Coincidence producing operators on a large fixed point structure

Abstract

In this paper we investigate the following problem (see I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006, pp. 193-194): Let X be a set with structure and (X, S(X), M) be a large fixed point structure on X. Let U ∈ S(X). An operator p: U→U is by definition a coincidence-producing operator on (X, S(X), M) if for each f 2 M(U) there exists u 2 U such that f(u) = p(u). The problem is to study the existence and the properties of these classes of operators. For the case of topological space with fixed point property with respect to continuous operators, the starting papers are given by W. Holsztynski (Une generalization du theorem Brouwer Sur Les points invariants, Bull. Acad. Pol. Sc., 12(1964), 603-606) and by H. Schirmer (Coincidence producing maps onto trees, Canad. Math. Bull., 10(1967), 417-423) and for the case of metric spaces with fixed point property with respect to nonexpansive operators, by W.A. Kirk (Universal nonexpansive maps, 95-101, in Proc. 8th ICFPTA (2007), Yokohama Publ., 2008).