## A new general iterative method for solution of a new general system of variational inclusions for nonexpansive semigroups in Banach spaces.(English)Zbl 1221.47110

Summary: We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed points of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is treated.

### MSC:

 47J22 Variational and other types of inclusions 47J25 Iterative procedures involving nonlinear operators 47H20 Semigroups of nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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