A projection method for approximating fixed points of quasinonexpansive mappings in Hadamard spaces
- Shuechin Huang^{1}Email author and
- Yasunori Kimura^{2}
https://doi.org/10.1186/s13663-016-0523-6
© Huang and Kimura 2016
Received: 17 November 2015
Accepted: 3 March 2016
Published: 16 March 2016
Abstract
This work is devoted to analyzing the feasibility study of a Moudafi viscosity projection method with a weak contraction for a finite family of quasinonexpansive mappings in a Hadamard space. To this end, we need to construct a countable family of nonexpansive mappings satisfying AKTT condition with a weak contraction by choosing an appropriate control sequence under certain conditions.
Keywords
MSC
1 Introduction
Let C be a nonempty subset of a metric space \((X,d)\). Suppose that, for each \(x \in X\), there exists a unique point \(P_{C}x \in C\) such that \(d(x,P_{C}x)=d(x,C)=\inf_{y \in C}d(x,y)\). Then, the mapping \(P_{C}\) of X onto C is called the metric projection.
Approximation methods for finding specific fixed points of a family of nonexpansive mappings in Hilbert, Banach, and geodesic metric spaces have been studied by many researchers; see, e.g., [3–9] and the references therein. One well-known method, called the shrinking projection method, was first proposed by Takahashi et al. [10] and has been applied to a variety of approximation problems; see, e.g., [11, 12]. In particular, Kimura and Takahashi [11] applied this method to the zero-point problem for a maximal monotone operator defined in a Banach space and obtained strong convergence theorems. To generate the iterative sequence by the shrinking projection method, they use the metric projection onto a closed convex set \(C_{n}\) for each \(n \in\mathbb{N}\). It is noticeable that the larger the integer n, the more complicated the shape of \(C_{n}\). Hence, the calculation of the projection is tedious as n gets larger. In 2011, Kimura et al. [2] overcome this difficulty and introduce the so-called averaged projection method of Halpern type for a family of quasinonexpansive mappings by combining the Halpern iteration. They still use the metric projection approach; nevertheless, the subsets corresponding to these projections have simpler shapes than the classical ones. Let us denote by \(\mathfrak{F}(\mathfrak{T})\) the common fixed point set of all mappings in a family \(\mathfrak{T}\). Their theorem is stated as follows.
Theorem 1.1
(Kimura et al. [2], Theorem 3.1)
- (i)
\(\liminf_{n \to\infty}\alpha_{n}<1\),
- (ii)
\(\beta^{j}_{n}>0\) for \(j=1,\ldots,N\), and \(\sum_{j=1}^{N}\beta^{j}_{n}=1\) for \(n \in\mathbb{N}\),
- (iii)
\(\sum_{k=1}^{n}\gamma_{n,k}=1\) for \(n \in\mathbb{N}\), \(\lim_{n \to\infty}\gamma_{n,k}>0\) for \(k \in\mathbb{N}\), and \(\sum_{n=1}^{\infty}\sum_{k=1}^{n}|\gamma_{n+1,k}-\gamma _{n,k}|<\infty\),
- (iv)
\(\lim_{n \to\infty}\delta_{n}=0\), \(\sum_{n=1}^{\infty}\delta_{n}=\infty\), and \(\sum_{n=1}^{\infty}|\delta_{n+1}-\delta_{n}|<\infty\).
The problem of whether or not we can construct a shrinking projection method analogous to that given in Theorem 1.1 for solving a common fixed point problem for a finite family of quasinonexpansive mappings in a geodesic metric space is still open. The purpose of this paper is to analyze the feasibility study of Moudafi viscosity type of projection method with a weak contraction for a finite family of quasinonexpansive mappings in a complete \(\operatorname{CAT}(0)\) space, also known as a Hadamard space.
This paper is organized as follows. In Section 2 we recall the definition of geodesic metric spaces and summarize some useful lemmas and the main properties of \(\operatorname{CAT}(0)\) spaces. Besides, without vector addition as in a Banach space, we present an inequality to estimate the distance between two elements defined by finite convex combination ‘⊕’ in a \(\operatorname{CAT}(0)\) space; see Lemma 2.2. In Section 3 we construct a sequence of nonexpansive mappings satisfying AKTT condition by choosing an appropriate control sequence under certain conditions; see Theorem 3.2. Therefore, a convergence theorem of a new Moudafi viscosity approximation follows from Theorem 3.2; see Theorem 3.3. Using Theorem 3.3, we also derive a strong convergence theorem by a Moudafi type viscosity approximation with a weak contraction for a family of quasinonexpansive mappings; see Theorem 3.4. As a particular case where a weak contraction is constant in Theorem 3.4, a strong convergence theorem by the averaged projection method of Halpern type is then obtained; see Theorem 3.5.
2 Preliminaries
Let \((X,d)\) be a metric space. For \(x,y \in X\), a geodesic path joining x to y (or a geodesic from x to y) is an isometric mapping \(c:[0,\ell] \subset\mathbb{R} \to X\) such that \(c(0)=x\), \(c(\ell)=y\), that is, \(d(c(t),c(t'))=|t-t'|\) for all \(t,t' \in[0,\ell]\). Therefore, \(d(x,y)=\ell\). The image of c is called a geodesic (segment) from x to y, and we shall denote a definite choice of this geodesic segment by \([x,y]\). A point \(z=c(t)\) in the geodesic \([x,y]\) will be written as \(z=(1-\lambda)x \oplus\lambda y\), where \(\lambda=t/\ell\), and so \(d(z,x)=\lambda d(x,y)\) and \(d(z,y)=(1-\lambda)d(x,y)\). A subset C of X is convex if every pair of points \(x,y \in C\) can be joined by a geodesic in X and the image of every such geodesic is contained in C.
A geodesic triangle \(\triangle(x_{1},x_{2},x_{3})\) in \((X,d)\) consists of three points \(x_{i} \in X\) (\(i=1,2,3\)), its vertices, and a geodesic segment between each pair of vertices, its sides. If a point \(x \in X\) lies in the union of \([x_{i},x_{j}]\), \(i,j \in\{ 1,2,3\}\), then we write \(x \in\triangle(x_{1},x_{2},x_{3})\). A comparison triangle for the geodesic triangle \(\triangle (x_{1},x_{2},x_{3})\) in X is a triangle \(\triangle(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})\) in the Euclidean plane \(\mathbb{E}^{2}\) such that \(d_{\mathbb{E}^{2}}(\bar{x}_{i},\bar{x}_{j})=d(x_{i},x_{j})\) for \(i,j \in\{1,2,3\}\).
Lemma 2.1
- (i)For \(x,y \in X\), we have$$d\bigl(\alpha x \oplus(1-\alpha)y,\beta x \oplus(1-\beta)y\bigr)=|\alpha- \beta|d(x,y). $$
- (ii)([13], Chapter II.2. Proposition 2.2) For \(x,y,p,q \in X\), we haveIn particular, if \(p=q\), this reduces to$$d\bigl(\alpha x \oplus(1-\alpha)y,\alpha p \oplus(1-\alpha)q\bigr) \le\alpha d(x,p)+(1-\alpha) d(y,q). $$$$d\bigl(\alpha x \oplus(1-\alpha)y,p\bigr) \le\alpha d(x,p)+(1-\alpha)d(y,p). $$
- (iii)([14], Lemma 2.5) For \(x,y,z \in X\), we have$$d\bigl(\alpha x \oplus(1-\alpha)y,z\bigr)^{2} \le\alpha d(x,z)^{2}+(1-\alpha )d(y,z)^{2}-\alpha(1- \alpha)d(x,y)^{2}. $$
Lemma 2.2
Proof
We will prove the result by induction.
Lemma 2.3
To verify our main results in Section 3, the following property is required and crucial.
Lemma 2.4
(Dhompongsa et al. [5], Lemma 3.8)
Let C be a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, \(\{T_{n}\}\) a sequence of nonexpansive mappings on C with \(\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n}) \ne\emptyset\), and \(\{\alpha_{n}\}\) a sequence in \((0,1)\) such that \(\sum_{n=1}^{\infty}\alpha_{n}=1\) and \(\lim_{n \to\infty}\sum_{k=n}^{\infty}\alpha'_{k}=0\). Define the mapping \(S:C \to C\) by \(Sx=\bigoplus_{n=1}^{\infty}\alpha _{n}T_{n}x\), \(x \in C\). Then S is nonexpansive, and \(\mathfrak{F}(S)=\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n})\).
3 Projection method
Theorem 3.1
(Huang [15], Theorem 4.11)
- (C1)
\(\lim_{n \to\infty}\alpha_{n}=0\);
- (C2)
\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
- (C3)
either \(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\), or \(\lim_{n \to\infty}(\alpha_{n+1}/\alpha_{n})=1\).
We now construct a sequence of nonexpansive mappings satisfying AKTT condition by choosing an appropriate control sequence under certain conditions.
Theorem 3.2
- (D1)
\(\sum_{k=1}^{n}\gamma_{n,k}=1\), \(\forall n \in\mathbb{N}\);
- (D2)
\(\lambda_{k}=\lim_{n \to\infty}\gamma_{n,k}>0\), \(\forall k \in\mathbb {N}\), and \(\lim_{n \to\infty}\sum_{k=n}^{\infty}\lambda'_{k}=0\);
- (D3)\(\sum_{n=1}^{\infty}\sum_{k=1}^{n+1}|\bar{\gamma}_{n+1,k}-\bar{\gamma }_{n,k}|<\infty\), where \(\gamma_{n,n+1}=0\) and$$\bar{\gamma}_{n,k}=\frac{\gamma_{n,k}}{\gamma_{n,1}+\cdots+\gamma _{n,k}}, \quad k=1,\ldots,n+1. $$
Proof
The following result follows immediately from Theorems 3.1 and 3.2.
Theorem 3.3
Proof
Using Theorem 3.3, we establish a strong convergence theorem by a Moudafi type of shrinking projection method for a family of quasinonexpansive mappings as follows.
Theorem 3.4
- (i)
\(\liminf_{n \to\infty}\delta_{n}<1\);
- (ii)
\(\sum_{j=1}^{N}\beta^{j}_{n}=1\) for \(n \in\mathbb{N}\).
Proof
Consequently, when f is constant in Theorem 3.4, we obtain the following strong convergence theorem by a new Halpern type of shrinking projection method.
Theorem 3.5
Declarations
Acknowledgements
The first author is supported by a grant MOST 104-2115-M-259-004 from the Ministry of Science and Technology of Taiwan and therefore thanks the MOST financial support. The second author is supported by JSPS KAKENHI Grant Number 15K05007 from Japan Society for the Promotion of Science. The authors would like to express the most sincere thanks to Professor Qamrul Hasan Ansari, Editor of Fixed Point Theory and Applications, and the anonymous referees for their careful reading of the manuscript and for giving insightful comments and the citation of [9].
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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