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Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

Abstract

In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12:259–274, 2004).

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Correspondence to Jen-Chih Yao.

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The authors wish to express their sincere appreciation to Professor Stephen M. Robinson, Department of Industrial and Systems Engineering, University of Wisconsin-Madison, for his valuable comments and suggestions. This research was partially supported by a grant from National Science Council of Taiwan, ROC.

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Kien, B.T., Yao, JC. Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces. Set-Valued Anal 16, 399–412 (2008). https://doi.org/10.1007/s11228-007-0058-4

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  • DOI: https://doi.org/10.1007/s11228-007-0058-4

Keywords

  • Generalized normal map
  • Variational inequality
  • Degree theory
  • Lower semicontinuity

Mathematics Subject Classifications (2000)

  • 47J20
  • 49J40
  • 49J53
  • 90C33