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 x ̃ ∈ H ; x ̃ = ( proj F ( T ) ⋅ S ) x ̃ , where H is a Hilbert space, S is some nonlinear operator and T is a nonexpansive mapping on a closed convex subset C and proj F ( T ) denotes the metric projection on the set of fixed points of T . 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.