On the convexity of the solution set of symmetric vector equilibrium problems

Abstract

In this paper, without assumption of monotonicity and boundedness, we study existence results for a solution and the convexity of the solution set to the symmetric vector equilibrium problem for setvalued mappings in the setting of topological vector spaces. Our results improve the corresponding results in [9, 18, 19, 22, 28, 33, 36, 37].

On a new generalized symmetric vector equilibrium problem

“…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.…”

“…Moreover, we show that Lemma 2.5 and Ky Fan’s lemma are not equivalent. 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 (see, e.g., [ 1 , 19 ] and the references therein), when we work with the scalarization mapping, while Lemma 2.5 is a useful tool for proving our existence results.…”

“…Recently, Farajzadeh [ 1 ] established some existence results for a solution of SVEP whose proof strongly depends on non-triviality of the dual cones, lower semicontinuity and quasiconvexity of the mappings (see Theorem 3.4 in [ 1 ]). In this paper, we first introduce a general form of SVEP (called GSVEP), and then we relax the non-triviality of the cones, lower semicontinuity and quasiconvexity that appeared in Theorem 3.4 in [ 1 ] by ‘mixing’ the properties of the nonlinear scalarization mapping and the maximal element lemma (see Lemma 2.5 and Theorem 3.1 ). It is worth noting that the hypotheses of lower semicontinuity and quasiconvexity on the mappings are common assumptions in most of papers that have appeared in this area.…”

“…GSVEP is a generalization of all the following problems: If we take , then GSVEP collapses to the symmetric vector equilibrium problem; see [ 1 ] and the references therein. If we take , ; and are two single-valued mappings, and and then ( 1.1 ) reduces to the symmetric vector equilibrium problem which was studied in [ 10 ].…”

“…If we take , then GSVEP collapses to the symmetric vector equilibrium problem; see [ 1 ] and the references therein.…”