Asfaw, T. (2019). A variational inequality theory for constrained problems in reflexive Banach spaces. Advances in Operator Theory, 4(2), 462-480. doi: 10.15352/aot.1809-1423

Teffera M. Asfaw. "A variational inequality theory for constrained problems in reflexive Banach spaces". Advances in Operator Theory, 4, 2, 2019, 462-480. doi: 10.15352/aot.1809-1423

Asfaw, T. (2019). 'A variational inequality theory for constrained problems in reflexive Banach spaces', Advances in Operator Theory, 4(2), pp. 462-480. doi: 10.15352/aot.1809-1423

Asfaw, T. A variational inequality theory for constrained problems in reflexive Banach spaces. Advances in Operator Theory, 2019; 4(2): 462-480. doi: 10.15352/aot.1809-1423

A variational inequality theory for constrained problems in reflexive Banach spaces

^{}Department of Mathematics, Virginia Polytechnic Institute and State University, USA

Receive Date: 26 September 2018,
Revise Date: 12 October 2018,
Accept Date: 14 October 2018

Abstract

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.