Some new classes of general quasi variational inequalities

Muhammad Aslam Noor; Khalida Inayat Noor; Bandar B. Mohsen; Muhammad Aslam Noor; Khalida Inayat Noor; Bandar B. Mohsen

doi:10.3934/math.2021376

AIMS Mathematics

2021, Volume 6, Issue 6: 6406-6421. doi: 10.3934/math.2021376

Previous Article Next Article

Research article

Some new classes of general quasi variational inequalities

1.
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
2.
Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia

Received: 29 December 2020 Accepted: 06 April 2021 Published: 13 April 2021
MSC : 49J40, 90C33

In this paper, we introduce and consider some new classes of general quasi variational inequalities, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We propose some new inertial projection methods for solving the general quasi variational inequalities and related problems. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inequalities include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inequalities for further research activities.
- quasi-variational inequality,
- nonsymmetric boundary value problems,
- projection operator,
- inertial methods,
- convergence
Citation: Muhammad Aslam Noor, Khalida Inayat Noor, Bandar B. Mohsen. Some new classes of general quasi variational inequalities[J]. AIMS Mathematics, 2021, 6(6): 6406-6421. doi: 10.3934/math.2021376

Related Papers:

Abstract

In this paper, we introduce and consider some new classes of general quasi variational inequalities, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We propose some new inertial projection methods for solving the general quasi variational inequalities and related problems. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inequalities include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inequalities for further research activities.

References

[1]	G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sci., 258 (1964), 4413–4416.
[2]	M. A. Noor, K. I. Noor, From representation theorems to variational inequalities, In: N. J. Daras, T. M. Rassiass, Computational Mathematics and Variational Analysis, Switzerland: Springer, 2020.
[3]	M. A. Noor, On Variational Inequalities, Ph.D Thesis, Brunel University, London, 1975.
[4]	R. Glowinski, J. L. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities, New York: North-Holland Publishing Company, 1981.
[5]	D. Kinderlehrer, G. Stampachia, An Introduction to Variational Inequalities and Their Applications, New York: Academic Press, 1980.
[6]	M. A. Noor, General variational inequalities, Appl. Math. Lett., 1 (1988), 119–122. doi: 10.1016/0893-9659(88)90054-7
[7]	M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199–277. doi: 10.1016/S0096-3003(03)00558-7
[8]	M. A. Noor, K. I. Noor, M. Th. Rassias, New trends in general variational inequalities, Acta Appl. Mathematica, 170 (2020), 981–1064. doi: 10.1007/s10440-020-00366-2
[9]	M. A. Noor, K. I. Noor, T. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math., 47 (1993), 285–312. doi: 10.1016/0377-0427(93)90058-J
[10]	A. Bensoussan, J. L. Lions, Application des Inéqualities Variationnelles en Contrôl Stochastique, Paris: Bordas(Dunod), 1978.
[11]	D. Chan, J. Pang, The generalized quasi-variational inequality problem, Math. Oper. Res., 7 (1982), 211–222. doi: 10.1287/moor.7.2.211
[12]	M. A. Noor, An iterative scheme for class of quasi variational inequalities, J. Math. Anal. Appl., 110 (1985), 463–468. doi: 10.1016/0022-247X(85)90308-7
[13]	M. A. Noor, Noor, General quasi complementarity problems, J. Math. Anal. Appl., 120 (1986), 321–327. doi: 10.1016/0022-247X(86)90219-2
[14]	M. A. Noor, Implicit dynamical systems and quasi variational inequalities, Appl. Math. Comput., 134 (2003), 69–81.
[15]	M. A. Noor, Merit functions for quasi variational inequalities, J. Math. Inequal., 1 (2007), 259–269.
[16]	M. Jacimovic, N. Mijajlovic, On a continuous gradient-type method for solving quasi variational inequalities, Proc. Mont. Acad. Sci Arts, 19 (2011), 16–27.
[17]	N. Mijajlovic, J. Milojica, M. A. Noor, Gradient-type projection methods for quasi variational inequalities, Optim. Lett., 13 (2019), 1885–1896. doi: 10.1007/s11590-018-1323-1
[18]	Y. Shehu, A. Gibali, S. Sagratella, Inertial projection-type method for solving quasi variational inequalities in real Hilbert space, J. Optim. Theory Appl., 184 (2020), 877–894. doi: 10.1007/s10957-019-01616-6
[19]	A. S. Antipin, M. Jacimovic, N. Mijajlovic, Extra gradient method for solving quasi variational inequalities, Optimization, 67 (2018), 103–112. doi: 10.1080/02331934.2017.1384477
[20]	S. Jabeen, M. A. Noor, K. I. Noor, Some new inertial projection methods for quasi variational inequalities, Appl. Math. E-Notes, 2021 (2021).
[21]	M. A. Noor, Quasi variational inequalities, Appl. Math. Lett., 1 (1988), 367–370. doi: 10.1016/0893-9659(88)90152-8
[22]	M. A. Noor, W. Oettli, On general nonlinear complementarity problems and quasi equilibria, Le Mathematiche, 49 (1994), 313–331.
[23]	M. A. Noor, On general quasi variational inequalities, J. King Saud Univ. Sci., 24 (2012), 81–88. doi: 10.1016/j.jksus.2010.07.002
[24]	S. Jabeen, M. A. Noor, K. I. Noor, Inertial iterative methods for general quasi variational inequalities and dynamical systems, J. Math. Anal., 11 (2020), 14–29.
[25]	S. Jabeen, B. Bin-Mohsen, M. A. Noor, K. I. Noor, Inertial projection methods for solving general quasi-variational inequalities, AIMS Math., 6 (2021), 1075–1086. doi: 10.3934/math.2021064
[26]	C. F. Lemke, Bimatrix equilibrium points and mathematical programming, Manage. Sci., 11 (1965), 681–689.
[27]	R. W. Cottle, J. S. Pang, R. E. Stone, The Linear Complementarity Problem, SIAM Publication, 1992.
[28]	S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl., 8 (1971), 161–168. doi: 10.1007/BF00932464
[29]	K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Berlin: Heldermann Verlag, 1988.
[30]	A. S. Antipin, Minimization of convex functions on convex sets by means of differential equations, Differ. Equations, 30 (2003), 1365–1357.
[31]	F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2003), 773–782.
[32]	F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. doi: 10.1023/A:1011253113155
[33]	H. Attouch, M. O. Czarnecki, Asymtotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Differ. Equatations, 179 (2002), 278–310. doi: 10.1006/jdeq.2001.4034
[34]	H. Attouch, X. Goudon, P. Redont, The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1–34. doi: 10.1142/S0219199700000025
[35]	P. E. Mainge, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl., 344 (2008), 876–887. doi: 10.1016/j.jmaa.2008.03.028
[36]	Y. Shehu, P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019), 4. doi: 10.1007/s10092-018-0300-5
[37]	F. Faraci, B. Jadamba, F. Raciti, On stochastic variational inequalities with mean value constraints, J. Optim. Theory Appl., 171 (2016), 675–693. doi: 10.1007/s10957-016-0888-z
[38]	B. Jadamba, A. A. Khan, F. Raciti, Regularization of stochastic variational inequalities and a comparison of an $Lp$ approach and a sample-path approach, Nonlinear Anal.: Theory Methods Appl., 94 (2014), 65–83. doi: 10.1016/j.na.2013.08.009
[39]	G. M. Koperlevich, The extragradient method for finding saddle points and other problems, Matecom, 12 (1976), 203–213.
[40]	B. T. Polyak, Some methods of speeding up the convergence of iterative methods, USSR Comput. Math. Math. Phys., 4 (1964), 1–17.
[41]	A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183–202. doi: 10.1137/080716542
[42]	A. Bnouhachem, M. A. Noor, Three-step projection method for general variational inequalities, Int. J. Modern Phys. B, 26 (2012), 1250066. doi: 10.1142/S021797921250066X
[43]	M. A. Noor, Extended general variational inequalities, Appl. Math. Lett., 22 (2009), 182–185. doi: 10.1016/j.aml.2008.03.007
[44]	M. A. Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput., 32 (2010), 83–95. doi: 10.1007/s12190-009-0234-9
[45]	V. M. Filippov, Variational Principles for Nonpotential Operators, Providence, USA, 1989.
[46]	E. Tonti, Variational formulation for every nonlinear problem, Int. J. Eng. Sci., 22 (1984), 1343–1371. doi: 10.1016/0020-7225(84)90026-0
[47]	M. A. Noor, Variational inequalities in physical oceanography, In: M. Rahman, Ocean Wave Engineering, Scotland: Printed Bell & Bain Ltd., 1994.
[48]	H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256. doi: 10.1112/S0024610702003332

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)