Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity

  • Nicuşor Costea
  • Daniel Alexandru Ion
  • Cezar Lupu


The aim of this paper is to establish existence results for some variational-like inequality problems involving set-valued maps, in reflexive and nonreflexive Banach spaces. When the set K, in which we seek solutions, is compact and convex, we no dot impose any monotonicity assumptions on the set-valued map A, which appears in the formulation of the inequality problems. In the case when K is only bounded, closed, and convex, certain monotonicity assumptions are needed: We ask A to be relaxed η-α monotone for generalized variational-like inequalities and relaxed η-α quasimonotone for variational-like inequalities. We also provide sufficient conditions for the existence of solutions in the case when K is unbounded, closed, and convex.


Variational-like inequalities Generalized variational-like inequalities Set-valued maps Generalized monotonicity Lower and upper semicontinuity 



This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-RU-PD-2011-3-0032.

We would like to thank the anonymous referees for valuable comments and suggestions which helped us improve the presentation of the paper.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Nicuşor Costea
    • 1
    • 2
  • Daniel Alexandru Ion
    • 3
  • Cezar Lupu
    • 3
    • 4
  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  2. 2.Department of Mathematics and Its ApplicationsCentral European UniversityBudapestHungary
  3. 3.Department of MathematicsUniversity of CraiovaCraiovaRomania
  4. 4.Department of MathematicsPolitehnica University of BucharestBucharestRomania

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