Approximating a common point of fixed points of a pseudocontractive mapping and zeros of sum of monotone mappings

Naseer Shahzad,Habtu Zegeye

doi:10.1186/1687-1812-2014-85

Abstract

Let C be a closed and convex subset of a real Hilbert space H. Let T be a Lipschitzian pseudocontractive mapping of C into itself, A be a γ-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. We introduce an iteration scheme for finding a minimum-norm point of . Application to a common element of the set of fixed points of a Lipschitzian pseudocontractive and solutions of variational inequality for α-inverse strongly monotone mappings is included. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. To the best of our knowledge, approximating a common fixed point of pseudocontractive mappings with explicit scheme has not been possible and our result is even the first result that states the solution of a variational inequality in the set of fixed points of pseudocontractive mappings. Our scheme which is explicit is the best to use for the problem under consideration. MSC:47H05, 47H09, 47J25.

Highlights

Let C be a closed convex subset of a real Hilbert space H
A natural question arises: can we obtain an iterative scheme which converges strongly to a common point of fixed points of the pseudocontractive mapping T and zeros of two monotone mappings? It is our purpose in this paper to introduce an iterative scheme which converges strongly to a common minimum-norm point of fixed points of a Lipschitzian pseudocontractive mapping and zeros of sum of two monotone mappings
Theorem . improves Theorem . of Takahashi et al [ ] in the sense that our convergence is to the common minimum-norm point of fixed points of a Lipschitzian pseudocontractive mapping and zeros of sum of two monotone mappings

Summary

Introduction

Let C be a closed convex subset of a real Hilbert space H. ) for A : C → H a γ -inverse strongly monotone mapping and fixed points of a nonexpansive mapping T : C → C by considering the following iterative algorithm: x ∈ H, xn+ = αnxn + ( – αn)TPC(xn – λnAxn), n = , , .

Results

Conclusion