Trifunction Bihemivariational Inequalities

  • Muhammad Aslam Noor Mathematics Department, COMSATS University Islamabad, Islamabad, Pakistan
  • Khalida Inayat Noor Mathematics Department, COMSATS University Islamabad, Islamabad, Pakistan
Keywords: hemivariational inequalities, auxiliary principle, proximal methods, convergence

Abstract

In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered.  Results proved in this paper can be viewed as a refinement and improvement of the known results.

References

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator damping, Set-Valued Anal. 9 (2001), 3-11. https://doi.org/10.1023/A:1011253113155

F. H. Calrke, Optimization and Nonsmooth Analysis, J. Wiley and Sons, NY, 1983.

G. Cristescu and L. Lupsa, Non-connected Convexities and Applications, Springer-Verlag, Berlin, 2002. https://doi.org/10.1007/978-1-4615-0003-2_1

F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, Holland, 2001.

R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

B. Martinet, Regularization d'inequations variationnelles par approximation successive, Rev. d'Autom. Inform. Rech. Oper., Serie Rouge 3 (1970), 154-159. https://doi.org/10.1051/m2an/197004R301541

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Boston, 1995.

M. A. Noor, On Variational Inequalities, PhD Thesis, Brunel University, London, UK, 1975.

M. A. Noor, General variational inequalities, Appl. Math. Letters 1 (1988), 119-121. https://doi.org/10.1016/0893-9659(88)90054-7

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042

M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl. 122 (2004), 371-386. https://doi.org/10.1023/B:JOTA.0000042526.24671.b2

M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7

M. A. Noor, Hemivariational inequalities, J. Appl. Math. Computing 17 (2005), 59-72. https://doi.org/10.1007/BF02936041

M. A. Noor, Hemivariational-like inequalities, J. Comput. Appl. Math. 182(2) (2005), 316-326. https://doi.org/10.1016/j.cam.2004.12.013

M. A. Noor, Fundamentals of equilibrium problems, Math. Inequal. Appl. 9 (2006), 529-566. https://doi.org/10.7153/mia-09-51

M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi equilibria, Le Mathematiche 49 (1994), 313-331.

M. A. Noor and K. Inayat Noor, Iterative schemes for trifunction hemivariational inequalities, Optim. Letters 5 (2011), 273-282. https://doi.org/10.1007/s11590-010-0206-x

M. A. Noor, K. I. Noor and Z. Y. Huang, Bifunction hemivariational inequalities, J. Appl. Math. Comput. 35 (2011), 595-605. https://doi.org/10.1007/s12190-010-0380-0

M. A, Noor, K. I. Noor and M. Th. Rassias, New trends in general variational inequalities, Acta Appl. Math. 170(1) (2020), 981-1046. https://doi.org/10.1007/s10440-020-00366-2

M. A. Noor, K. I. Noor and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), 285-312. https://doi.org/10.1016/0377-0427(93)90058-J

M. A. Noor and K. I. Noor, Higher order strongly exponentially biconvex functions and bivariational inequalities, J. Math. Anal. 12(2) (2021), 23-43.

M. A. Noor and K. I. Noor, Higher order strongly biconvex functions and biequilibrium problems, Advanc. Lin. Algebr. Matrix Theory 11(2) (2021), 31-53. https://doi.org/10.4236/alamt.2021.112004

M. A. Noor and K. I. Noor, Strongly log-biconvex functions and applications, Earthline J. Math. Sci. 7(1) (2021), 1-23. https://doi.org/10.34198/ejms.7121.123

M. A. Noor, K. I. Noor and M. Th. Rassias, Strongly biconvex functions and bivariational inequalities, in: Mathematical Analysis, Optimization, Approximation and Applications (Edited: Panos M. Pardalos and Th. M. Rassias), World Scientific Publishing Company, Singapore, 2021.

M. A. Noor, K. I. Noor, A. Hamdi and E. H. El-Shemas, On difference of two monotone operators, Optim. Letters 3 (2009), 329-335. https://doi.org/10.1007/s11590-008-0112-7

M. A. Noor, K. Inayat Noor and M. Lotayif, Biconvex functions and mixed bivariational inequalities, Inform. Sci. Lett. 10(3) (2021), 469-475.

M. A. Noor, K. Inayat Noor and H. M. Y. Al-Bayatti, Higher order variational inequalities, Inform. Sci. Lett. 11(1) (2022).

P. D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationarity principles, Acta Mechanica 42 (1983), 160-183.

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications to Mechanics and Engineering, Springer Verlag, Berlin, 1993.

M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publishers, Dordrecht, Holland, 1999. https://doi.org/10.1007/978-1-4757-2991-7

G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.

D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6 (1996), 714-726. https://doi.org/10.1137/S1052623494250415

Published
2021-08-30
How to Cite
Noor, M. A., & Noor, K. I. (2021). Trifunction Bihemivariational Inequalities. Earthline Journal of Mathematical Sciences, 7(2), 287-313. https://doi.org/10.34198/ejms.7221.287313
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Articles