Abstract
This work is concerned with several questions which arise in the study of selfmaps of metric and Banach spaces. If (X, d) is a compact metric space and f a continuous selfmap, the behavior of the set of subsequential limit points is studied. Several results pertaining to the structure and invariance (under f) of the set of subsequential limit points for a given element of the space are obtained. An example which provides an answer to a question posed by Metcalf and Rogers is included; namely, it is shown that the derived set, L'(x), of a given set of subsequential limit points, L(x), is not necessarily invariant. Sufficient conditions to insure that L'(x) will be invariant are provided. Some questions posed by Nadler and by Bryant and Guseman are studied. We generalize a result of Nadler by showing that the iterative test of Edelstein is conclusive for the class of chainable metric spaces which are locally compact. This theorem is employed to provide an answer to a question of Bryant and Guseman; specifically, we show that the iterative test conclusive does not imply the contractive extension property. By introducing the remetrization contractive extension property, we are able to characterize those dense subsets of locally compact, complete chainable metric spaces which have the Iterative test conclusive. Finally, the concept of an attractor, as defined by Nussbaum, for compact sets in a topological space is introduced. It is shown that an attractor for compact sets under a continuous selfmap f of a closed convex subset G of a Banach space has a unique invariant (under f) component. It is then shown that for any x in G, L(x) [does not equal] 0 and furthermore, L(x) is contained in the invariant component. The unique invariant component of an attractor for compact sets is shown to be an attractor for points.
Solomon, Jimmy Lloyd (1972). Some results in nonlinear fixed point theory. Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -186155.