A Resolvent Algorithm for System of General Mixed Variational Inequalities

Abdellah Bnouhachem, Muhammad Aslam Noor, Zineb Sayl


In this paper, we suggest and analyze a new resolvent algorithm for finding the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point of a nonexpansive mapping in Hilbert spaces. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. In this respect, our results present a refinement and improvement of the previously known results.


[1]Aslam Noor, M., & Inayat Noor, K. (2009). Projection algorithms for solving a system of general variational inequalities. Nonlinear Analysis: Theory, Methods & Applications, 70(7), 2700–2706.

[2] Bnouhachem, A. (2005). A self-adaptive method for solving general mixed variational inequalities. Journal of Mathematical Analysis and Applications, 309(1), 136–150.

[3] Brezis, H. (1973). Operateurs Maximaux Monotone et Semigroupes de Contractions dans les Espace dHilbert. North-Holland, Amsterdam, Holland.

[4] Chang, S., Joseph Lee, H., & Chan, C. (2007). Generalized system for relaxed cocoercive variational inequalities in hilbert spaces. Applied Mathematics Letters, 20(3), 329–334.

[5] He, Z., & Gu, F. (2009). Generalized system for relaxed cocoercive mixed variational inequalities in hilbert spaces. Applied Mathematics and Computation, 214(1), 26–30.

[6] Huang, Z., & Aslam Noor, M. (2007). An explicit projection method for a system of nonlinear variational inequalities with different (γ , r)-cocoercive mappings. Applied Mathematics and Computation, 190(1), 356–361.

[7] Lions, J., & Stampacchia, G. (1967). Variational inequalities. Comm. Pure Appl. Math, 20, 493–512.

[8] Noor (2007-2009). Variational Inequalities and Applications. Lecture Notes, Mathematics Department, COMSATS Institute of information Technology, Islamabad, Pakistan.

[9] Noor, M. A. (2002). Proximal methods for mixed quasivariational inequalities. Journal of optimization theory and applications, 115(2), 453–459.

[10] Noor, M. A. (2003). Mixed quasi variational inequalities. Applied mathematics and computation, 146(2), 553–578.

[11] Noor, M. A. (2004). Fundamentals of mixed quasi variational inequalities. International Journal of Pure and Applied Mathematics, 15(2), 137–258.

[12] Petrot, N. (2010). A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems. Applied Mathematics Letters, 23(4), 440–445.

[13] Stampacchia, G. (1964). Formes bilin´eaires coercitives sur les ensembles convexes.(french). CR Acad. Sci. Paris, 258, 4413–4416.

[14] Tr´emoli`eres, R., Lions, J.-L., & Glowinski, R. (1981). Numerical analysis of variational inequalities, volume 8. North Holland.

[15] Verma, R. (2001). Projection methods, algorithms, and a new system of nonlinear variational inequalities. Computers & Mathematics with Applications, 41(7), 1025–1031.

[16] Verma, R. (2004). Generalized system for relaxed cocoercive variational inequalities and projection methods. Journal of Optimization Theory and Applications, 121(1), 203–210.

[17] Verma, R. U. (2005). General convergence analysis for two-step projection methods and applications to variational problems. Applied Mathematics Letters, 18(11), 1286–1292.

[18] Weng, X. (1991). Fixed point iteration for local strictly pseudo-contractive mapping. 113(3), 727– 731.

[19] Yang, H., Zhou, L., & Li, Q. (2010). A parallel projection method for a system of nonlinear variational inequalities. Applied Mathematics and Computation, 217(5), 1971–1975.

DOI: http://dx.doi.org/10.3968%2Fj.ans.1715787020130601.2333


  • There are currently no refbacks.


If you have already registered in Journal A and plan to submit article(s) to Journal B, please click the CATEGORIES, or JOURNALS A-Z on the right side of the "HOME".

We only use the following emails to deal with issues about paper acceptance, payment and submission of electronic versions of our journals to databases:
caooc@hotmail.com; office@cscanada.net; office@cscanada.org

Copyright © Canadian Research & Development Centre of Sciences and Cultures (CRDCSC)
Address:758, 77e AV, Laval, Quebec, H7V 4A8, Canada

Telephone: 1-514-558 6138
Http://www.cscanada.net; Http://www.cscanada.org
E-mail:caooc@hotmail.com; office@cscanada.net