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

Year 2016, Volume: 4 Issue: 2, 193 - 202, 01.03.2016

İbrahim Karahan , Murat Özdemir

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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