Existence of solutions for nonlinear inequalities in \(G\)-convex spaces
For a nonempty set \(Y\), \(2^Y\) and \(\left\langle Y\right\rangle\) denote the class of all subsets of \(Y\) and the class of all nonempty finite subsets of \(Y\), respectively. A generalized convex space (briefly, a \(G\)-convex X space) (\(X, \Gamma\)) consists of a topological space \(X\) and a map \(\Gamma : \left\langle X\right\rangle\rightarrow 2^X\) satisfying the following condition: for each \(A\in\left\langle X\right\rangle\) with \(|A|=n+1\), there exists a continuous mapping \(\Phi_A:\Delta_n\rightarrow\Gamma (A)\) such that for each \(J\in\left\langle A\right\rangle\), \(\Phi_A(\Delta_J)\subset\Gamma (J)\), where \(\Delta_n\) is the standard \(n\)-dimensional simplex and \(\Delta_J\) is the face of \(\Delta_n\) corresponding to \(J\). Given a \(G\)-convex space (\(X, \Gamma\)) and two bifunctions \(\Phi,\Psi:X\times X\rightarrow\mathbb R\), the authors of this paper study the following abstract nonlinear problem, of finding \(\overline{x}\in X\) such that \(\Phi (\overline{x},y)\geq\Psi (\overline{x},y)\) for all \(y\in Y\). Using a fixed-point theorem in \(G\)-convex spaces, the authors establish two existence theorems of solutions for the abstract nonlinear problem above. As consequence, an existence theorem for the perturbed saddle point problem in noncompact \(G\)-convex spaces is obtained.
scientific article; zbMATH DE number 1794939