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

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn} : C → H be a sequence of nearly nonexpansive mappings such that F := T ∞=1 F i Ti ¢ 6= ;. Let V : C → H be a °-Lipschitzian mapping and F : C → H be a L-Lipschitzian and ´-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 {xn} converges strongly to x ∗ ∈ F which is also the unique solution of the following variationalinequality: ­i ½V −µF ¢ x ∗ ,x −x ∗ ® ≤0, ∀x ∈F. 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 ∗ = argmin x∈F kxk 2 . The results here improve and extend some recent corresponding results of other authors.