On an open question concerning fixed points

Abstract

A space X is said to have the f.p.p. (fixed point property) if every continuous function f from X to X has a fixed point. Whether if X and Y have the f.p.p. then XX Y has the f.p.p. is an open question. A space X is said to have the F.p.p. (fixed point property for multivalued functions) if every continuous multi-valued function F from X to X has a fixed point, i.e., a point x such that xGF(x). Interest in fixed points for multi-valued functions leads one to question under what conditions on the spaces X and Y and on the multi-valued function F on X to Y there will exist a continuous trace f of F, that is, a continuous function f on X to Y such that f(x) E F(x) for all x. For some specific multi-valued functions F it is possible to produce a continuous trace. It is by use of these traces that most of the fixed point theorems in the literature for multi-valued functions are proved. In fact the open question mentioned above (which is concerned only with single-valued functions) can be answered if one can produce continuous traces of two particular multi-valued functions. This paper proves some fixed point theorems by producing continuous traces, shows that a continuous multi-valued function need not have a continuous trace, and gives an example which indicates that a general theorem on the existence of a continuous trace is not likely to be established without strong conditions on F regardless of what conditions are placed on X and Y. This example answers in the negative the generalization of the above open question to the multi-valued case, exhibits a continuous multi-valued function which has no continuous trace, shows that the general Tychonoff cube does not have the F.p.p., and shows that a space with the f.p.p. need not have the F.p.p. NOTATION. By {1x }-*xo we denote a sequence of points indexed by a directed set A and converging to xo. The directing relation in A will be denoted by *. DEFINITION 1. Continuous. A multi-valued function on a space X to a space Y is said to be continuous at xo if {Xa } -+x0 implies that F(xo) = cofinal limit { F(xa) } = residual limit { F(Xa) }. F is said to be