Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds
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 mono...
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2025-03-01
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| author | Balendu Bhooshan Upadhyay Shivani Sain Priyanka Mishra Ioan Stancu-Minasian |
| author_facet | Balendu Bhooshan Upadhyay Shivani Sain Priyanka Mishra Ioan Stancu-Minasian |
| author_sort | Balendu Bhooshan Upadhyay |
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| description | 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. |
| format | Article |
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| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-c2df8b3cb9cc47eabafccd05f70522602025-08-20T01:49:04ZengMDPI AGMathematics2227-73902025-03-0113699510.3390/math13060995Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard ManifoldsBalendu Bhooshan Upadhyay0Shivani Sain1Priyanka Mishra2Ioan Stancu-Minasian3Department of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, IndiaDepartment of Mathematics, Indian Institute of Technology Patna, Patna 801103, Bihar, IndiaMathematics Division, School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal-Indore Highway, Kothrikalan, Sehore 466114, Madhya Pradesh, India“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, RomaniaIn 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.https://www.mdpi.com/2227-7390/13/6/995vector variational-hemivariational inequalitygap functionsKKM lemmaHadamard manifoldserror bounds |
| spellingShingle | Balendu Bhooshan Upadhyay Shivani Sain Priyanka Mishra Ioan Stancu-Minasian Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds Mathematics vector variational-hemivariational inequality gap functions KKM lemma Hadamard manifolds error bounds |
| title | Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds |
| title_full | Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds |
| title_fullStr | Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds |
| title_full_unstemmed | Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds |
| title_short | Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds |
| title_sort | existence results and gap functions for nonsmooth weak vector variational hemivariational inequality problems on hadamard manifolds |
| topic | vector variational-hemivariational inequality gap functions KKM lemma Hadamard manifolds error bounds |
| url | https://www.mdpi.com/2227-7390/13/6/995 |
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