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.
Citation
T. M. Asfaw. "A variational inequality theory for constrained problems in reflexive Banach spaces." Adv. Oper. Theory 4 (2) 462 - 480, Spring 2019. https://doi.org/10.15352/aot.1809-1423
Information