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Nonemptiness and Compactness of Solution Sets to Weakly Homogeneous Generalized Variational Inequalities

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Abstract

In this paper, we deal with the weakly homogeneous generalized variational inequality, which provides a unified setting for several special variational inequalities and complementarity problems studied in recent years. By exploiting weakly homogeneous structures of involved map pairs and using degree theory, we establish a result which demonstrates the connection between weakly homogeneous generalized variational inequalities and weakly homogeneous generalized complementarity problems. Subsequently, we obtain a result on the nonemptiness and compactness of solution sets to weakly homogeneous generalized variational inequalities by utilizing Harker–Pang-type condition, which can lead to a Hartman–Stampacchia-type existence theorem. Last, we give several copositivity results for weakly homogeneous generalized variational inequalities, which can reduce to some existing ones.

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Acknowledgements

The authors are very thankful to the editor and anonymous reviewers for their useful comments and constructive advice. The second author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11871051).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300350, People’s Republic of China

    Meng-Meng Zheng & Zheng-Hai Huang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Xue-Li Bai

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  1. Meng-Meng Zheng
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  2. Zheng-Hai Huang
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  3. Xue-Li Bai
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Correspondence to Zheng-Hai Huang.

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Communicated by Antonino Maugeri.

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Zheng, MM., Huang, ZH. & Bai, XL. Nonemptiness and Compactness of Solution Sets to Weakly Homogeneous Generalized Variational Inequalities. J Optim Theory Appl 189, 919–937 (2021). https://doi.org/10.1007/s10957-021-01866-3

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  • DOI: https://doi.org/10.1007/s10957-021-01866-3

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