On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hil
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RESEARCH
Open Access
On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces Yuan Hecai* *
Correspondence: [email protected] School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Abstract In this paper, a monotone inclusion problem and a fixed point problem of nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces. MSC: 47H05; 47H09; 47J25 Keywords: maximal monotone operator; nonexpansive mapping; Hilbert space; fixed point
1 Introduction Variational inclusion has become rich of inspiration in pure and applied mathematics. In recent years, classical variational inclusion problems have been extended and generalized to study a large variety of problems arising in image recovery, economics, and signal processing; for more details, see [–]. Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems. This alternative formulation has played a fundamental and significant part in developing several numerical methods for solving variational inclusion problems and related optimization problems. The purposes of this paper is to study the zero point problem of the sum of a maximal monotone mapping and an inverse-strongly monotone mapping, and the fixed point problem of a nonexpansive mapping. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In Section , a Mann-type iterative algorithm with mixed errors is investigated. A weak convergence theorem is established. Applications of the main results are also discussed in this section. 2 Preliminaries Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm · , respectively. Let C be a nonempty closed convex subset of H and let PC be the metric projection from H onto C. Let S : C → C be a mapping. F(S) stands for the fixed point set of S; that is, F(S) := {x ∈ C : x = Sx}. © 2013 Hecai; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hecai Fixed Point Theory and Applications 2013, 2013:155 http://www.fixedpointtheoryandapplications.com/content/2013/1/155
Page 2 of 13
Recall that S is said to be nonexpansive iff Sx – Sy ≤ x – y,
∀x, y ∈ C.
If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see []. Let A : C → H be a mapping. Recall that A is said to be monotone iff Ax – Ay, x – y ≥ ,
∀x, y ∈ C.
A is said to be strongly monotone iff there exists a constant α > such that Ax – Ay, x – y ≥ αx – y ,
∀x, y ∈ C
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