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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700, Bucharest, Romania

    Nicuşor Costea

  2. Department of Mathematics and Its Applications, Central European University, Nador u. 9, 1051, Budapest, Hungary

    Nicuşor Costea

  3. Department of Mathematics, University of Craiova, Al. I. Cuza Str. 13, 200585, Craiova, Romania

    Daniel Alexandru Ion & Cezar Lupu

  4. Department of Mathematics, Politehnica University of Bucharest, Str. Splaiul Independenţei 313, 060042, Bucharest, Romania

    Cezar Lupu

Authors
  1. Nicuşor Costea
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  2. Daniel Alexandru Ion
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  3. Cezar Lupu
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Correspondence to Nicuşor Costea.

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Communicated by Qamrul Hasan Ansari.

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Costea, N., Ion, D.A. & Lupu, C. Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity. J Optim Theory Appl 155, 79–99 (2012). https://doi.org/10.1007/s10957-012-0047-0

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