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Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces. (English) Zbl 1020.47050
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 $$E$$ over $$\mathbb{R}$$ and let $$F:C\to E$$ be an inner map such that $$F(C)$$ is a convex subset of $$E$$. Assume that the following conditions hold: (i) for each $$c\in C$$, $$F(c)$$ is nonempty and convex; (ii) for each $$y\in F(C)$$, $$F^-(y)$$ is open in $$C$$. Then there exists $$u\in C$$ such that $$u\in F(u)$$.
The main result of Section 4 is Theorem 4.1: Let $$E$$ be a Hausdorff topological vector space over $$\mathbb{R}$$ and let $$n\geq 2$$. Let $$C_1,\dots,C_n$$ be nonempty (not necessarily convex or compact) subsets of $$E$$, let $$K_1,\dots,K_n$$ be compact and convex subset of $$E$$, let $$S_1,\dots, S_n$$ be nonempty subsets of $$E^n$$, and let $$C=\prod^n_{j=1} C_j$$, $$K=\prod^n_{j=1}K_j$$, $$S=\bigcup^n_{j=1} S_j$$. Assume that the following properties hold: (i) $$C\subset K=S$$; (ii) for each $$i,1\leq i\leq n$$, and for each point $$(y_1,\dots,y_{i-1},y_{i+1},\dots,y_n)$$ of $$\prod^n_{j\neq i}K_j$$, the section $$S_i(y_1, \dots,y_{i-1},y_{i+1}, \dots, y_n)$$, formed by all points $$c_i\in C_i$$ such that $$(y_1,\dots, y_{i-1}, ci$$, $$y_{i+1}, \dots,y_n)\in S_i$$, is a nonempty convex subset of $$C_i$$; (iii) for each $$i$$, $$0\leq i\leq n$$, and each point $$c_i\in C_i$$, the section $$S_i(c_i)$$, formed by all points $$(y_1,\dots, y_{i-1}, y_{i+1}, \dots,y_n)$$ of $$\prod^n_{j \neq i}K_j$$ such that $$(y_1,\dots, y_{i-1},c_i, y_{i+1},\dots, y_n)\in S_i$$ is an open subset of $$\prod^n_{j\neq i}K_j$$. Then $$C\cap \bigcap^n_{i=1} S_i\neq \emptyset$$.
Reviewer: V.Popa (Bacau)

##### MSC:
 47H10 Fixed-point theorems 47H04 Set-valued operators
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