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Existence and Uniqueness of Solutions of Generalized Mixed Variational Inequalities

Author

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  • Jian-Xun Liu

    (Guangxi Minzu University
    Tianjin University)

  • Zhao-Feng Lan

    (Guangxi Minzu University)

  • Zheng-Hai Huang

    (Tianjin University)

Abstract

In this paper, we study the generalized mixed variational inequality, which encompasses both the generalized variational inequality and the mixed variational inequality. The core contribution of this paper is twofold. Firstly, by utilizing the principles of degree theory, we establish certain sufficient conditions for the existence of solutions to the generalized mixed variational inequality. Additionally, we formulate a sufficient condition that ensures the uniqueness of these solutions. Secondly, we recognize that the conditions outlined in our theorem are inapplicable to the generalized mixed polynomial variational inequality, a subclass within the broader family of generalized mixed variational inequalities. To address this, we employ an exceptional family of elements and establish an existence and uniqueness theorem specifically tailored for the generalized mixed polynomial variational inequality.

Suggested Citation

  • Jian-Xun Liu & Zhao-Feng Lan & Zheng-Hai Huang, 2025. "Existence and Uniqueness of Solutions of Generalized Mixed Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 205(1), pages 1-21, April.
    • Handle: RePEc:spr:joptap:v:205:y:2025:i:1:d:10.1007_s10957-025-02636-1
    DOI: 10.1007/s10957-025-02636-1
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    References listed on IDEAS

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    1. Lu-Chuan Zeng & Jen-Chih Yao, 2005. "Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(2), pages 211-224, November.
    2. Vu Trung Hieu, 2020. "Solution maps of polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 77(4), pages 807-824, August.
    3. Wenjie Mu & Jianghua Fan, 2022. "Existence results for solutions of mixed tensor variational inequalities," Journal of Global Optimization, Springer, vol. 82(2), pages 389-412, February.
    4. Xue-Li Bai & Zheng-Hai Huang & Yong Wang, 2016. "Global Uniqueness and Solvability for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 72-84, July.
    5. Vu Trung Hieu, 2019. "On the R0-Tensors and the Solution Map of Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 163-183, April.
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