Generalized Variational and Quasi-Variational Inequalities with Discontinuous Operators

Abstract

We study the existence of solutions to generalized variational and generalized quasi-variational inequalities with discontinuous operators. We obtain results which generalize and extend previously known theorems. We also compare our new continuity condition on the operator in the variational inequality to previously used continuity conditions. We then apply our results to generalized variational inequalities which involve pseudo-monotone operators. ᮊ 1997 Academic Press

Optimal Control of Generalized Quasi-Variational Hemivariational Inequalities and Its Applications

“…3) In recent years the theory of variational and quasi variational inequalities provided us with a convenient mathematical apparatus for studying a wide range of problems arising in numerous applications (see [8,9,11,15]). Optimal control problem for equations, variational and quasi variational inequalities is an expanding and vibrant branch of applied mathematics that has found numerous applications (see [1,[5][6][7][16][17][18]).…”

“…Lemma 2.6 ([15], Theorem 3.4) Let E be a separable Banach space and K be a nonempty closed convex subset of E. Suppose that F : K ⇒ E * and A : K ⇒ K and C is a nonempty compact convex subset of K such that the following hold:…”

Discontinuous Generalized Quasi-Variational Inequalities With Application to Fixed Points

“…In this direction, we refer to the papers [8, 9, 11-15, 26, 29, 32, 34-37] and to references therein. In particular, we refer to the paper [26,35] for discussions and comparisons about the basic regularity assumptions imposed on T .…”