Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces

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]

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“…In 2007, Khanh and Luu [12] obtained semi(upper)continuous of the approximate solution sets and solution sets of parametric multivalued QVI in topological vector spaces. Chen et al [13] first obtained a scalarization result of the ε-weakly efficient solution for a class of vector equilibrium problems under the Hausdorff topological vector space. Then, they proved the connectedness of the ε -efficient solution sets and ε-weakly efficient solution sets for this problem by applying the scalarization result.…”

Section: Introductionmentioning

confidence: 99%

An Approximation Theorem and Generic Convergence of Solutions of Inverse Quasivariational Inequality Problems

Liu

Jia

2022

Journal of Function Spaces

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In this paper, we mainly obtain an approximation theorem and generic convergence of solutions for inverse quasivariational inequality problems. First, we define the concept of the approximate solution to inverse quasivariational inequality problems under bounded rationality theory. Afterward, an approximation theorem that satisfies fairly mild assumptions is proved. Moreover, we establish a function space and discuss the convergence properties of solutions for inverse quasivariational inequality problems by the method of set-valued analysis. Finally, we prove that most of inverse quasivariational inequality problems are stable in the case of perturbation of the objective function. These results are new, which improve the corresponding outcomes of the recent literatures.

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Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces

Cited by 10 publications

References 29 publications

Vector Equilibrium Problems—A Unified Approach and Applications

Vector Equilibrium Problems—A Unified Approach and Applications

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

An Approximation Theorem and Generic Convergence of Solutions of Inverse Quasivariational Inequality Problems

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