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

Author

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  • Nicuşor Costea

    (Institute of Mathematics “Simion Stoilow” of the Romanian Academy
    Central European University)

  • Daniel Alexandru Ion

    (University of Craiova)

  • Cezar Lupu

    (University of Craiova
    Politehnica University of Bucharest)

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.

Suggested Citation

  • Nicuşor Costea & Daniel Alexandru Ion & Cezar Lupu, 2012. "Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 79-99, October.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:1:d:10.1007_s10957-012-0047-0
    DOI: 10.1007/s10957-012-0047-0
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    References listed on IDEAS

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    1. I. V. Konnov & S. Schaible, 2000. "Duality for Equilibrium Problems under Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 395-408, February.
    2. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    3. M. Bianchi & N. Hadjisavvas & S. Schaible, 2004. "Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 1-17, July.
    4. Y.P. Fang & N.J. Huang, 2003. "Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 327-338, August.
    5. Nicuşor Costea & Vicenţiu Rădulescu, 2012. "Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term," Journal of Global Optimization, Springer, vol. 52(4), pages 743-756, April.
    6. Q. H. Ansari & J. C. Yao, 2001. "Iterative Schemes for Solving Mixed Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 108(3), pages 527-541, March.
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    Cited by:

    1. Bijaya Kumar Sahu & Ouayl Chadli & Ram N. Mohapatra & Sabyasachi Pani, 2020. "Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 810-823, March.

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