EXISTENCE THEOREMS FOR ELLIPTIC QUASI-VARIATIONAL INEQUALITIES IN BANACH SPACES

Abstract

A class of quasi-variational inequalities (QVI) of the elliptic type is studied in Banach spaces. The concept of QVI was ealier introduced by A. Bensoussan and J. L. Lions [2] and its general theory was evolved by many mathematicians, for instance, see [7,9,13] and a monograph [1]. In this paper we give a generalization of the existence theorem due to J. L. Joly and U. Mosco [6,7] from not only the view-point of the nonlinear operator theory, but also the application to nonlinear variational inequalities including partial differential operators. In fact, employing the compactness argument based on the Mosco convergence (cf.[11]) for convex sets and the graph convergence for nonlinear operators, we shall prove an abstract existence result for our class of QVI’s. Moreover we shall give some new applications to QVI’s arising in the material science.