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.
"Variational inequalities for perturbations of maximal monotone operators in reflexive Banach spaces." Tohoku Math. J. (2) 66 (2) 171 - 203, 2014. https://doi.org/10.2748/tmj/1404911860