A variational inequality theory for constrained problems in reflexive Banach spaces

Abstract

‎‎Let $X$ be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space $X^*$‎, ‎and let $K$ be a nonempty‎, ‎closed‎, ‎and convex subset of $X$‎. ‎Let $T‎: ‎X\supseteq D(T)\to 2^{X^*}$ be maximal monotone‎, ‎let $S‎: ‎K\to 2^{X^*}$ be bounded and of type $(S_+)$‎, ‎and let $C‎: ‎X\supseteq D(C)\to X^*$ with $D(T)\cap D(\partial \phi)\cap K\subseteq D(C)$‎. ‎Let $\phi‎ : ‎X\to (-\infty‎, ‎\infty]$ be a proper‎, ‎convex‎, ‎and lower semicontinuous function‎. ‎New existence theorems are proved for solvability of variational inequality problems of the type $\rm{VIP}(T+S+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $C$ is compact and $\rm{VIP}(T+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $T$ is of compact resolvent and $C$ is bounded and continuous‎. ‎Various improvements and generalizations of the existing results for $T+S$ and $\phi$ are obtained‎. ‎The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems‎.