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

Lam Quoc Anh; Nguyen Van Hung

doi:10.3934/jimo.2017037

Article Contents

2018, Volume 14, Issue 1: 65-79. Doi: 10.3934/jimo.2017037

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

Lam Quoc Anh^1, , and
Nguyen Van Hung^2,3,

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
* Corresponding author

Received: October 31, 2015

Revised: January 31, 2017

Early access: March 2017

Published: January 2018

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C31, 91B50.

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