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

Abstract

Let  be a nonempty closed convex subset of a real Hilbert space . Let  be a sequence of nearly nonexpansive mappings such that . Let  be a -Lipschitzian mapping and  be a -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  converges strongly to  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  to the quadratic minimization problem: . The results here improve and extend some recent corresponding results of other authors.

Keywords

• Variational inequality,hierarchical fixed point,nearly nonexpansivemappings

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