Journal of Optimization Theory and Applications

, Volume 167, Issue 3, pp 1136–1161 | Cite as

Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities

  • Akhtar A. Khan
  • Dumitru Motreanu


This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge’s fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.


Quasi-variational inequalities Variational inequalities Hemivariational inequalities Monotone Pseudo-monotone Generalized pseudo-monotone Coercivity  Asymptotic recessivity 

Mathematics Subject Classification

49J20 90C51 90C30 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Center for Applied and Computational Mathematics, School of Mathematical SciencesRochester Institute of TechnologyRochesterUSA
  2. 2.Département de MathématiquesUniversité de PerpignanPerpignanFrance

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