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

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  • Akhtar A. Khan

    (Rochester Institute of Technology)

  • Dumitru Motreanu

    (Université de Perpignan)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-015-0825-6
    DOI: 10.1007/s10957-015-0825-6
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    References listed on IDEAS

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    1. Z. H. Liu, 2004. "Existence Results for Evolution Noncoercive Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 417-427, February.
    2. Baasansuren Jadamba & Akhtar A. Khan & Fabio Raciti & Behzad Djafari Rouhani, 2010. "Generalized Solutions of Multi-valued Monotone Quasi-variational Inequalities," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 227-240, Springer.
    3. F. Giannessi & A.A. Khan, 2010. "On the Envelope of a Variational Inequality," Springer Optimization and Its Applications, in: Panos M. Pardalos & Themistocles M. Rassias & Akhtar A. Khan (ed.), Nonlinear Analysis and Variational Problems, chapter 0, pages 285-293, Springer.
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    Cited by:

    1. Vy Khoi Le, 2022. "On Inclusions with Monotone-Type Mappings in Nonreflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 484-509, February.
    2. Dumitru Motreanu & Van Thien Nguyen & Shengda Zeng, 2020. "Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 391-407, November.
    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. Bing Tan & Xiaolong Qin & Jen-Chih Yao, 2022. "Strong convergence of inertial projection and contraction methods for pseudomonotone variational inequalities with applications to optimal control problems," Journal of Global Optimization, Springer, vol. 82(3), pages 523-557, March.
    5. 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.

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