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})?Ø. 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 {x_{n}} converges strongly to x^{*}?F which is also the unique solution of the following variational inequality: ?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}?x??. The results here improve and extend some recent corresponding results of other authors.