Abstract
In this paper, we study the existence of a solution for a system of quasi-variational relation problems (in short, (SQVR)). Moreover, we discuss the existence of essentially connected components of the solution set for (SQVR). Then the obtained results are applied to systems of quasi-variational inclusions and to systems of weak vector quasi-equilibrium problems. The results presented in the paper improve and extend many results from the literature. Some examples are given to illustrate our results.MSC:47J20, 49J40.
Highlights
1 Introduction and preliminaries Variational relation problems were first introduced and studied by Luc in [ ]. These problems include as special cases variational inclusion problems, vector equilibrium problems, vector variational inequality problems and vector optimization problems, etc
Motivated by the research works mentioned above, in this paper, we introduce the system of generalized quasi-variational relation problems
([ ]) Let X and Z be two topological vector spaces and A ⊆ X be a nonempty convex set, and C ⊂ Z be a nonempty, closed, and convex cone
Summary
Introduction and preliminariesVariational relation problems were first introduced and studied by Luc in [ ]. Let Fi : Ki × Ki × Ki → Yi be a set-valued mapping, Ci ⊂ Yi be a nonempty, closed, and convex cone, and the relation Ri be defined as follows: Ri(xi, xi, yi) holds iff Fi(xi, xi, yi) ⊂ Ci. (SQVR) becomes the system of strong vector quasi-equilibrium problem (in short, (SSVQEP)) studied in [ ].
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