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.
Peters, Burnis Charles (1973). Fixed point theorems in linear topological spaces. Doctoral dissertation, Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -157619.