Abstract
In the first part of this chapter, we investigate the problem of existence of solutions of quasi-hemivariational inequalities. Some concepts of semicontinuity and hemicontinuity on subsets for functions as well as for set-valued mappings are developed and applied for solving quasi-hemivariational inequalities. Generalizations of some old results on the existence of solutions of equilibrium problems are obtained and applications to quasi-hemivariational inequalities are derived. Next, we are concerned with set-valued equilibrium problems under mild conditions of continuity and convexity on subsets recently introduced in the literature. We obtain that neither semicontinuity nor convexity are needed on the whole domain when solving set-valued and single-valued equilibrium problems. As applications, we derive some existence results for Browder variational inclusions, and we extend the well-known Berge maximum theorem in order to obtain versions of the Kakutani and Schauder fixed point theorems.
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