Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces
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Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces Jia-Wei Chen · Zhongping Wan
Received: 2 September 2012 / Revised: 12 May 2013 / Accepted: 17 May 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract This paper is devoted to the semicontinuity of solutions of a parametric generalized Minty vector quasivariational inequality problem with set-valued mappings [(in short (PGMVQVI)] in Hausdorff topological vector spaces, when the mapping and the constraint sets are perturbed by different parameters. The upper (lower) semicontinuity and closedness of the solution set mapping for (PGMVQVI) are established under some appropriate assumptions. The sufficient and necessary conditions of the Hausdorff lower semicontinuity and Hausdorff continuity of the solution set mapping for (PGMVQVI) are also derived without monotonicity. As an application, we discuss the upper semicontinuity for the solution set mapping of a special case of the (PGMVQVI). Keywords Lower (upper) semicontinuity · Closedness · Hausdorff continuity · Nonlinear scalarization function · Gap function · Parametric generalized Minty vector quasivariational inequality Mathematics Subject Classification
49J40 · 90C33
1 Introduction Vector variational inequality problems [in short (VVI)] were firstly introduced by Giannessi (1980) in finite-dimensional spaces. Since then, extensive study of (VVI) has been done in finite and infinite spaces (see Giannessi 1998, 2000; Facchinei and Pang 2003; Chen et al.
J.-W. Chen (B) School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China e-mail: [email protected] Z. Wan School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China e-mail: [email protected]
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2000, 2005a,b; Yang and Yao 2002; Chen and Wan 2011 and the references therein). (VVI) has been proved to be a very powerful tool of the current mathematical technology, which has been widely applied to transportation, finance and economics, mathematical physics, engineering sciences and so forth. In many situations, we always want to know the behavior of solution sets of VVI in practical problems when the problems’ data vary. The stability of the solution set mappings for vector variational inequality and optimization with perturbed data is of great importance in variational inequalities and optimization theory. Several variants of stability, such as semicontinuity, continuity, Hölder continuity, Lipschitz continuity and some kinds of differentiability of the solution set mapping, have been studied for variational inequalities and equilibrium problems (see Zhong and Huang 2010; Li and Chen 2009; Barbagallo and Cojocaru 2009; Yen 1995; Huang et al. 2006; Chen et al. 2011, 2012a, 2013; Wong 2010 and the references therein). Barbagallo and Cojocaru (2009) considered a class of scalar-type pseudo-monotone parametric variational inequalities in Banach space and sh
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