Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Multivalued Usco Functions and Stegall Spaces

In this article we consider the study of the -differentiability and -ifferentiability for convex functions, not only in the general context of topological vector spaces (), but also in the context of Banach spaces. We study a special class of Banach spaces named Stegall spaces, denoted by , which is located between the Asplund -spaces and Asplund -spaces (-Asplund). We present a self-contained proof of the Stegall theorem, without appealing to the huge number of references required in some proofs available in the classical literature (4). This requires a thorough study of a very special type of multivalued functions between Banach spaces known as usco multi-functions.