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Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control

Author

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  • Shengda Zeng

    (Yulin Normal University
    Jagiellonian University in Krakow)

  • Dumitru Motreanu

    (Yulin Normal University
    Université de Perpignan)

  • Akhtar A. Khan

    (Rochester Institute of Technology)

Abstract

We study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving a set-valued pseudo-monotone map. The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational problem by employing a surjectivity theorem for the sum of operators, combined with Minty’s formulation and techniques from the nonsmooth analysis. Then, an existence theorem for (QVHVI) is established by using Kluge’s fixed point theorem for set-valued operators. As an application, an abstract optimal control problem for the (QVHVI) is investigated. We prove the existence of solutions for the optimal control problem and the weak sequential compactness of the solution set via the Weierstrass minimization theorem and the Kuratowski-type continuity properties.

Suggested Citation

  • Shengda Zeng & Dumitru Motreanu & Akhtar A. Khan, 2022. "Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 950-970, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01963-3
    DOI: 10.1007/s10957-021-01963-3
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    References listed on IDEAS

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    1. Guo-ji Tang & Nan-jing Huang, 2013. "Existence theorems of the variational-hemivariational inequalities," Journal of Global Optimization, Springer, vol. 56(2), pages 605-622, June.
    2. Yi-bin Xiao & Xinmin Yang & Nan-jing Huang, 2015. "Some equivalence results for well-posedness of hemivariational inequalities," Journal of Global Optimization, Springer, vol. 61(4), pages 789-802, April.
    3. Akhtar A. Khan & Dumitru Motreanu, 2018. "Inverse problems for quasi-variational inequalities," Journal of Global Optimization, Springer, vol. 70(2), pages 401-411, February.
    4. Akhtar A. Khan & Dumitru Motreanu, 2015. "Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1136-1161, December.
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