On the Analysis and Solution Structure of Generalized Hemivariational Inclusion Problems

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-10
Published: Jan 12, 2026
Cite as:
Salahuddin, On the Analysis and Solution Structure of Generalized Hemivariational Inclusion Problems, Int. J. Anal. Appl., 24 (2026), 10.

Main Article Content

Salahuddin

Abstract

This paper introduces and analyzes a new class of generalized hemivariational inclusion problems. We establish the existence and uniqueness of solutions under mild assumptions and develop an efficient iterative algorithm for their numerical approximation. To demonstrate the practical utility of our theoretical framework, we apply it to a frictional contact problem in elasticity. The model involves an elastic body in contact with a rigid foundation, governed by a nonmonotone friction law that depends on both normal and tangential displacements. Our results provide a comprehensive solution, from theory to computation, for this challenging class of nonsmooth systems.

Article Details

References

  1. P.D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer, Berlin, 1993.
  2. Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.
  3. J. Haslinger, M. Miettinen, P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Springer, 1999. https://doi.org/10.1007/978-1-4757-5233-5.
  4. F.H. Clarke, Optimization and nonsmooth analysis, Interscience, 1983.
  5. A. Bagirov, N. Karmitsa, M.M. Mäkelä, Introduction to Nonsmooth Optimization: Theory, Practice and Software, Springer, 2014.
  6. M. Barboteu, W. Han, S. Migórski, On Numerical Approximation of a Variational–Hemivariational Inequality Modeling Contact Problems for Locking Materials, Comput. Math. Appl. 77 (2019), 2894–2905. https://doi.org/10.1016/j.camwa.2018.08.004.
  7. W. Han, M. Sofonea, D. Danan, Numerical Analysis of Stationary Variational-Hemivariational Inequalities, Numer. Math. 139 (2018), 563–592. https://doi.org/10.1007/s00211-018-0951-9.
  8. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.
  9. S. Migórski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Springer, 2013. https://doi.org/10.1007/978-1-4614-4232-5.
  10. M. Jureczka, A. Ochal, A Nonsmooth Optimization Approach for Hemivariational Inequalities with Applications to Contact Mechanics, Appl. Math. Optim. 83 (2019), 1465–1485. https://doi.org/10.1007/s00245-019-09593-y.
  11. I. Hlaváček, J. Haslinger, J. Nečas, J. Lovíšek, Solution of Variational Inequalities in Mechanics, Springer New York, 1988. https://doi.org/10.1007/978-1-4612-1048-1.
  12. N. Kikuchi, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, (1988).
  13. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer New York, 2002. https://doi.org/10.1007/978-1-4757-3658-8.
  14. S. Chang, Salahuddin, A.a.h. Ahmadini, L. Wang, G. Wang, The Penalty Method for Generalized Mixed Variational-Hemivariational Inequality Problems, Carpathian J. Math. 38 (2022), 357–381. https://doi.org/10.37193/cjm.2022.02.08.
  15. S. Chang, Salahuddin, L. Wang, G. Wang, Y. Zhao, Existence and Convergence Results for Generalized Mixed Quasi-Variational Hemivariational Inequality Problem, Symmetry 13 (2021), 1882. https://doi.org/10.3390/sym13101882.
  16. J.P. Penot, Mean Value Theorems for Correspondences, Acta Math. Vietnam. 28 (2001), 365–376.