On solutions of inclusion problems and fixed point problems
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RESEARCH
Open Access
On solutions of inclusion problems and fixed point problems Yuan Hecai* *
Correspondence: [email protected] School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Abstract An inclusion problem and a fixed point problem are investigated based on a hybrid projection method. The strong convergence of the hybrid projection method is obtained in the framework of Hilbert spaces. Variational inequalities and fixed point problems of quasi-nonexpansive mappings are also considered as applications of the main results. MSC: 47H05; 47H09; 47J25 Keywords: nonexpansive mapping; inverse-strongly monotone mapping; maximal monotone operator; fixed point
1 Introduction and preliminaries Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators. Splitting methods for linear equations were introduced by Peaceman and Rachford [] and Douglas and Rachford []. Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg [] and Lions and Mercier []. The central problem is to iteratively find a zero of the sum of two monotone operators A and B in a Hilbert space H. In this paper, we consider the problem of finding a solution to the following problem: find an x in the fixed point set of the mapping S such that x ∈ (A + B)– (), where A and B are two monotone operators. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [–] and the references therein. Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and norm · , respectively. Let C be a nonempty closed convex subset of H and PC be the metric projection from H onto C. Let S : C → C be a mapping. In this paper, we use F(S) to denote the fixed point set of S; that is, F(S) := {x ∈ C : x = Sx}. Recall that S is said to be nonexpansive iff Sx – Sy ≤ x – y,
∀x, y ∈ C.
© 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, 2013:11 http://www.fixedpointtheoryandapplications.com/content/2013/1/11
Page 2 of 11
If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see []. S is said to be quasi-nonexpansive iff F(S) = ∅ and Sx – y ≤ x – y,
∀x ∈ C, y ∈ F(S).
It is easy to see that nonexpansive mappings are Lipschitz continuous; however, the quasi-nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasinonexpansive mapping i
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