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The modified Siteration process for nonexpansive mappings in \(\operatorname{CAT}(\kappa)\) spaces
Fixed Point Theory and Applications volume 2016, Article number: 25 (2016)
Abstract
We establish Δconvergence results of a sequence generated by the modified Siteration process for two nonexpansive mappings in complete \(\operatorname {CAT}(\kappa)\) spaces. Some numerical examples are also provided to compare with the Ishikawatype iteration process. Our main result extends the corresponding results in the literature.
Introduction
Let C be a nonempty subset of a metric space \((X, d)\). A mapping \(T : C\rightarrow C\) is said to be nonexpansive if
for all \(x, y\in C\). We say that \(x\in C\) is a fixed point of T if
We denote the set of all fixed points of T by \(\operatorname {Fix}(T)\); for more details see [1].
The concept of Δconvergence 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 Ishikawatype iteration process involving two mappings in \(\operatorname {CAT}(0)\) spaces.
The Mann iteration process [7] was defined by \(x_{0}\in C\) and
where \(\{a_{n}\}\) is a sequence in \((0,1)\). He et al. [7] proved the convergence results in \(\operatorname {CAT}(\kappa)\) spaces.
The Ishikawa iteration process [8] was defined by \(x_{0}\in C\) and
where \(\{a_{n}\}\) and \(\{b_{n}\}\) are sequences in \((0,1)\). Jun [8] proved that the sequence \(\{x_{n}\}\) generated by (1.2) Δconverges to a fixed point of T in \(\operatorname {CAT}(\kappa)\) spaces.
The Siteration process [9] was defined by \(x_{0}\in C\) and
where \(\{a_{n}\}\) and \(\{b_{n}\}\) are sequences in \((0,1)\). This scheme has a better convergence rate than those of (1.1) and (1.2) for a contraction in metric space (see [9]).
In 2011, Khan and Abbas [6] studied the iteration (1.3) in \(\operatorname {CAT}(0)\) spaces and proved the Δconvergence. Khan and Abbas [6] also studied the following Ishikawatype iteration process: \(x_{0}\in C\) and
where \(\{a_{n}\}\) and \(\{b_{n}\}\) are sequences in \((0,1)\). This iteration was introduced by Das and Debata [10]. They proved some results on Δconvergence in \(\operatorname {CAT}(0)\) spaces for two nonexpansive mappings of the sequence defined by (1.4).
There have been, recently, many convergence and existence results established in \(\operatorname {CAT}(0)\) and \(\operatorname {CAT}(\kappa)\) spaces (see [11–19]).
Motivated by [6] and [9], in this paper, we study the following modified Siteration process: \(x_{0}\in C\) and
where \(\{a_{n}\}\) and \(\{b_{n}\}\) are sequences in \((0,1)\). We prove some results on Δconvergence for two nonexpansive mappings in \(\operatorname {CAT}(\kappa)\) spaces with \(\kappa\geq0\) under suitable conditions. We finally provide some examples and numerical results to support our main result.
Remark 1.1
We note that this scheme reduces to the iteration process (1.3) when \(S=T\). The iteration process (1.5) is quite different from (1.4).
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 \((1t)x\oplus ty\) for the unique point z in the geodesic segment joining x and y such that \(d(x,z)=td(x,y)\) and \(d(y,z)=(1t)d(x,y)\) 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 Dgeodesic 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 Dconvex. For a constant κ, we denote \(M_{\kappa}\) by the 2dimensional, complete, simply connected spaces of curvature κ.
In the following, we assume that \(\kappa\geq0\) and define the diameter \(D_{\kappa}\) of \(M_{\kappa}\) by \(D_{\kappa}=\frac{\pi}{\sqrt{\kappa}}\) for \(\kappa>0\) and \(D_{\kappa}=\infty\) for \(\kappa=0\). It is well known that any ball in X with radius less than \(D_{\kappa}/2\) is convex [20]. A geodesic triangle \(\Delta(x, y, z)\) in the metric space \((X, d)\) consists of three points x, y, z in X (the vertices of Δ) and three geodesic segments between each pair of vertices. For \(\Delta(x, y, z)\) in a geodesic space X satisfying
there exist points \(\bar{x},\bar{y},\bar{z}\in M_{\kappa}\) such that \(d(x,y)=d_{\kappa}(\bar{x},\bar{y})\), \(d(y,z)=d_{\kappa}(\bar{y},\bar{z})\), and \(d(z,x)=d_{\kappa}(\bar{z},\bar{x})\) where \(d_{\kappa}\) is the metric of \(M_{\kappa}\). We call the triangle having vertices \(\bar{x},\bar{y},\bar{z}\in M_{\kappa}\) a comparison triangle of \(\Delta(x, y, z)\). A geodesic triangle \(\Delta(x, y, z)\) in X with \(d(x, y) + d(y, z) + d(z, x) < 2D_{\kappa}\) is said to satisfy the \(\operatorname {CAT}(\kappa)\) inequality if, for any \(p,q\in\Delta(x, y, z)\) and for their comparison points \(\bar{p},\bar{q}\in\bar{\Delta}(\bar{x},\bar{y},\bar{z})\), we have \(d(p,q)\leq d_{\kappa}(\bar{p},\bar{q})\).
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 πconvex 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]
Let \((X,d)\) be a \(\operatorname {CAT}(1)\) space. Then there is a constant \(M>0\) such that
for any \(t\in[0,1]\) and any point \(x,y,z\in X\) such that \(d(x,y)\leq\pi/4\), \(d(x,z)\leq\pi/4\), and \(d(y,z)\leq\pi/2\).
Let \(\{x_{n}\}\) be a bounded sequence in X. For \(x\in X\), we set
The asymptotic radius \(r(\{x_{n}\})\) of \(\{x_{n}\}\) is given by
and the asymptotic center \(A(\{x_{n}\})\) of \(\{x_{n}\}\) is the set
Definition 2.4
A sequence \(\{x_{n}\}\) in X is said to Δconverge 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 Δlimit of \(\{x_{n}\}\).
Definition 2.5
For a sequence \(\{x_{n}\}\) in X, a point \(x\in X\) is a Δcluster point of \(\{x_{n}\}\) if there exists a subsequence of \(\{x_{n}\}\) that Δconverges to x.
Lemma 2.6
[7]
Let \((X,d)\) be a complete \(\operatorname {CAT}(\kappa)\) space and let \(p\in X\). Suppose that a sequence \(\{x_{n}\}\) in X Δconverges to x such that \(r(p,\{x_{n}\})< D_{\kappa}/2\). Then
Definition 2.7
Let \((X,d)\) be a complete metric space and let K be a nonempty subset of X. Then a sequence \(\{x_{n}\}\) in X is Fejér monotone with respect to K if
for all \(n\geq0\) and all \(q\in K\).
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 Δcluster point x of \(\{x_{n}\}\) belongs to K, then \(\{x_{n}\}\) Δconverges to a point in K.
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
By Theorem 3.4 in [22] and Lemma 2.2, there exists a unique point p in F such that \(d(x_{0},F)=d(x_{0},p)\). From \(d(Tx_{0},p)\leq d(x_{0},p)\leq\pi/4\) and \(B_{\pi/4}[p]\) is convex, we have
Suppose that \(d(y_{k},p)\leq d(x_{k},p)\leq\pi/4\) for \(k\geq1\). Since \(d(Sy_{k},p)\leq d(y_{k},p)\leq\pi/4\) and \(B_{\pi/4}[p]\) is convex, we have
and
It follows that \(d(y_{k+1},p)\leq d(x_{k+1},p)\leq\pi/4\). By mathematical induction, hence \(d(y_{n},p)\leq d(x_{n},p)\leq\pi/4\) for all \(n\geq0\). □
Lemma 3.2
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 \(\{ a_{n}\}\) 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 \(d(x_{0},F)\leq\pi/4\), then

(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
By Lemma 2.3 and Lemma 3.1, there exist \(p\in F\) and \(M>0\) such that
and
Hence
This shows that \(\{d(x_{n},p)\}\) is decreasing and this proves part (i). Let
We next prove part (ii). From (3.2), we get
from which it follows that
This implies that
So
On the other hand, (3.3) gives
so that
We see that
thus
Using (3.4) and (3.5), we can conclude that
Next, we know that
from which it follows that
This yields
Using (1.5), we obtain
This implies by (3.6),
Since
Finally, we see that
hence, by (3.8) and (3.9), we get
This completes the 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 \(\{x_{n}\}\) Δconverges to a point in F.
Proof
Without loss of generality, we assume that \(\kappa=1\). Set \(F_{0}:=F\cap B_{\pi/2}(x_{0})\). Let \(q\in F_{0}\). Since \(d(Tx_{0},q)\leq d(x_{0},q)\) and since the open ball \(B_{\pi/2}(q)\) in C with radius \(r<\pi/2\) is convex, we have
Since \(d(Sy_{0},q)\leq d(y_{0},q)\) and since the open ball \(B_{\pi /2}(q)\) in C with radius \(r<\pi/2\) is convex, we have
By mathematical induction, we can show that
for all \(n\geq0\). Hence \(\{x_{n}\}\) is a Fejér monotone sequence with respect to \(F_{0}\). Let \(p\in F\) such that \(d(x_{0},p)\leq\pi/4\). Then \(p\in F_{0}\). Also we have
for all \(n\geq0\). This shows that \(r(\{x_{n}\})<\pi/4\). By Lemma 2.8, let \(x\in C\) be a Δcluster point of \(\{x_{n}\}\). Then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) which Δconverges to x. From (3.10), we get
By Lemma 2.6, we obtain
This implies that \(x\in B_{\pi/2}(x_{0})\). By Lemma 3.2, we have
and
Hence \(Tx,Sx\in A(\{x_{n_{k}}\})\) and \(Tx=x=Sx\). Therefore \(x\in F_{0}\). By Lemma 2.8, we thus complete the 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 \(\{x_{n}\}\) Δconverges 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 Ishikawatype 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 \(\{x_{n}\}\) Δconverges 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 \(\{x_{n}\}\) Δconverges to a point in F.
Numerical examples
In this section, we consider the msphere \(\mathbb{S}^{m}\), which is a \(\operatorname {CAT}(\kappa)\) space.
The msphere \(\mathbb{S}^{m}\) is defined by
where \(\langle\cdot,\cdot\rangle\) denotes the Euclidean scalar product.
Next, the normalized geodesic \(c : \mathbb{R}\rightarrow\mathbb {S}^{m}\) starting from \(x\in\mathbb{S}^{m}\) is given by
where \(v\in T_{x}\mathbb{S}^{m}\) is the unit vector; while the distance d on \(\mathbb{S}^{m}\) is
Then iteration process (1.4) has the form
and iteration process (1.5) has the form
where
Example 4.1
Let \(C=\mathbb{S}^{3}\) and let T and S be two nonexpansive mappings of C be defined by
For any \(x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{S}^{3}\). Then \(\operatorname {Fix}(T)=\{ (1,0,0,0)\}=\operatorname {Fix}(S)\).
Choose \(x_{0}=(0.5,0.5,0.5,0.5)\) and let \(a_{n}=\frac{n}{20n+1}\) and \(b_{n}=\frac{n}{10n+1}\). Then we obtain the numerical results in Table 1 and Figure 1.
We next consider the hyperbolic mspace \(\mathbb{H}^{m}\).
The hyperbolic mspace \(\mathbb{H}^{m}\) is defined by
where
Next, the normalized geodesic \(c : \mathbb{R}\rightarrow\mathbb {H}^{m}\) starting from \(x\in\mathbb{H}^{m}\) is given by
where \(v\in T_{x}\mathbb{H}^{m}\) is the unit vector; while the distance d on \(\mathbb{H}^{m}\) is
Then iteration process (1.4) has the form
and iteration process (1.5) has the form
where
Example 4.2
Let \(C=\mathbb{H}^{3}\) and let T and S be two nonexpansive mappings of C be defined by
for any \(x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{H}^{3}\). Then \(\operatorname {Fix}(T)=\{ (0,0,0,1)\}=\operatorname {Fix}(S)\).
Choose \(x_{0}=(2,2,4,5)\) and let \(a_{n}=\frac{n}{5n+1}\) and \(b_{n}=\frac{n}{7n+1}\). Then we obtain the numerical results in Table 2 and Figure 2.
From the numerical experience, we observe that the convergence rate of Siteration 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].
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Acknowledgements
The authors wish to thank the referees for valuable suggestions. This research was supported by University of Phayao.
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MSC
 47H09
 47H10
Keywords
 Δconvergence
 Siteration process
 nonexpansive mapping
 common fixed point
 \(\operatorname {CAT}(\kappa)\) space