Set-Valued Analysis

, Volume 16, Issue 4, pp 399–412 | Cite as

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

  • Bui Trong Kien
  • Jen-Chih YaoEmail author


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).


Generalized normal map Variational inequality Degree theory Lower semicontinuity 

Mathematics Subject Classifications (2000)

47J20 49J40 49J53 90C33 


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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan

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