General Variational Inclusions and Nonexpansive Mappings

  • Muhammad Aslam Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
  • Khalida Inayat Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Keywords: common elements, nonexpansive mappings, iterative methods, convergence analysis, nonlinear variational inclusions, Hilbert spaces

Abstract

In this paper, we introduce a new class of variational inclusions involving three operators. We suggest and analyze three-step iterations for finding the common element of the set of fixed points of a nonexpansive mappings and the set of the solutions of the variational inclusions using the resolvent operator technique. We also study the convergence criteria of three-step iterative method under some mild conditions. Inertial type methods are suggested and investigated for general variational inclusions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.

References

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. https://doi.org/10.1137/S1052623403427859

W. F. Ames, Numerical Methods for Partial Differential Equations, Third Edition, Academic Press, New York, 1992.

C. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities, J. Wiley and Sons, New York, London, 1984.

A. Bnouhachem, M. A. Noor and T. M. Rassias, Three-step iterative algorithms for mixed variational inequalities, Appl. Math. Comput. 183 (2006), 322-327. https://doi.org/10.1016/j.amc.2006.05.086

A. Bnouhachem and M. A. Noor, Numerical comparison between prediction-correction methods for general variational inequalities, Appl. Math. Comput. 186 (2007), 496-505. https://doi.org/10.1016/j.amc.2006.08.001

H. Brezis, Operateurs Maximaux Monotone et Semigroups de Contractions dan Espaces de Hilbert, North-Holland, Amsterdam, 1973. https://doi.org/10.1112/blms/6.2.221

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, J. Wiley and Sons, New York, 1980.

J. Eckstein and B. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Prog. 55 (1992), 293-318. https://doi.org/10.1007/BF01581204

M. Erturk, F. Gursoy and N. Simsek, S-iterative algorithm for solving variational inequalities, Int. J. Computer Math. 98(3) (2021), 435-448. https://doi.org/10.1080/00207160.2020.1755430

M. Fukushima, The primal Douglas-Rachford splitting algorithm for a class of monotone operators with applications to the traffic equilibrium problem, Math. Prog. 72 (1996), 1-15. https://doi.org/10.1007/BF02592328

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

S. Gupta, Generalized system of relaxed cocoercive variational inequalities governed by nonexpansive mappings, IOP Conf. Ser.: Mater. Sci. Eng. 1116 (2021), 012149. https://doi.org/10.1088/1757-899X/1116/1/012149

S. Haubruge, V. H. Nguyen and J. J. Strodiot, Convergence analysis and applications of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl. 97 (1998), 645-673. https://doi.org/10.1023/A:1022646327085

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 69-76. https://doi.org/10.1137/0716071

S. Jabeen, B. Bin-Mohsen, M. A. Noor and K. I. Noor, Inertial projection methods for solving general quasi-variational inequalities, AIMS Math. 6 (2021), 1075-1086. https://doi.org/10.3934/math.2021064

A. Moudafi and M. Thera, Finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl. 97 (1997), 425-448. https://doi.org/10.1023/A:1022643914538

K. Muangchoo, Modified subgradient extragradient method to solve variational inequalities, J. Math. Computer Sci. 25 (2022), 133-149. https://doi.org/10.22436/jmcs.025.02.03

M. A. Noor, On variational inequalities, PhD Thesis, Brunel University, London, U.K., 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, Quasi variational inequalities, Appl. Math. Letters 1(4) (1988), 367-370. https://doi.org/10.1016/0893-9659(88)90152-8

M. A. Noor, An extraresolvent method for monotone mixed variational inequalities, Math. Computer Modelling 29(1999), 95-100. https://doi.org/10.1016/S0895-7177(99)00033-3

M. A. Noor, Some recent advances in variational inequalities, Part II, other concepts, New Zealand J. Math. 26 (1997), 229-255.

M. A. Noor, Generalized set-valued variational inclusions and resolvent equations, J. Math. Anal. Appl. 228 (1998), 206-220. https://doi.org/10.1006/jmaa.1998.6127

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, 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, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl. 331 (2007), 810-822. https://doi.org/10.1016/j.jmaa.2006.09.039

M. A. Noor, Differentiable nonconvex functions and general variational inequalities, Appl. Math. Comput. 199(2,1) (2008), 623-630. https://doi.org/10.1016/j.amc.2007.10.023

M. A. Noor and K. I. Noor, Absolute value variational inclusions, Earthline. J. Math. Sci. 8(1) (2022), 121-153. https://doi.org/10.34198/ejms.8122.121153

M. A. Noor, K. I. Noor and B. B. Mohsen, Some new classes of general quasi variational inequalities, AIMS Math. 6(6) (2021), 6406-6421. https://doi.org/10.3934/math.2021376

M. A. Noor, M. Akhter and K. I. Noor, Inertial proximal method for mixed quasi variational inequalities, Nonlinear Funct. Anal. Appl. 8 (2003), 489-496.

M. A. Noor and Z. Y. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput. 187(2) (2007), 680-685. https://doi.org/10.1016/j.amc.2006.08.088

M. A. Noor, K. I. Noor and M. Th. Rassias, New trends in general variational inequalities, Acta Math. Applicandae 170(1) (2021), 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

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, USSR Comput. Math. Math. Phys. 4 (1964), 1-17. https://doi.org/10.1016/0041-5553(64)90137-5

S. Rathee and M. Swami, On solving variational inequalities by SM-iteration, J. Math. Computer Sci. 11(4) (2021), 4139-4154.

R. T. Rockafellar, Monotone operators and the proximal point algorithms, SIAM J. Control Optim. 14 (1976), 877-898. https://doi.org/10.1137/0314056

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

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118(2) (2003), 417-428. https://doi.org/10.1023/A:1025407607560

S. Ullah and M. A. Noor, An efficient method for solving new general mixed variational inequalities, J. Inequal. Special Funct. 11(3) (2019), 1-9.

X. L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 113 (1991), 727-731. https://doi.org/10.1090/S0002-9939-1991-1086345-8

Published
2022-03-23
How to Cite
Noor, M. A., & Noor, K. I. (2022). General Variational Inclusions and Nonexpansive Mappings. Earthline Journal of Mathematical Sciences, 9(2), 145-164. https://doi.org/10.34198/ejms.9222.145164
Section
Articles