Coincidence Producing Operators on a Large Fixed Point Structure

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 coincidence producing operator on (X,S(X),M) if for each f ∈M(U) there exists u ∈ U such that f(u) = p(u). The problem is to study the existence and the properties of these class 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 généralization du théorème de 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. 8 ICFPTA (2007), Yokohama Publ., 2008). 2000 Mathematics Subject Classification: 47H10, 54H25, 55M20.