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In this paper, a scalarization result of approximate weak efficient solution for a set-vector equilibrium problem (SVEP) is given. With this scalarization result, the connectedness of ε -weak efficient sets for the SVEP is proved under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper improve and generalize some known results in the literature [6,7,10]. 1.Introduction Vector equilibrium problems were first introduced by Blum and Oettli, It is well known that several problems such as vector optimization problems, fixed point problems, vector variational problems, and Nash economic equilibrium problems can be studied as particular cases of vector equilibrium problems. Vector equilibrium problems research mainly includes the existence of the solution, the stability of the solution sets, sensitivity analysis, topological properties of the set of solutions etc. The solution of the connectedness is an important aspect of topological properties, because it provides the possibility of continuously moving from one solution to any other solution. Recently, Lee et al. [2],Cheng [3] have studied the connectedness of weak efficient solutions set for vector variational inequalities in finite dimensional Euclidean space. Gong [4-6] has studied the connectedness of the various solutions set for VEPs in infinite dimension space. Chen et al. [7] studied the connectedness and the compactness of the weak efficient solutions set for set-valued VEPs and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space. Gong and Yao [8] have studied the connectedness of the set of efficient solutions for generalized systems. Zhong et al. [9] have studied the connectedness and path-connectedness of solutions set for symmetric VEPs. Chen et al. [10] studied the connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces .and Chen Bin et al have studied the connected of the single value of vector equilibrium problems. This paper inspired by the paper[6,7,10],studies the connectedness and compactness of approximate weak efficient sets for the set-valued vector equilibrium problem under different conditions ,thus promoting the conclusion of paper[6,7,10] 2nd International Conference on Electronics, Network and Computer Engineering (ICENCE 2016) © 2016. The authors Published by Atlantis Press 751

Connectedness of approximate weak efficient sets for the set-vector equilibrium problem