Iterative algorithms for hierarchical fixed points problems and variational inequalities

This paper deals with a method for approximating a solution of the fixed point problem: find x̃∈H; x̃=(projF(T)⋅S)x̃, where HH is a Hilbert space, SS is some nonlinear operator and TT is a nonexpansive mapping on a closed convex subset CC and projF(T) denotes the metric projection on the set of fixed points of TT. First, we prove a strong convergence theorem by using a projection method which solves some variational inequality. As a special case, this projection method also solves some minimization problems. Secondly, under different restrictions on parameters, we obtain another strong convergence result which solves the above fixed point problem.