## System of nonlinear set-valued variational inclusions involving a finite family of $$H(\cdot, \cdot)$$-accretive operators in Banach spaces.(English)Zbl 1252.49024

Summary: We study a new system of nonlinear set-valued variational inclusions involving a finite family of $$H(\cdot, \cdot)$$-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family of $$H(\cdot, \cdot)$$-accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.

### MSC:

 49J53 Set-valued and variational analysis 47H06 Nonlinear accretive operators, dissipative operators, etc.
Full Text:

### References:

 [1] A. Hassouni and A. Moudafi, “A perturbed algorithm for variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 185, no. 3, pp. 706-712, 1994. · Zbl 0809.49008 [2] S. Adly, “Perturbed algorithms and sensitivity analysis for a general class of variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 201, no. 2, pp. 609-630, 1996. · Zbl 0856.65077 [3] X. P. Ding, “Perturbed proximal point algorithms for generalized quasivariational inclusions,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 88-101, 1997. · Zbl 0902.49010 [4] K. R. Kazmi, “Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions,” Journal of Mathematical Analysis and Applications, vol. 209, no. 2, pp. 572-584, 1997. · Zbl 0898.49007 [5] Q. H. Ansari and J.-C. Yao, “A fixed point theorem and its applications to a system of variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 433-442, 1999. · Zbl 0944.47037 [6] Q. H. Ansari, S. Schaible, and J. C. Yao, “System of vector equilibrium problems and its applications,” Journal of Optimization Theory and Applications, vol. 107, no. 3, pp. 547-557, 2000. · Zbl 0972.49009 [7] S. Plubtieng and K. Sombut, “Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space,” Journal of Inequalities and Applications, vol. 2010, Article ID 246237, 12 pages, 2010. · Zbl 1189.47067 [8] S. Plubtieng and K. Sitthithakerngkiet, “On the existence result for system of generalized strong vector quasiequilibrium problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 475121, 9 pages, 2011. · Zbl 1215.49010 [9] S. Plubtieng and T. Thammathiwat, “Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems in G-convex spaces,” Computers & Mathematics with Applications, vol. 62, no. 1, pp. 124-130, 2011. · Zbl 1250.47059 [10] R. U. Verma, “A-monotonicity and applications to nonlinear variational inclusion problems,” Journal of Applied Mathematics and Stochastic Analysis, no. 2, pp. 193-195, 2004. · Zbl 1064.49012 [11] W.-Y. Yan, Y.-P. Fang, and N.-J. Huang, “A new system of set-valued variational inclusions with H-monotone operators,” Mathematical Inequalities & Applications, vol. 8, no. 3, pp. 537-546, 2005. · Zbl 1070.49007 [12] S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 567147, 20 pages, 2009. · Zbl 1186.47075 [13] R. U. Verma, “General nonlinear variational inclusion problems involving A-monotone mappings,” Applied Mathematics Letters, vol. 19, no. 9, pp. 960-963, 2006. · Zbl 1114.49013 [14] Y. J. Cho, H.-Y. Lan, and R. U. Verma, “Nonlinear relaxed cocoercive variational inclusions involving (A,\eta )-accretive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1529-1538, 2006. · Zbl 1207.49011 [15] R. U. Verma, “Approximation solvability of a class of nonlinear set-valued variational inclusions involving (A,\eta )-monotone mappings,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 969-975, 2008. · Zbl 1140.49008 [16] Y.-Z. Zou and N.-J. Huang, “H(\cdot ,\cdot )-accretive operator with an application for solving variational inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 204, no. 2, pp. 809-816, 2008. · Zbl 1170.65043 [17] Y. Z. Zou and N. J. Huang, “A new system of variational inclusions involving H(\cdot , \cdot )-accretive operator in Banach spaces,” Applied Mathematics and Computation, vol. 212, pp. 135-144, 2009. · Zbl 1188.65092 [18] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127-1138, 1991. · Zbl 0757.46033 [19] Y.-P. Fang, N.-J. Huang, and H. B. Thompson, “A new system of variational inclusions with (H,\eta )-monotone operators in Hilbert spaces,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 365-374, 2005. · Zbl 1068.49003 [20] S. B. Nadler,, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475-488, 1969. · Zbl 0187.45002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.