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.

In this paper, first we prove some lemmas, then by using the nonlinear scalarization function, we prove that there is no difference between the topological boundedness and the ordered boundedness. As an application, a fixed point theorem which is a new version of the main result obtained by Zangenehmehr et al (Positivity, 19, 333-340, 2015), by relaxing or weakening some assumptions, is established.

In this paper, a system of generalized operator equilibrium problems(for short, SGOEP) in the setting of topological vector spaces is introduced. Applying some properties of the nonlinear scalarization mapping and the maximal element lemma an existence theorem for SGOEP is proved. Moreover, using Ky Fan's lemma an existence result for the generalized operator equilibrium problem(for short, GOEP) is established. The results of the paper can be viewed as a generalization and improvement of the corresponding results given in [1, 2, 5, 8].