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Minimax Theory and its Applications 09 (2024), No. 1, 019--040
Copyright Heldermann Verlag 2024

Brézis-Pseudomonotone Mixed Equilibrium Problems Involving a Set-Valued Mapping with Application

Ouayl Chadli
(1) Dept. of Mathematics, University of Central Florida, Orlando, U.S.A.
(2) Ibn Zohr University, Agadir, Morocco

Ram N. Mohapatra
Dept. of Mathematics, University of Central Florida, Orlando, U.S.A.

Bijaya Kumar Sahu
Dept. of Mathematics, Chandbali College, Bhadrak, Odisha, India

We study the existence of solutions for quasi mixed equilibrium problems involving a set-valued mapping in topological spaces. In case of Banach spaces, we find its strong solutions using (η,g,f)-pseudomonotone mappings. The approach developed in this paper is completely different from most of the techniques used in literature for the study of similar problems, it is based on the notion of pseudomonotonicity in the sense of Brézis for bifunctions in addition to standard use of finite intersection property of compact sets and fixed point theorems. A recent paper by D. Steck [Brezis pseudomonotonicity is strictly weaker than Ky-Fan hemicontinuity, J. Optim. Theory Appl. 181 (2019) 318--323] has applied this notion and showed that it is strictly weaker than the notion called Ky Fan hemicontinuity, which has been used in many recent works in the literature related to the problem studied in this paper. The results obtained in this paper are new and they improve considerably many existing results in the literature. As an application, we study the existence of solutions of a generalized nonlinear hemivariational inequality problem involving a set-valued operator.

Keywords: Set-valued variational inequalities, equilibrium problems, set-valued mapping, pseudomonotonicity, hemivariational inequalities.

MSC: 47H04, 47J20, 49J40, 47H10, 46N10.

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