English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps. The aim of this note is to extend the Leray-Schauder type nonlinear alternative presented in [BI] to a condensing upper semicontinuous approximable set-valued map F: X -> E when X is a closed subset with nonempty interior of a locally convex topological vector space E. The proof presented here is even shorter and simpler than the one given in [BI]. In what follows E stands for a Hausdorff locally convex topological vector space with a fundamental basis JK. of convex, symmetric neighborhoods of the origin; if X, Y are nonempty subsets of E, then F: X -Y is a set-valued map with nonempty values (simply called map). The boundary, the interior, the closure, and the convex hull of a subset A in E are denoted by AA, int A, A, and co A respectively. Definition 1. F is said to be upper semicontinuous (u.s.c.) on X if and only if for any open subset V of Y, the set {x E X: F(x) C V} is open in X. Definitions 2 ([BD], [BI], see also [GGK] for metric spaces). (1) Given U, V E X, a function s: X -+ Y is said to be a (U, V)-approximative selection of F if for any xE X, S(x) E (F[(x+ U) nX] +V) n Y. (2) F: X -* Y is said to be approachable if it has a continuous (U, V)approximative selection for any (U, V) E KV x JK. A(X, Y) denotes the class of such maps. We write .A(X) for A(X, X). (3) F is said to be approximable if its restriction FIK to any compact subset K of X is approachable. Note that an approachable map is approximable (cf. [B]). Examples. It is well-known that if F is u.s.c. with nonempty convex values, then F is approachable provided X is paracompact and Y is convex (cf. [DG]). Obviously, F is approximable without conditions on X (see [C]). Received by the editors August 12, 1996 and, in revised form, January 16, 1997. 1991 Mathematics Subject Classification. Primary 47H04, 47H10, 54C60.

A Leray-Schauder type theorem for approximable maps: A simple proof