Abstract
In this paper, a new form of the symmetric vector equilibrium problem is introduced and, by mixing properties of the nonlinear scalarization mapping and the maximal element lemma, an existence theorem for it is established. We show that Ky Fan’s lemma, as a usual technique for proving the existence results for equilibrium problems, implies the maximal element lemma, while it is useless for proving the main theorem of this paper. Our results can be viewed as an extension and improvement of the main results obtained by Farajzadeh (Filomat 29(9):2097-2105, 2015) and some corresponding results that appeared in this area by relaxing the lower semicontinuity, quasiconvexity on the mappings and being nontrivial of the dual cones. Finally, some examples are given to support the main results.
Highlights
Introduction and preliminariesExistence results for vector equilibrium problems have been extensively studied in recent years
We first introduce a general form of SVEP, and we relax the non-triviality of the cones, lower semicontinuity and quasiconvexity that appeared in Theorem . in [ ] by ‘mixing’ the properties of the nonlinear scalarization mapping and the maximal element lemma
It is a remarkable fact that we cannot apply Ky Fan’s lemma, which plays an important role in the study of the existence results of equilibrium problems, when we work with the scalarization mapping, while Lemma . is a useful tool for proving our existence results
Summary
Introduction and preliminariesExistence results for vector equilibrium problems (in short, VEP) have been extensively studied in recent years. One of the important symmetric vector equilibrium problems is to investigate the existence theorems in order to guarantee its solution set is nonempty (see, e.g., [ – ] and the references therein).
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