Tohoku Mathematical Journal 2014 June SECOND SERIES VOL. 66, NO. 2

 Tohoku Math. J. 66 (2014), 171-203

Title VARIATIONAL INEQUALITIES FOR PERTURBATIONS OF MAXIMAL MONOTONE OPERATORS IN REFLEXIVE BANACH SPACES

Author Teffera M. Asfaw and Athanassios G. Kartsatos

(Received May 17, 2012, revised May 1, 2013)
Abstract. Let $X$ be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space $X^*,$ and let $K$ be a nonempty, closed and convex subset of $X$ with $0$ in its interior. Let $T$ be maximal monotone and $S$ a possibly unbounded pseudomonotone, or finitely continuous generalized pseudomonotone, or regular generalized pseudomonotone operator with domain $K$. Let $\phi$ be a proper, convex and lower semicontinuous function. New results are given concerning the solvability of perturbed variational inequalities involving the operator $T+S$ and the function $\phi$. The associated range results for nonlinear operators are also given, as well asextensions and/or improvements of known results of Kenmochi, Le, Browder, Browder and Hess, De Figueiredo, Zhou, and others.

Mathematics Subject Classification. Primary 47H05.

Key words and phrases. Nonlinear, maximal monotone, pseudomonotone and strongly quasibounded operators, variational inequalities, existence problems.