Existence results for a class of hemivariational-like inequality problems

Kartikeswar Mahalik; Chandal Nahak; Ravi P. Agarwal

doi:10.2478/auom-2026-0023

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Existence results for a class of hemivariational-like inequality problems

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 34 (2026): Issue 2 (July 2026)

By: Kartikeswar Mahalik, Chandal Nahak and Ravi P. Agarwal

Open Access

|Jul 2026

Abstract

In this work, we establish existence results for a class of hemivariational-like inequality problems driven by a trifunction and perturbed by a bifunction. Initially, we establish solvability results based on the weak compactness of the constraint set. Subsequently, under appropriate coercivity conditions and in the absence of boundedness of the constraint set, we show the existence results and boundedness properties of the solution set. Furthermore, several theoretical applications are provided, and a necessary and sufficient condition for the existence of solutions to a particular subclass of these problems is derived.

References

P. D. Panagiotopoulos. Nonconvex energy functions. Hemivariational inequalities and substationarity principles. Acta Mech., 48 (1983) 111–130.
Search in Google Scholar Back to article
F. H. Clarke. Optimization and Nonsmooth Analysis. Wiley, New York, 1983.
Search in Google Scholar Back to article
N. Costea and V. Rădulescu. Hartman–Stampacchia results for stably pseudomonotone operators and non-linear hemivariational inequalities. Appl. Anal., 89 (2) (2010) 175–188.
Search in Google Scholar Back to article
G. J. Tang and N. J. Huang. Existence theorems of the variationalhemivariational inequalities. J. Glob. Optim., 56 (2) (2013) 605–622.
Search in Google Scholar Back to article
G. J. Tang, X. Wang, and Z. B. Wang. Existence of variational quasi-hemivariational inequalities involving a set-valued operator and a nonlinear term. Optim. Lett., 9 (1) (2015) 75–90.
Search in Google Scholar Back to article
Y. Zhang and Y. He. On stably quasimonotone hemivariational inequalities. Nonlinear Anal., 74 (10) (2011) 3324–3332.
Search in Google Scholar Back to article
N. Costea and A. Pitea. Existence results for mixed hemivariational-like inequalities involving set-valued maps. Optimization, 70 (2) (2021) 269–305.
Search in Google Scholar Back to article
G. J. Tang, L. W. Zhou, and N. J. Huang. Existence results for a class of hemivariational inequality problems on Hadamard manifolds. Optimization, 65 (7) (2016) 1451–1461.
Search in Google Scholar Back to article
D. Motreanu and P. D. Panagiotopoulos. Minimax Theorems and Qualitative properties of the Solutions of Hemivariational Inequalities. Kluwer Academic Publishers, Dordrecht, 1999.
Search in Google Scholar Back to article
P. D. Panagiotopoulos. Hemivariational inequalities, Applications in Mechanics and Engineering. Springer, Berlin, 1993.
Search in Google Scholar Back to article
P. D. Panagiotopoulos. Coercive and semicoercive hemivariational inequalities. Nonlinear Anal., 16 (3) (1991) 209–231.
Search in Google Scholar Back to article
S. Carl, V. K. Le, and D. Motreanu. The sub-supersolution method and extremal solutions for quasilinear hemivariational inequalities. Differ. Integral Equ., 17 (1–2) (2004) 165–178.
Search in Google Scholar Back to article
S. Carl, V. K Le, and D. Motreanu. Evolutionary variational– hemivariational inequalities: existence and comparison results. J. Math. Anal. Appl., 345 (1) (2008) 545–558.
Search in Google Scholar Back to article
Z. Liu. Existence results for quasilinear parabolic hemivariational inequalities. J. Differ. Equ., 244(6) (2008) 1395–1409.
Search in Google Scholar Back to article
D. Goeleven, D. Motreanu, and P. D. Panagiotopoulos. Eigenvalue problems for variational-hemivariational inequalities at resonance. Nonlinear Anal., 33 (2) (1998) 161–180.
Search in Google Scholar Back to article
S. Migórski. On existence of solutions for parabolic hemivariational inequalities. J. Comput. Appl. Math., 129 (1–2) (2001) 77–87.
Search in Google Scholar Back to article
N. Costea and C. Lupu. On a class of variational-hemivariational inequalities involving set valued mappings. Adv. Pure Appl. Math., 1 (2) (2010) 233–246.
Search in Google Scholar Back to article
N. Costea. Existence and uniqueness results for a class of quasi-hemivariational inequalities. J. Math. Anal. Appl., 373 (1) (2011) 305–315.
Search in Google Scholar Back to article
Y. Huang and Y. Zhou. Existence of solutions for a class of hemivariational inequality problems. Comput. Math. Appl., 57 (9) (2009) 1456–1462.
Search in Google Scholar Back to article
Z. H. Liu. Elliptic variational hemivariational inequalities. Appl. Math. Lett., 16 (6) (2003) 871–876.
Search in Google Scholar Back to article
R. Wangkeeree and P. Preechasilp. Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces. J. Glob. Optim., 57 (4) (2013) 1447–1464.
Search in Google Scholar Back to article
L. C. Ceng, J. C. Yao, and Y. H. Yao. Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces. Filomat, 32 (10) (2018) 3609–3622.
Search in Google Scholar Back to article
K. Mahalik and C. Nahak. Existence results for a class of variational quasi-mixed hemivariational-like inequalities. Bull. Malays. Math. Sci. Soc., 45 (5) (2022) 1877–1901.
Search in Google Scholar Back to article
K. Mahalik, C. Nahak, and R. P. Agarwal. On well-posedness of inverse mixed variational inequality problems. J. Nonlinear Convex Anal., 24 (11) (2023) 2451–2472.
Search in Google Scholar Back to article
K. Mahalik and C. Nahak. Well-posedness characterizations for system of mixed hemivariational inequalities. Miskolc Math. Notes, 2024 (2) (2024) 1787–2413.
Search in Google Scholar Back to article
K. Mahalik and C. Nahak. On inverse variational–hemivariational inequalities and its regularization. Optimization, 74 (9) (2024) 2057-2071.
Search in Google Scholar Back to article
K. Fan. Some properties of convex sets related to fixed point theorems. Math. Ann., 266 (4) (1984) 519–537.
Search in Google Scholar Back to article
K. Fan. A minimax inequality and applications. Inequalities, 3 (1972) 103–113.
Search in Google Scholar Back to article
Q. H. Ansari, N. C. Wong, and J. C. Yao. The existence of nonlinear inequalities. Appl. Math. Lett., 12 (5) (1999) 89–92.
Search in Google Scholar Back to article
P. D. Panagiotopoulos, M. Fundo, and V. Rădulescu. Existence theorems of Hartman–Stampacchia type for hemivariational inequalities and applications. J. Glob. Optim., 15 (1) (1999) 41–54.
Search in Google Scholar Back to article
F. Facchinei and J. S. Pang. Finite-dimensional variational inequalities and complementarity problems. Springer, New York, 2003.
Search in Google Scholar Back to article
L. C. Ceng, X. Qin, J. C. Yao, and Y. Yao. On existence results for a new class of variational-hemivariational-like inequalities in Banach spaces. Commun. Optim. Theory, (2018) 2018 Article ID 7.
Search in Google Scholar Back to article
L. C. Ceng and J. C. Yao. Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems. Nonlinear Anal., 69 (12) (2008) 4585–4603.
Search in Google Scholar Back to article
M. A. Noor and W. Oettli. On general nonlinear complementarity problems and quasi-equilibria. Matematiche (Catania), 49 (2) (1994) 313–331.
Search in Google Scholar Back to article
S. Z. Németh. Variational inequalities on Hadamard manifolds. Nonlinear Anal., 52 (5) (2002) 1491–1498.
Search in Google Scholar Back to article
S. Ranjbar. Variational inequality for a vector field on Hadamard spaces. An. S¸t. Univ. Ovidius Constant¸a, 31 (1) (2023) 241–250.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2026-0023 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 139 - 162

Submitted on: Jul 3, 2025

Accepted on: Sep 16, 2025

Published on: Jul 11, 2026

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Keywords:

Hemivariational-like inequality,

KKM mapping,

H-Hemicontinuity,

Coercivity conditions,

Existence results

Related subjects:

Mathematics,

General mathematics

© 2026 Kartikeswar Mahalik, Chandal Nahak, Ravi P. Agarwal, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 34 (2026): Issue 2 (July 2026)