Fixed point theorems in linear topological spaces

Abstract

In this work some notions involving continuous mappings of a linear topological space are developed. It is shown that if K is a compact, starshaped subset of a linear topological space and f: K a?Ç K is nonexpansive with respect to a translation invariant, strictly monotone metric, then f has a fixed point. If the metric d of the space is monotone, but not strictly monotone, then the condition d(tf(x), tf(y)) [less than or equal to] d(tx, ty), for all x,y [epsilon] K and t [epsilon] [0, 1] insures that f is nonexpansive with respect to an equivalent strictly monotone metric. The first result mentioned is a consequence of a lemma which generalizes a theorem of D. R. Smart on approximating the identity mapping of a compact metric space by strictly contraceptive mappings. A theorem of W. G. Dotson, Jr. follows as a corollary of these results. In order to extend these results to spaces which are not necessarily metrizable, the ideas of an absorbing base U of neighborhoods of [theta], U-nonexpansive mappings, and radical contractions are introduced. An analog of the Branch contraction mapping theorem is obtained for radical contractions defined on a subset of a linear topological space. It is shown that U-nonexpansive mappings of countably compact, starshaped subsets of linear topological spaces have fixed points. Given a bounded subset B of a linear topological space E with an absorbing base U of neighborhoods of [theta], we define the subsets L(B) and M(B) of U so that they have a close relationship to Kuratowski's set measure of noncompactness. Using these concepts we are able to extend the ideas of set condensing maps and ball condensing maps to linear topological spaces. A number of Banach space results due to Furi and Vignoli, Sadovskii, Darbo and others are then extended.