Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {T n } : C → H be a sequence of nearly nonexpansive mappings such that F := ∞ i=1 F T i = .Let V : C → H be a γ-Lipschitzian mapping and F : C → H be a L-Lipschitzian and η-strongly monotone operator.This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem.It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence {x n } converges strongly to x * ∈ F which is also the unique solution of the following variational inequality:As a special case, this projection method can be used to find the minimum norm solution of above variational inequality; namely, the unique solution x * to the quadratic minimization problem: x * = ar g mi n x∈F x 2 .The results here improve and extend some recent corresponding results of other authors.