New collectively fixed point theorems and applications in \(G\)-convex spaces
In the first part of the present paper, the authors establish several fixed point theorems for families of set-valued mappings defined on the product space of noncompact \(G\)-convex spaces. For the sake of illustration, we give one of these theorems. To this purpose, we need to recall two concepts. If \(A\) is a nonempty subset of a topological space \(X\), the compact interior of \(A\) is defined by cint\(A=\cup\{B\subset X:B\subset A\) and \(B\) is compactly open in \(X\}\). A subset \(Y\) of a \(G\)-convex space \((X,\Gamma)\) is said to be \(c\)-compact \(G\)-convex if for each nonempty finite subset \(N\) of \(X\) there is a compact \(G\)-convex subset of \(X\) containing \(Y\cup N\). In a simplified version, the aforementioned theorem can be formulated as follows: Let \(\{(X_i,\Gamma_i)\}_{i\in I}\) be a family of \(G\)-convex spaces. For each \(i\in I\), let \[ F_i,G_i:X^i= \prod_{j\not= i}X_j\rightarrow 2^{X_i} \] be two set-valued mappings such that: (a) for each \(x^i\in X^i\) and any nonempty finite subset \(N_i\) of \(F_i(x^i)\), \(\Gamma_i(N_i)\subset G_i(x^i)\); (b) \(X^i\cup_{y_i\in X_i} \operatorname {cint} F^{-1}_i(y_i)\); (c) there exists a \(c\)-compact \(G\)-convex set \(X_{i_0}\subset X_i\) such that \(X^i\setminus\cap_{y_i\in X_{i_0}} \operatorname{cint} F_i^{-1}(y_i)\) is empty or compact in \(X^i\). Then there exists a point \(\hat{x}\in X=\prod_{i\in I}X_i\) such that \(\hat{x}_i\in G_i(\hat{x}^i)\) for each \(i\in I\), where \(\hat{x}_i\) and \(\hat{x}^i\) denote the projection of \(\hat{x}\) onto \(X_i\) and \(X^i\), respectively. As applications of the obtained fixed point theorems, the authors obtain nonempty intersection theorems of Ky Fan type in \(G\)-convex spaces. Also they improve and extend to \(G\)-convex spaces known results concerning the compatibility of some systems of nonlinear inequalities, or the existence of the equilibrium points for some abstract economies.
scientific article; zbMATH DE number 1963363
scientific article