- Volume 25 Issue 1
In this paper, we introduce and study a system of nonlinear set-valued implicit variational inclusions (SNSIVI) with relaxed cocoercive mappings in real Banach spaces. By using resolvent operator technique for M-accretive mapping, we construct a new class of iterative algorithms for solving this class of system of set-valued implicit variational inclusions. The convergence of iterative algorithms is proved in q-uniformly smooth Banach spaces. Our results generalize and improve the corresponding results of recent works.
system of nonlinear set-valued implicit variational inclusion;resolvent operator, M-accretive mapping;iterative algorithm;convergence
- C. Z. Bai and J. X. Fang, A system of nonlinear variational inclusions in real Banach spaces, Bull. Korean Math. Soc. 40 (2003), no. 3, 385–397. https://doi.org/10.4134/BKMS.2003.40.3.385
- S. S. Chang, H. W. Joseph Lee, and C. K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (2007), no. 3, 329–334. https://doi.org/10.1016/j.aml.2006.04.017
- K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
- Y. P. Fang and N. J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004), no. 6, 647–653. https://doi.org/10.1016/S0893-9659(04)90099-7
- Y. P. Fang and N. J. Huang, H-Monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145 (2003), no. 2-3, 795–803. https://doi.org/10.1016/S0096-3003(03)00275-3
- H. Nie, Z. Liu, K. H. Kim, and S. M. Kang, A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings, Adv. Nonlinear Var. Inequal. 6 (2003), no. 2, 91–99.
- M. Aslam Noor, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math. Anal. Appl. 255 (2001), no. 2, 589–604. https://doi.org/10.1006/jmaa.2000.7298
- R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. Optim. Theory Appl. 121 (2004), no. 1, 203–210. https://doi.org/10.1023/B:JOTA.0000026271.19947.05
- R. U. Verma, General convergence analysis for two-step projection methods and applications to variational problem, Appl. Math. Lett. 18 (2005), no. 11, 1286–1292. https://doi.org/10.1016/j.aml.2005.02.026
- H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1985. https://doi.org/10.1016/0362-546X(91)90200-K
- Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces vol.2013, 2013, https://doi.org/10.1155/2013/461371