Generalized Solutions of Quasi-Variational-Like Problems

This paper deals with abstract quasi-variational-like problems, which cover vector quasi-variational problems with set-valued mappings, for instance. The main objective of this paper is to study existence results for generalized solutions of quasi-variational-like problems extending the approach by Jadamba, Khan and Sama, see (Optim. Lett. 6, 1221–1231, 2012). It is well-known that a quasi-variational-like problem can conveniently be formulated as a set-valued fixed-point problem by using the so-called variational selection. Since the values of the corresponding set-valued mapping are non-convex in general and since the existing fixed-point theorems for non-convex mappings require very stringent conditions on the data of the problem, a few existence results have been obtained in the case where the underlying constraining set is bounded or even compact. In this paper, we focus on the relationship between solutions of a quasi-variational-like problem and generalized solutions of an optimization problem minimizing the difference between inputs and outputs of the variational selection. Doing so, we can solve the latter problem by the well-known Weierstraß theorem with much less restrictive assumptions imposed on the given data. This approach allows to consider problems where the variational selection is non-convex and/or the underlying constraining set is unbounded.