Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces
- Shih-sen Chang^{1}Email author,
- Jen-Chih Yao^{1},
- Lin Wang^{2} and
- Li Juan Qin^{3}
https://doi.org/10.1186/s13663-016-0559-7
© Chang et al. 2016
Received: 8 January 2016
Accepted: 1 June 2016
Published: 8 June 2016
Abstract
In this paper, a new modified proximal point algorithm involving fixed point iterates of asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces is proposed and the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a common fixed point of asymptotically nonexpansive mappings is proved.
Keywords
1 Introduction
Recently, many convergence results by the proximal point algorithm (shortly, the PPA) which was initiated by Martinet [1] in 1970 for solving optimization problems have been extended from the classical linear spaces such as Euclidean spaces, Hilbert spaces, and Banach spaces to the setting of manifolds (see [1–9]).
Also in 2015, Cholamjiak-Abdou-Cho [10] established the strong convergence of the sequence to minimizers of a convex function and to fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces.
Motivated and inspired by the research going on in this direction, it is naturally to put forward the following.
Open question
Can we establish the strong convergence of the sequence to minimizers of a convex function and to a common fixed point of asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces?
The purpose of this paper is to propose the modified proximal point algorithm using the S-type iteration process for four asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces and to prove some △- and strong convergence theorems of the proposed processes under suitable conditions.
Our results not only give an affirmative answer to the above open question but also generalize the corresponding results of Bačák [6], Ariza-Ruiz et al. [7], Cholamjiak-Abdou-Cho [10], Agarwal et al. [11], Dhompongsa-Panyanak [12], Khan-Abbas [13], and many others.
2 Preliminaries
Recall that a metric space \((X, d)\) is called a \(\operatorname{CAT}(0)\) space, if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A subset K of a \(\operatorname{CAT}(0)\) space X is convex if, for any \(x, y \in K\), we have \([x, y] \subset K\), where \([x, y]: = \{\lambda x \oplus(1-\lambda)y: 0 \le\lambda\le 1 \}\) is the unique geodesic joining x and y.
In order to save space, we will not repeat the geometric properties, some conclusions, and the △-convergence of \(\operatorname {CAT}(0)\) space here. The interested reader may refer to (for example) [12, 14–16].
In the sequel, we denote by \(F(T)\) the fixed point set of a mapping T.
Examples of convex functions in \(\operatorname{CAT}(0)\) space X:
Example 1
The function \(y \mapsto d(x, y): X \to[0, \infty)\) is convex.
Example 2
Example 3
The function \(y \mapsto d^{2}(z, y): X \to[0, \infty )\) is convex.
Let \(f : X \to(- \infty, \infty]\) be a proper convex and lower semi-continuous function. It was shown in [11] that the set \(F(J_{\lambda})\) of fixed points of the resolvent associated with f coincides with the set \(\operatorname{arg\,min}y_{X} f (y)\) of minimizers of f. Also for any \(\lambda> 0\), the resolvent \(J_{\lambda}\) of f is nonexpansive [17].
Lemma 2.1
(Sub-differential inequality [18])
Lemma 2.2
(Demi-closed principle [16])
Assume C is a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X and \(T : C \to C\) be an asymptotically nonexpansive mapping. Let \(\{ x_{n}\}\) be a bounded sequence in C such that \(\bigtriangleup\mbox{-} \lim x_{n} = p\) and \(\lim_{n\to\infty} d(x_{n}, Tx_{n}) = 0\). Then \(Tp = p\).
Lemma 2.3
[19]
Lemma 2.4
Lemma 2.5
(The resolvent identity [17])
3 Some △-convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces
We are now in a position to give the main results of the paper.
Theorem 3.1
- (1)
\((X,d)\) is a complete \(\operatorname{CAT}(0)\) space, and C is a nonempty, closed, and convex subset of X;
- (2)
\(f: C \to(-\infty, \infty]\) is a proper convex and lower continuous function;
- (3)\(T_{i}:C\rightarrow C\) and \(S_{i}:C\rightarrow C\), \(i=1,2\) all are \(\{k_{n}\}\)-asymptotically nonexpansive mappings with \(k_{n} \in[1, \infty )\), \(k_{n} \to1\) and \(\sum_{i = 1}^{\infty}(k_{n} - 1) < \infty\) such that$$ \Omega: = F(T_{1})\cap F(T_{2})\cap F(S_{1})\cap F(S_{2})\cap \mathop{\operatorname{arg\,min}}_{y \in C}f(y) \neq \emptyset; $$(3.1)
- (4)\(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\), \(\{\delta_{n}\}\), \(\{ \eta_{n}\}\), \(\{\xi_{n}\}\) are sequences in \([0, 1]\) withwhere a is a positive constant in \((0, 1)\);$$ \begin{aligned} &\alpha_{n} + \beta_{n} + \gamma_{n} = 1, \\ &\delta_{n} + \eta_{n} + \xi_{n} = 1,\quad 0 < a \le \alpha_{n}, \beta_{n}, \gamma_{n}, \delta_{n}, \eta_{n}, \xi_{n} < 1, \forall n \ge1, \end{aligned} $$(3.2)
- (5)
\(\{\lambda_{n}\}\) is a sequence such that \(\lambda_{n} \ge\lambda> 0\) for all \(n \ge1\) and some λ.
Proof
(I) First we prove that the limit \(\lim_{n \to\infty} d(x_{n}, q)\) exists.
(II) Now we prove that \(\lim_{n \to\infty} d(x_{n}, z_{n}) = 0\).
Finally, we show that the sequence \(\{x_{n}\}\) △-converges to a point in Ω. To this end, it suffices to show that \(w_{\bigtriangleup}(x_{n})\) consists of exactly one point. Let \(\{u_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\}) = \{u\}\) and let \(A(\{x_{n}\}) = \{x\}\). Since \(u \in w_{\bigtriangleup}(x_{n}) \subset\Omega\) and \(\{d(x_{n},u)\}\) converges, we have \(x = u\) [12]. Hence \(w_{\bigtriangleup}(x_{n}) = \{x\}\).
This completes the proof of Theorem 3.1. □
Remark 3.2
- 1.
Theorem 3.1 generalizes the main results in Agarwal et al. [11] and Khan-Abbas [13] from one nonexpansive mapping to four asymptotically nonexpansive mappings involving the convex and lower semi-continuous function in \(\operatorname{CAT}(0)\) spaces.
- 2.
Theorem 3.1 extends the main result in Bačák [6], and the corresponding results in Ariza-Ruiz et al. [7] and Cholamjiak et al. [10]. In fact, we present a new modified proximal point algorithm for solving the convex minimization problem as well as the fixed point problem of asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces.
Since every real Hilbert space H is a complete \(\operatorname {CAT}(0)\) space, the following result can be obtained from Theorem 3.1 immediately.
Corollary 3.3
4 Some strong convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces
Let \((X,d)\) be a \(\operatorname{CAT}(0)\) space, and C be a nonempty, closed, and convex subset of X.
Recall that a mapping \(T: C \to C\) is said to be demi-compact, if for any bounded sequence \(\{x_{n}\}\) in C such that \(d(x_{n}, Tx_{n}) \to 0\) (as \(n \to\infty\)), there exists a subsequence \(\{x_{n_{i}}\} \subset \{x_{n}\}\) such that \(\{x_{n_{i}}\}\) converges strongly (i.e., in metric topology) to some point \(p \in C\).
Theorem 4.1
Under the assumptions of Theorem 3.1, if, in addition, one of \(S_{1}\), \(S_{2}\), \(T_{1}\), and \(T_{2}\) is demi-compact, then the sequence \(\{ x_{n}\}\) defined by (3.3) converges strongly (i.e., in metric topology) to a point \(x^{*} \in\Omega\).
Proof
This completes the proof of Theorem 4.1. □
Theorem 4.2
Proof
By virtue of the definition of \(\{x_{n}\}\) and (4.5), it is easy to prove that \(\{x_{n}\}\) is a Cauchy sequence in C. Since C is a closed subset in a complete \(\operatorname{CAT}(0)\) space X, it is complete. Without loss of generality, we can assume that \(\{x_{n}\}\) converges strongly to some point \(p^{*}\). It is easy to see that \(F(J_{\lambda})\), \(F(T_{i})\), and \(F(S_{i})\), \(i = 1, 2\), all are closed subsets in C, so is Ω. Since \(\lim_{n \to \infty} d(x_{n}, \Omega) = 0\), \(p^{*} \in\Omega\). This completes the proof of Theorem 4.2. □
Declarations
Acknowledgements
The authors would like to express their thanks to the referees for their helpful comments and advices. This study was supported by the National Natural Science Foundation of China (Grant No. 11361070) and the Natural Science Foundation of China Medical University, Taiwan.
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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