Variational-like inequality problems involving set-valued maps and generalized monotonicity
The authors extend results of the paper by \textit{J. Parida, M. Sahoo} and \textit{S. Kumar} in [Bull. Aust. Math. Soc. 39, No. 2, 225--231 (1989; Zbl 0649.49007)] concerning variational-like inequalities generalizing the set \(K\) for the solutions to a nonempty, closed and convex set in the bidual of a Banach space \(X\), generalizing the operator \(A\) in the duality pairing into a set-valued map from \(K\) to the dual of \(X\) and generalizing the continuous map in the right side of the duality product to a mapping having its images in the bidual of \(X\). To prove existence of solutions Ky-Fan's version of the KKM principle, Mosco's alternative and a fixed-point theorem are used. The operator \(A\) must fulfill lower semicontinuity or upper semicontinuity conditions. If some compactness conditions hold, then existence of solutions follows for general non-reflexive Banach spaces. If there are no compactness conditions, special additional monotonicity conditions for the dates of the inequality are necessary. So the last big part of the paper is devoted to study the application of suitable generalized monotonicity conditions to prove existence of solutions in reflexive Banach spaces. Examples and remarks give hints that the results extend theorems of other papers.
scientific article; zbMATH DE number 6132205
scientific article