The modified S-iteration process for nonexpansive mappings in \(\operatorname{CAT}(\kappa)\) spaces
- Raweerote Suparatulatorn^{1} and
- Prasit Cholamjiak^{1}Email author
https://doi.org/10.1186/s13663-016-0515-6
© Suparatulatorn and Cholamjiak 2016
Received: 20 October 2015
Accepted: 28 February 2016
Published: 9 March 2016
Abstract
We establish Δ-convergence results of a sequence generated by the modified S-iteration process for two nonexpansive mappings in complete \(\operatorname {CAT}(\kappa)\) spaces. Some numerical examples are also provided to compare with the Ishikawa-type iteration process. Our main result extends the corresponding results in the literature.
Keywords
MSC
1 Introduction
The concept of Δ-convergence in general metric spaces was introduced by Lim [2]. Kirk [3] has proved the existence of fixed point of nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces. Kirk and Panyanak [4] specialized this concept to \(\operatorname {CAT}(0)\) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. Dhompongsa and Panyanak [5] continued to work in this direction. Their results involved the Mann and Ishikawa iteration process involving one mapping. After that Khan and Abbas [6] studied the approximation of common fixed point by the Ishikawa-type iteration process involving two mappings in \(\operatorname {CAT}(0)\) spaces.
There have been, recently, many convergence and existence results established in \(\operatorname {CAT}(0)\) and \(\operatorname {CAT}(\kappa)\) spaces (see [11–19]).
2 Preliminaries and lemmas
In this section, we provide some basic concepts, definitions, and lemmas which will be used in the sequel and can be found in [20].
Let \((X,d)\) be a metric space and \(x,y\in X\) with \(d(x,y)=l\). A geodesic path from x to y is an isometry \(c:[0,l]\rightarrow X\) such that \(c(0)=x\), \(c(l)=y\). The image of a geodesic path is called geodesic segment. The space \((X,d)\) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is a uniquely geodesic space if every two points of X are joined by only one geodesic segment. We write \((1-t)x\oplus ty\) for the unique point z in the geodesic segment joining x and y such that \(d(x,z)=td(x,y)\) and \(d(y,z)=(1-t)d(x,y)\) for \(t\in[0,1]\). A subset E of X is said to be convex if E includes every geodesic segment joining any two of its points.
Let D be a positive number. A metric space \((X, d)\) is called a D-geodesic space if any two points of X with the distance less than D are joined by a geodesic. If this holds in a convex set E, then E is said to be D-convex. For a constant κ, we denote \(M_{\kappa}\) by the 2-dimensional, complete, simply connected spaces of curvature κ.
Definition 2.1
A metric space \((X, d)\) is called a \(\operatorname {CAT}(\kappa)\) space if it is \(D_{\kappa}\)-geodesic and any geodesic triangle \(\Delta(x, y, z)\) in X with \(d(x, y) + d(y, z) + d(z, x) <2 D_{\kappa}\) satisfies the \(\operatorname {CAT}(\kappa)\) inequality.
Since the results in \(\operatorname {CAT}(\kappa)\) spaces can be deduced from those in \(\operatorname {CAT}(1)\) spaces, we now sufficiently state lemmas on \(\operatorname {CAT}(1)\) spaces.
Lemma 2.2
[20]
Let \((X,d)\) be a \(\operatorname {CAT}(1)\) space and let K be a closed and π-convex subset of X. Then for each point \(x\in X\) such that \(d(x,K)<\pi/2\), there exists a unique point \(y\in K\) such that \(d(x,y)=d(x,K)\).
Lemma 2.3
[21]
Definition 2.4
A sequence \(\{x_{n}\}\) in X is said to Δ-converge to \(x\in X\) if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\).
In this case we write Δ-\(\lim_{n\rightarrow\infty}x_{n}=x\) and call x the Δ-limit of \(\{x_{n}\}\).
Definition 2.5
For a sequence \(\{x_{n}\}\) in X, a point \(x\in X\) is a Δ-cluster point of \(\{x_{n}\}\) if there exists a subsequence of \(\{x_{n}\}\) that Δ-converges to x.
Lemma 2.6
[7]
Definition 2.7
Lemma 2.8
[7]
Let \((X,d)\) be a complete \(\operatorname {CAT}(1)\) space and let K be a nonempty subset of X. Suppose that the sequence \(\{x_{n}\}\) in X is Fejér monotone with respect to K and the asymptotic radius \(r(\{x_{n}\})\) of \(\{x_{n}\}\) is less than \(\pi/2\). If any Δ-cluster point x of \(\{x_{n}\}\) belongs to K, then \(\{x_{n}\}\) Δ-converges to a point in K.
3 Main results
Lemma 3.1
Let \((X,d)\) be a complete \(\operatorname {CAT}(1)\) space and let C be a nonempty, closed, and convex subset of X. Let T and S be two nonexpansive mappings of C such that \(F := \operatorname {Fix}(T)\cap \operatorname {Fix}(S)\neq\emptyset\). Let \(\{ x_{n}\}\) be defined by (1.5) for \(x_{0}\in C\) such that \(d(x_{0},F)\leq\pi/4\). Then there exists a unique point p in F such that \(d(y_{n},p)\leq d(x_{n},p)\leq\pi/4\) for all \(n\geq0\).
Proof
Lemma 3.2
- (i)
\(\lim_{n\to\infty}d(x_{n},p)\) exists;
- (ii)
\(\lim_{n\to\infty}d(Tx_{n},x_{n})=0=\lim_{n\to\infty}d(Sx_{n},x_{n})\).
Proof
Theorem 3.3
Let \((X,d)\) be complete a \(\operatorname {CAT}(\kappa)\) space and let C be a nonempty, closed, and convex subset of X. Let T and S be two nonexpansive mappings of C such that \(F:=\operatorname {Fix}(T)\cap \operatorname {Fix}(S)\neq \emptyset\). Let \(\{a_{n}\}\) and \(\{b_{n}\}\) be such that \(0< a\leq a_{n},b_{n}\leq b<1\) for all \(n\geq0\) and for some a, b. If \(\{x_{n}\} \) is defined by (1.5) for \(x_{0}\in C\) such that \(d(x_{0},F)< D_{\kappa}/4\), then \(\{x_{n}\}\) Δ-converges to a point in F.
Proof
We immediately obtain the following results in \(\operatorname {CAT}(0)\) spaces.
Corollary 3.4
Let \((X,d)\) be a complete \(\operatorname {CAT}(0)\) space and let C be a nonempty, closed, and convex subset of X. Let T and S be two nonexpansive mappings of C such that \(F:=\operatorname {Fix}(T)\cap \operatorname {Fix}(S)\neq\emptyset\). Let \(\{ a_{n}\}\) and \(\{b_{n}\}\) be such that \(0< a\leq a_{n},b_{n}\leq b<1\) for all \(n\geq0\) and for some a, b. If \(\{x_{n}\}\) is defined by (1.5), then \(\{x_{n}\}\) Δ-converges to a point in F.
Remark 3.5
When \(S=T\), we obtain Theorem 1 of Khan and Abbas [6].
Along a similar proof line, we can obtain the following result for the Ishikawa-type iteration process.
Theorem 3.6
Let \((X,d)\) be a complete \(\operatorname {CAT}(\kappa)\) space and let C be a nonempty, closed, and convex subset of X. Let T and S be two nonexpansive mappings of C such that \(F:=\operatorname {Fix}(T)\cap \operatorname {Fix}(S)\neq \emptyset\). Let \(\{a_{n}\}\) and \(\{b_{n}\}\) be such that \(0< a\leq a_{n},b_{n}\leq b<1\) for all \(n\geq0\) and for some a, b. If \(\{x_{n}\} \) is defined by (1.4) for \(x_{0}\in C\) such that \(d(x_{0},F)< D_{\kappa}/4\), then \(\{x_{n}\}\) Δ-converges to a point in F.
Corollary 3.7
[6] Let \((X,d)\) be a complete \(\operatorname {CAT}(0)\) space and let C be a nonempty, closed, and convex subset of X. Let T and S be two nonexpansive mappings of C such that \(F:=\operatorname {Fix}(T)\cap \operatorname {Fix}(S)\neq\emptyset\). Let \(\{ a_{n}\}\) and \(\{b_{n}\}\) be such that \(0< a\leq a_{n},b_{n}\leq b<1\) for all \(n\geq0\) and for some a, b. If \(\{x_{n}\}\) is defined by (1.4), then \(\{x_{n}\}\) Δ-converges to a point in F.
4 Numerical examples
In this section, we consider the m-sphere \(\mathbb{S}^{m}\), which is a \(\operatorname {CAT}(\kappa)\) space.
Example 4.1
n | \(x_{n}\) is defined by (4.1) | \(x_{n}\) is defined by (4.2) |
---|---|---|
1 | (0.64798180, 0.43974219, 0.43974219, 0.43974219) | (0.72059826, 0.40030744, 0.40030744, 0.40030744) |
2 | (0.75902424, 0.37589103, 0.37589103, 0.37589103) | (0.85117540, 0.30304039, 0.30304039, 0.30304039) |
3 | (0.83727864, 0.31568152, 0.31568152, 0.31568152) | (0.92240530, 0.22298615, 0.22298615, 0.22298615) |
4 | (0.89102247, 0.26209347, 0.26209347, 0.26209347) | (0.95999322, 0.16167149, 0.16167149, 0.16167149) |
5 | (0.92740383, 0.21596460, 0.21596460, 0.21596460) | (0.97950207, 0.11629804, 0.11629804, 0.11629804) |
6 | (0.95181217, 0.17706270, 0.17706270, 0.17706270) | (0.98953675, 0.08330070, 0.08330070, 0.08330070) |
7 | (0.96809244, 0.14468014, 0.14468014, 0.14468014) | (0.99467153, 0.05952183, 0.05952183, 0.05952183) |
8 | (0.97890884, 0.11795122, 0.11795122, 0.11795122) | (0.99729071, 0.04247057, 0.04247057, 0.04247057) |
9 | (0.98607580, 0.09601130, 0.09601130, 0.09601130) | (0.99862397, 0.03027739, 0.03027739, 0.03027739) |
10 | (0.99081568, 0.07806896, 0.07806896, 0.07806896) | (0.99930171, 0.02157235, 0.02157235, 0.02157235) |
⋮ | ⋮ | ⋮ |
55 | (1.00000000, 0.00000620, 0.00000620, 0.00000620) | (1.00000000, 0.00000000, 0.00000000, 0.00000000) |
We next consider the hyperbolic m-space \(\mathbb{H}^{m}\).
Example 4.2
n | \(x_{n}\) is defined by (4.3) | \(x_{n}\) is defined by (4.4) |
---|---|---|
1 | (1.77547237, 0.98692599, 1.18530561, 2.55563582) | (1.08924010, 0.51696725, 0.46166164, 1.63304335) |
2 | (0.86101141, 1.01967810, 0.66960079, 1.79706686) | (0.09467395, 0.45823859, 0.19274032, 1.12075626) |
3 | (0.50793301, 0.63730440, 0.65740263, 1.44787122) | (0.09214727, −0.02529962, 0.22462285, 1.02936224) |
4 | (0.45512909, 0.39762058, 0.47219289, 1.26024234) | (0.11301324, 0.05817553, −0.05672644, 1.00964067) |
5 | (0.35116401, 0.33063402, 0.31268711, 1.15343324) | (−0.05463911, 0.05458569, 0.04470058, 0.04470058) |
6 | (0.24499408, 0.26285320, 0.24775268, 1.09109821) | (0.03652253, −0.04228414, 0.02306040, 1.00182515) |
7 | (0.18906704, 0.19095564, 0.19822904, 1.05427945) | (0.00627063, 0.02939078, −0.02899485, 1.00087154) |
8 | (0.15056381, 0.14567561, 0.14817634, 1.03239870) | (−0.01800599, −0.00210393, 0.02266815, 1.00042115) |
9 | (0.11463049, 0.11504761, 0.11272592, 1.01935432) | (0.01662614, −0.00999700, −0.00565079, 1.00020413) |
10 | (0.08734712, 0.08852508, 0.08831015, 1.01156557) | (−0.00652282, 0.01156060, −0.00466467, 1.00009897) |
⋮ | ⋮ | ⋮ |
50 | (0.00000284, 0.00000284, 0.00000284, 1.00000000) | (0.00000000, 0.00000000, 0.00000000, 1.00000000) |
From the numerical experience, we observe that the convergence rate of S-iteration process is much quicker than that of the Ishikawa iteration process.
Remark 4.3
The convergence behavior of Mann and Halpern iterations in Hadamard manifolds can be found in the work of Li et al. [18].
Declarations
Acknowledgements
The authors wish to thank the referees for valuable suggestions. This research was supported by University of Phayao.
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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