Existence of Minimizers

Abstract

In this chapter, we present sufficient conditions for an extended real-valued function to have minimizers. After discussing the main concepts, we begin by addressing the existing issue in abstract Hausdorff spaces, under certain (one-sided) continuity and compactness hypotheses. We also present Ekeland’s Variational Principle, providing the existence of approximate minimizers that are strict in some sense. Afterward, we study the minimization of convex functions in reflexive spaces, where the verification of the hypothesis is more practical. Although it is possible to focus directly on this setting, we preferred to take the long path. Actually, the techniques used for the abstract framework are useful for problems that do not fit in the convex reflexive setting, but where convexity and reflexivity still play an important role.