Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces.

*(English)*Zbl 1020.47050Let \(E\) 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 \(C\cap \bigcap^n_{i=1} S_i\neq \emptyset\).

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)