# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020002

## On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set

 College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing 400074, China

* Corresponding author: Zaiyun Peng

Received  June 2018 Revised  March 2019 Published  January 2020

Fund Project: The first and the fourth authors are supported by the Graduate Innovation Foundation of Chongqing Jiaotong University (2019S0123). The second author is supported by the National Natural Science Foundation of China (11301571), the Basic and Advanced Research Project of Chongqing (cstc2018jcyjAX0337), the Program for University Innovation Team of Chongqing (CXTDX201601022), the Innovation Project for Returned Overseas Scholars in Chongqing (cx2019148), the open project funded by the Chongqing Key Lab on ORSE (CSSXKFKTZ201801) and the Education Committee Project Foundation of Bayu Scholar. The third author is supported by the National Natural Science Foundation of China (11271389).

In this paper, we mainly discuss the stability of generalized vector quasi-equilibrium problems (GVQEPs) where the ordering relations are defined by free-disposal set. Firstly, by virtue of the oriented distance function $(\triangle)$, gap functions for (GVQEPs) are given and some properties of them are studied. Then, under some types of continuity assumption, the sufficient conditions of the upper semicontinuity and the upper Painlevé-Kuratowski convergence of solutions for (GVQEPs) are talked about. Moreover, sufficient and necessary conditions of the lower semicontinuity and the lower Painlevé-Kuratowski convergence of solutions for (GVQEPs) are obtained in normed linear spaces. Some examples are given to illustrate the results, and our results are new and extend some known results in the literature.

Citation: Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020002
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