Inequality Problems Governed by Set-valued Maps of Monotone Type

Abstract

Various classes of inequality problems driven by set-valued maps are investigated throughout this chapter via topological methods such as the Fan-KKM lemma, Tarafdar’s fixed point theorem or Mosco’s alternative. The standard approach is to establish the existence of at least one solution for the case of bounded closed convex constraint subsets of (not necessarily reflexive) Banach spaces, then to derive coercivity conditions that ensure the existence of solution for unbounded subsets. We also consider some variational-like inequality problems for which the KKM approach fails, but we are still able to prove the existence of at least solution.