, we use Suzuki-type contraction to prove three fixed point theorems for generalized contractions in an ordered space equipped with two metrics; we obtain some generalizations of the Kannan fixed point theorem. Our results on partially ordered metric spaces generalize and extend some results of Ran and Reurings as well as of Nieto and Rodríguez-López. To illustrate the effectiveness of our main result, we give an application to matrix equations which involves monotone mappings.

The aim of this paper is to discuss Penot’s problem on a generalization of Caristi’s fixed point theorem. We settle this problem in the negative and we present some new theorems on the existence of fixed points of set-valued mappings in ordered metric spaces and reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some fixed point results for noncyclic monotone ρ -nonexpansive mappings in uniformly convex modular spaces and uniformly convex in every direction modular spaces. As an application, we prove the existence of the solution of an integral equation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish fixed point results for a single mapping in a partially ordered modular metric space, and to prove a common fixed point theorem for two self-maps satisfying some weak contractive inequalities. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a generalized version of Caristi fixed point theorems in pseudo-metric spaces. Our results generalize and improve many of well-known theorems. As an application of our results, we give a new existence theorem to the generalized nonlinear complementarity problem and a solution of differential inclusion in the distributions setting. [ABSTRACT FROM AUTHOR]

In this paper, we present some related Caristi type fixed point theorems for multivalued maps in complete metric spaces. As application, we establish a new version of the ε-variational principle. Examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

We prove a fixed point theorem for some contraction mapping in complete quasi-metric space with w-distance, and a common fixed point theorem for two and three self mappings. [ABSTRACT FROM AUTHOR]

We give some generalizations of the B. Fischer fixed point theorem (B. Fisher [7]) for two mappings on metric spaces by using a function α defined from [0,+∞[ into [0, 1[ and satisfies lim supt→to α(t) < 1; for all to ≤ 0: We study also the existence of solutions for a functional equation arising in dynamic programming. [ABSTRACT FROM AUTHOR]