Maximal elements for \(G_B\)-majorized mappings in product \(G\)-convex spaces and applications. I (Q1888907)

Let \(Y\) be a topological space and denote by \(\mathcal{F}(Y)\) the set of nonempty finite subsets of \(Y\). If \(\Gamma:\mathcal{F}(Y)\to\mathcal{P}(Y)\setminus\{\emptyset\}\), then the pair \((Y,\Gamma)\) is called a \(G\)-convex space if for every \(M\in\mathcal{F}(Y)\) with \(\#(M)=n+1\) there is a continuous map \(\phi_M\) of the standard \(n\)-simplex \(\Delta_n\) into \(\Gamma(M)\) with the following property: If \(B\in\mathcal{F}(M)\) with \(\#(B)=\#(J)+1\) then \(\phi_M(\Delta_J)\subset\Gamma(B)\) where \(\Delta_J\) is the face of \(\Delta_n\) corresponding to \(J\). Let then \((Y,\Gamma)\) be \(G\)-convex and let \(X\) be a topological space. An upper semicontinuous map with compact values \(F:Y\to\mathcal{P}(Y)\) is said to belong to \(B(Y,X)\) if for any \(A\in\mathcal{F}(Y)\) with \(\#(A)=n+1\) and any continuous \(\psi:F(\Gamma(A))\to\Delta_n\) the mapping \(\psi\circ F| \;\phi_A(\Delta_n):\Delta_n\to\mathcal{P}(\Delta_n)\) has a fixed point. In this situation, a map \(A:X\to\mathcal{P}(Y)\) is said to be a \(G_B\)-mapping if \(F(\Gamma(N))\cap\bigcap_{y\in N}A^{-1}(\{y\})=\emptyset\) whenever \(N\in\mathcal{F}(Y)\) and if the intersection of \(A^{-1}(\{y\})\) with each compact subset of \(X\) is open whenever \(y\in Y\). The author proves the following theorem: Let \(X\) be a topological space, \((Y,\Gamma)\) \(G\)-convex, \(F\in B(Y,X)\) and \(A:X\to\mathcal{P}(Y)\) a \(G_B\)-mapping. Assume that there is a nonempty set \(Y_0\subset Y\) and that for each \(N\in\mathcal{F}(Y)\) there exists a \(G\)-convex subset \(L\subset Y\) with \(Y_0\cup N\subset L\) such that \(\bigcap_{y\in Y_0}[X\setminus A^{-1}(\{y\})]\) is compact. Then there is an \(x\in X\) such that \(A(x)=\emptyset\).