Advances in Operator Theory

A variational inequality theory for constrained problems in reflexive Banach spaces

T. M. ‎‎Asfaw

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‎‎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.

Article information

Adv. Oper. Theory, Volume 4, Number 2 (2019), 462-480.

Received: 26 September 2018
Accepted: 14 October 2018
First available in Project Euclid: 1 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]
Secondary: 47H14: Perturbations of nonlinear operators [See also 47A55, 58J37, 70H09, 70K60, 81Q15]

variational inequality‎ ‎ compact resolvent‎ ‎ ‎constrained problems‎‎ ‎ ‎elliptic and parabolic problems


‎‎Asfaw, T. M. A variational inequality theory for constrained problems in reflexive Banach spaces. Adv. Oper. Theory 4 (2019), no. 2, 462--480. doi:10.15352/aot.1809-1423.

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