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Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

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Abstract

The purpose of this paper is to establish some new results on the Levitin–Polyak well-posedness to a class of split hemivariational inequality problems on Hadamard manifolds. We first consider a new class of split hemivariational inequality problems (for short, SHIP) on Hadamard manifolds and introduce the regularized gap functions for these problems. Then, we study the notion of Levitin–Polyak well-posedness by perturbations to SHIP and show the equivalence between the Levitin–Polyak well-posedness by perturbations and the existence of solutions for SHIP under suitable conditions. Furthermore, based on the regularized gap functions for the perturbed SHIP, we establish the criterion for the Levitin–Polyak well-posedness by perturbations for SHIP via the split optimization problems on Hadamard manifolds. Our main results presented in paper are new even in the special case of hemivariational inequality problems.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable remarks which improved the results and presentation of this article. This research was supported by Ministry of Education and Training of Vietnam under Grant No. B2021.SPD.03, NNSF of China Grant No. 12071413 and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH.

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Authors and Affiliations

  1. Faculty of Mathematics - Information Technology Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap, Vietnam

    Vo Minh Tam

  2. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam

    Nguyen Van Hung

  3. Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University, Nanning, 530006, Guangxi, People’s Republic of China

    Zhenhai Liu

  4. Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Jen Chih Yao

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  1. Vo Minh Tam
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Correspondence to Nguyen Van Hung.

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Communicated by Alexandru Kristály.

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Tam, V.M., Van Hung, N., Liu, Z. et al. Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds. J Optim Theory Appl 195, 684–706 (2022). https://doi.org/10.1007/s10957-022-02111-1

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