Existence criteria for the solutions of two types of variational relation problems (Q1949601)

Let \(X\) and \(Y\) be convex subsets of Hausdorff topological vector spaces and \(Z\) a topological space. Let \(S_1,S_2: X \multimap X\); \(T: X \multimap Y\) and \(P:X \multimap Z\) be multimaps and \(R(x,y,z)\) a ternary relation connecting elements \(x \in X,\) \(y \in Y\), and \(z \in Z\). The authors study the solvability of the following problems: [\((VRP1)\)] Find \((\overline{x},\overline{y}) \in X \times Y\) such that \(\overline{x} \in S_1(\overline{x}),\) \(\overline{y} \in T(\overline{x})\) and, for each \(u \in S_2(\overline{x}),\) there is \(z \in P(\overline{x})\) for which \(R(u,\overline{y},z)\) holds; [\((VRP2)\)] Find \((\overline{x},\overline{y}) \in X \times Y\) such that \(\overline{x} \in S_1(\overline{x}),\) \(\overline{y} \in T(\overline{x})\) and \(R(u,\overline{y},z)\) holds for each \(u \in S_2(\overline{x})\) and \(z \in P(\overline{x})\).