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Let X be a uniformly convex and q-uniformly smooth Banach space with $1< q\leq 2$. In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators $\{S_{n}\}^{\infty }_{n=0}$. Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by $\{S_{n}\}^{\infty }_{n=0}$.

Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems