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 {T n } : C → H be a sequence of nearly nonexpansive mappings such thatC → 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.

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