Existence and algorithm of solutions for a new system of generalised variational inclusions involving relaxed Lipschitz mappings

We consider a new system for generalized variational inclusions in Hilbert spaces and define an iterative algorithm for finding the approximate solutions of this class of system of variational inclusions. We also establish that the approximate solutions obtained by our algorithm converge to the exact solution of new system of generalized variational inclusions. One can explore the role of our system of generalized variational inclusions for solving various known equilibrium problem and other related problems. References R. Ahmad and Q. H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett. 13(5), 23--26, (2000). J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. X. P. Ding and C. L. Luo, Perturbed proximal point algorithms for general quasi-variational-like inclusions, J. Comput. Appl. Math. 113, 153--165, (2000). C. H. Lee, Q. H. Ansari and J. C. Yao, A perturbed algorithm for strongly nonlinear variational inclusions, Bull. Austral. Math. Soc. 62, 417--426, (2000). J. S. Pang, Asymmetric variational inequality problem over product of sets:Applications and iterative methods, Math Prog. 31, 206--219, (1985). R. U. Verma, Partially relaxed monotone mappings and a new system of nonlinear variational inequalities, Nonlinear Funct. Anal. Appl. 5(1), 65--72, (2000). R. U. Verma, Projection method, Algorithm and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41(7-8), 1025--1031, (2001). R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl. 121(1), 203--210, (2004). R. P. Agarwal, Y. J. Cho and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13(6), 19--24, (2000). R. P. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl. 7(6), 807--828, (2002). R. P. Agarwal, N. J. Huang and M. Y. Tan, Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions, Appl. Math. Lett. 17, 345--352, (2004). Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc. 59(3), 433--442, (1999)