A variational inequality theory for constrained problems in reflexive Banach spaces

Document Type: Original Article


Department of Mathematics, ‎Virginia Polytechnic Institute and State University, USA


‎Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^*$ and $K$ be a nonempty‎, ‎closed and convex subset of $X$‎. ‎Let $T‎: ‎X\supseteq D(T)\to 2^{X^*}$ be maximal monotone‎, ‎$S‎: ‎K\to 2^{X^*}$ be bounded and of type $(S_+)$ and $C‎: ‎X\supseteq D(C)\to X^*$ 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.


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