Ekeland’s variational principle in weak and strong systems of arithmetic

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to Π11- CA0, a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma (WKL0) and to arithmetical comprehension (ACA0). We also find that the localized version of Ekeland’s variational principle is equivalent to Π11- CA0, even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

Caristi-Like Condition and the Existence of Minima of Mappings in Partially Ordered Spaces

In this paper, we study mappings acting in partially ordered spaces. For these mappings, we introduce a condition, analogous to the Caristi-like condition, used for functions defined on metric spaces. A proposition on the achievement of a minimal point by a mapping of partially ordered spaces is proved. It is shown that a known result on the existence of the minimum of a lower semicontinuous function defined on a complete metric space follows from the obtained proposition. New results on coincidence points of mappings of partially ordered spaces are obtained. © 2018, Springer Science+Business Media, LLC, part of Springer Nature