Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces
type: Item
instance of: scholarly article
title: Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces
zbMATH Open document ID: 1020.47050
DOI: 10.1155/S1085337503207028
author: Kazimierz Włodarczyk
author: Dorota Klim
published in: Abstract and Applied Analysis
publication date: 2003-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/50305
review text: Let \(E\) be a Hausdorff topological vector space, \(C\subset E\), \(C\neq \emptyset\). A multivalued map \(F:C\to E\) is said to be expansive (resp. inner) if \(C\subset F(C)\) (resp. \(F(C)\subset C)\). In this paper, using a technique based on the investigation of the image of maps, the authors obtain a number of new fixed point, coincidence, intersection, and section theorems of Fan-Browder type. Examples and counterexamples show a fundamental difference between the author's results and the known results of other authors. The paper contains 7 sections: 1. Introduction; 2. Fixed points and coincidences of expansive set-valued maps on not necessarily convex or compact sets in topological vector spaces; 3. Fixed points and coincidences of set-valued inner maps on not necessarily convex sets in topological vector spaces; 4. Intersection theorem with applications on not necessarily convex or compact sets in topological vector spaces; 5. Coincidence theorems for set-valued maps and section theorems on not necessarily convex sets in topological vector spaces; 6. Coincidences for upper semicontinuous set-valued maps on not necessarily convex sets in locally convex spaces; 7. Coincidences and fixed points for continuous single-valued maps on not necessarily convex sets in locally convex spaces. Many theorems are proved. For example, the main result of Section 2 is Theorem 2.1: Let \(C\) be a nonempty subset of a Hausdorff topological vector space \(E\) over \(\mathbb{R}\), let \(F:C\to E\), and let \(K\) be a convex subset of \(E\). Assume that the following conditions hold: (i) \(C\subset K\subset F(C)\); (ii) \(F(C)\) is a compact subset of \(E\); (iii) for each \(c\in C\), \(F(c)\) is open in \(F(C)\); (iv) for each \(y\in K\), \(F^{-1}(y)= \{c\in C:y \in F(c)\}\) is nonempty and convex. Then there exists \(u\in C\) such that \(u\in F(u)\). The main result of Section 3 is Theoren 3.1: Let \(C\) be a nonempty compact subset of a Hausdorff topological vector space \(E\)…
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 1883731
zbMATH Keywords: fixed point
zbMATH Keywords: coincidence point
zbMATH Keywords: set valued mapping
zbMATH Keywords: topological vector space
MaRDI profile type: Publication
OpenAlex ID: W2049974190
description: scientific article; zbMATH DE number 1883731