# American Institute of Mathematical Sciences

January  2018, 14(1): 65-79. doi: 10.3934/jimo.2017037

## Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

 1 Department of Mathematics, Teacher College, Can Tho University, Can Tho, Viet Nam 2 Center of Research and Development Duy Tan University K7/25, Quang Trung, Danang, VietNam 3 Department of Mathematics, Dong Thap University, Dong Thap, Viet Nam

* Corresponding author

Received  November 2015 Revised  February 2017 Published  April 2017

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.18.

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

Citation: Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
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