Stability of solution mappings for parametric bilevel vector equilibrium problems
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Stability of solution mappings for parametric bilevel vector equilibrium problems Lam Quoc Anh1 · Nguyen Van Hung2,3
Received: 28 February 2016 / Revised: 29 June 2016 / Accepted: 9 December 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this paper, we first revisit the parametric bilevel vector equilibrium problems in Hausdorff topological vector spaces. Then we study the stability conditions such as (Hausdorff) upper semicontinuity, (Hausdorff) lower semicontinuity, outer-continuity and outer-openness of solutions for such problems. Many examples are provided to illustrate the essentialness of the imposed assumptions. For the applications, we obtain the stability results for the parametric vector variational inequality problems with equilibrium constraints and parametric vector optimization problems with equilibrium constraints. Keywords Bilevel vector equilibrium problem · Variational inequality with equilibrium constraints · Optimization problems with equilibrium constraints · Upper (lower) semicontinuity · Outer-continuity · Outer-openness Mathematics Subject Classification 90C31 · 49J40 · 49J53
1 Introduction Equilibrium problem was first introduced and investigated by Blum and Oettli (1994). Then this problem has been focused and intensively studied in many topics such as the existence of solutions, the stability of solutions and the (unique) well posedness of approximate solutions
Communicated by José Mario Martínez.
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Nguyen Van Hung [email protected] Lam Quoc Anh [email protected]
1
Department of Mathematics, Teacher College, Can Tho University, Can Tho, Vietnam
2
Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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L. Q. Anh, N. V. Hung
in the sense of Hadamard and Tikhonov. In recent years, there has been an increasing interest in studying the existence conditions, the well posedness and the solution methods for bilevel equilibrium problems (see, e.g., Ding 2012, 2010; Moudafi 2010; Chen et al. 2014; Outrata 2000; Mordukhovich 2009; Bao et al. 2007; Wangkeeree and Yimmuang 2012; Kimura et al. 2008; Anh et al. 2012). Stability of solutions is an important topic in optimization theory and applications. Up to now, there have been many works dealing with stability conditions for optimization-related problems as optimization problems (see, e.g., Loridan 1984; Zhao 1997), vector variational inequality problems (see, e.g., Khanh and Luu 2005, 2007; Lalitha and Bhatia 2011), vector quasiequilibrium problems (see, e.g., Anh and Khanh 2004, 2006, 2007, 2008a, b, c, 2009, 2010; Kimura and Yao 2008; Li and Li 2011; Bianchi and Pini 2003, 2006; Anh et al. 2014), variational relation problems (see, e.g., Hung 2012a, b; Khanh and Luc 2008). To the best of our knowledge, there have not been any works on stability conditions for bilevel vector equilibrium problems. In this paper, we consider t
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