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Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds

Author

Listed:
  • Balendu Bhooshan Upadhyay

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, India)

  • Shivani Sain

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, India)

  • Priyanka Mishra

    (Mathematics Division, School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal-Indore Highway, Kothrikalan, Sehore 466114, Madhya Pradesh, India)

  • Ioan Stancu-Minasian

    (“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania)

Abstract

In this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (abbreviated as, WVVHVIP) in the framework of Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence of the solutions for WVVHVIP without utilizing any monotonicity assumptions. Moreover, a uniqueness result for the solutions of WVVHVIP is established by using generalized geodesic strong monotonicity assumptions. We formulate Auslender, regularized, and Moreau-Yosida regularized type gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of the solutions to WVVHVIP. In addition to this, by employing the Auslender, regularized, and Moreau-Yosida regularized type gap functions, we derive the global error bounds for the solution of WVVHVIP under the generalized geodesic strong monotonicity assumptions. Several non-trivial examples are furnished in the Hadamard manifold setting to illustrate the significance of the established results. To the best of our knowledge, this is the first time that the existence results, gap functions, and global error bounds for WVVHVIP have been investigated in the framework of Hadamard manifolds via Clarke subdifferentials.

Suggested Citation

  • Balendu Bhooshan Upadhyay & Shivani Sain & Priyanka Mishra & Ioan Stancu-Minasian, 2025. "Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds," Mathematics, MDPI, vol. 13(6), pages 1-34, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:6:p:995-:d:1615023
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    References listed on IDEAS

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    1. Savin Treanţă & Priyanka Mishra & Balendu Bhooshan Upadhyay, 2022. "Minty Variational Principle for Nonsmooth Interval-Valued Vector Optimization Problems on Hadamard Manifolds," Mathematics, MDPI, vol. 10(3), pages 1-15, February.
    2. 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.
    3. X. Q. Yang & C. J. Goh, 1997. "On Vector Variational Inequalities: Application to Vector Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 431-443, November.
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