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Strong and △-convergence theorems for two asymptotically nonexpansive mappings in the intermediate sense in \(\operatorname{CAT}(0)\) spaces
- Gurucharan S Saluja^{1} and
- Mihai Postolache^{2}Email author
https://doi.org/10.1186/s13663-015-0259-8
© Saluja and Postolache; licensee Springer. 2015
- Received: 28 August 2014
- Accepted: 2 January 2015
- Published: 1 February 2015
Abstract
In this paper, we study strong and △-convergence for a newly defined two-step iteration process involving two asymptotically nonexpansive mappings in the intermediate sense which is wider than the class of asymptotically nonexpansive mappings in the setting of \(\operatorname{CAT}(0)\) spaces. Our results generalize, unify and extend many known results from the existing literature.
Keywords
- asymptotically nonexpansive mapping in the intermediate sense
- Δ-convergence
- strong convergence
- two-step iteration process
- common fixed point
- \(\operatorname{CAT}(0)\) space
MSC
- 54H25
- 54E40
1 Introduction
A metric space \((X,d)\) is 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. It is well known that any complete, simply connected Riemannian manifold having non-positive sectional curvature is a \(\operatorname{CAT}(0)\) space. Fixed point theory in \(\operatorname{CAT}(0)\) spaces was first studied by Kirk (see [1, 2]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete \(\operatorname{CAT}(0)\) space always has a fixed point. It is worth mentioning that the results in \(\operatorname{CAT}(0)\) spaces can be applied to any \(\operatorname{CAT}(k)\) space with \(k\leq0\) since any \(\operatorname{CAT}(k)\) space is a \(\operatorname{CAT}(m)\) space for every \(m\geq k\) (see [3]).
Very recently, Şahin and Başarir [7] modified iteration process (1.6) in a \(\operatorname{CAT}(0)\) space as follows.
Inspired and motivated by the work of Şahin and Başarir [7] and some others, we further modify iteration scheme (1.7) for two mappings in a \(\operatorname{CAT}(0)\) space as follows.
Remark 1.1
If we take \(S=I\), where I is the identity mapping and \(\beta_{n}=0\) for all \(n\geq1\), then (1.8) reduces to the modified Mann iteration process in a \(\operatorname{CAT}(0)\) space.
In this paper, we study the newly defined two-step iteration process (1.8) involving two asymptotically nonexpansive mappings in the intermediate sense and investigate the existence and convergence theorems for the above said mappings and iteration scheme in the framework of \(\operatorname{CAT}(0)\) spaces. Our results generalize, unify and extend several comparable results in the existing literature.
2 Preliminaries and lemmas
In order to prove the main results of this paper, we need the following definitions, concepts and lemmas.
Let \((X,d)\) be a metric space and K be its nonempty subset. Consider \(T\colon K\to K\) to be a mapping. A point \(x\in K\) is called a fixed point of T if \(Tx=x\). We will also denote by F the set of common fixed points of S and T, that is, \(F=\{x\in K:Sx=Tx=x\}\).
The concept of asymptotically nonexpansive mapping was introduced by Goebel and Kirk [9] in 1972. The iterative approximation problems for asymptotically nonexpansive and asymptotically quasi-nonexpansive mappings were studied by many authors in a Banach space and a \(\operatorname{CAT}(0)\) space (see, e.g., [6, 10–18]).
Definition 2.1
- (1)
nonexpansive if \(d(Tx,Ty)\leq d(x,y)\) for all \(x,y\in K\);
- (2)
asymptotically nonexpansive if there exists a sequence \(\{u_{n}\}\subset[0,\infty)\), with \(\lim_{n\to\infty}u_{n}=0\), such that \(d(T^{n}x,T^{n}y)\leq(1+u_{n})d(x,y)\) for all \(x,y\in K\) and \(n\geq 1\);
- (3)
uniformly L-Lipschitzian if there exists a constant \(L>0\) such that \(d(T^{n}x,T^{n}y)\leq L d(x,y)\) for all \(x,y\in K\) and \(n\geq1\);
- (4)
semi-compact if for a sequence \(\{x_{n}\}\) in K with \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\to p\in K\).
From the above definitions, it follows that an asymptotically nonexpansive mapping must be an asymptotically nonexpansive mapping in the intermediate sense. But the converse does not hold as the following example shows.
Example 2.1
(see [20])
Remark 2.1
It is clear that the class of asymptotically nonexpansive mappings includes nonexpansive mappings, whereas the class of asymptotically nonexpansive mappings in the intermediate sense is larger than that of asymptotically nonexpansive mappings.
- (i)
a geodesic space if any two points of X are joined by a geodesic;
- (ii)
uniquely geodesic if there is exactly one geodesic joining x and y for each \(x,y\in X\), which we will denote by \([x,y]\), called the segment joining x to y.
A geodesic triangle \(\triangle(x_{1},x_{2},x_{3})\) in a geodesic metric space \((X,d)\) consists of three points in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle \(\triangle(x_{1},x_{2},x_{3})\) in \((X,d)\) is a triangle \(\overline{\triangle}(x_{1},x_{2},x_{3}):= \triangle(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})\) in \(\mathbb{R}^{2}\) such that \(d_{\mathbb{R}^{2}}(\overline{x}_{i},\overline{x}_{j})=d(x_{i},x_{j})\) for \(i,j\in\{1,2,3\}\). Such a triangle always exists (see [3]).
\(\boldsymbol {\operatorname {CAT}(0)}\) space
A geodesic metric space is said to be a \(\operatorname{CAT}(0)\) space if all geodesic triangles of appropriate size satisfy the following \(\operatorname{CAT}(0)\) comparison axiom.
A subset K of a \(\operatorname{CAT}(0)\) space X is convex if, for any \(x,y\in K\), we have \([x,y]\subset K\).
For the development of our main results, we recall some definitions, and some key results are listed in the form of lemmas.
Lemma 2.1
([14])
- (i)For \(x, y\in X\) and \(t\in[0,1]\), there exists a unique point \(z\in[x, y]\) such that$$ d(x, z)=t d(x, y) \quad\textit{and} \quad d(y, z)=(1-t) d(x, y). $$(A)
- (ii)For \(x, y\in X\) and \(t\in[0,1]\), we have$$d\bigl((1-t)x\oplus ty,z\bigr)\leq(1-t)d(x,z)+td(y,z). $$
It is known that, in a \(\operatorname{CAT}(0)\) space, \(A(\{x_{n}\})\) consists of exactly one point; please see [23, Proposition 7].
We now recall the definition of △-convergence and weak convergence (⇀) in a \(\operatorname{CAT}(0)\) space.
Definition 2.2
([24])
A sequence \(\{x_{n}\}\) in a \(\operatorname{CAT}(0)\) space X is said to △-converge to \(x\in X\) if x is the unique asymptotic center of \(\{x_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\).
In this case we write \(\triangle\mbox{-}\!\lim_{n}x_{n}=x\) and call x the △-limit of \(\{x_{n}\}\).
Now, let us recall that a bounded sequence \(\{x_{n}\}\) in X is said to be regular if \(r(\{x_{n}\})=r(\{u_{n}\})\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In the Banach space it is known that every bounded sequence has a regular subsequence; please see [25, Lemma 15.2].
Since in a \(\operatorname{CAT}(0)\) space every regular sequence △-converges, we see that every bounded sequence in X has a △-convergent subsequence, also it is noticed in [24, p.3690].
Lemma 2.2
([26])
In a Banach space the above condition is known as the Opial property.
Now, recall the definition of weak convergence in a \(\operatorname{CAT}(0)\) space.
Definition 2.3
([27])
Let K be a closed convex subset of a \(\operatorname{CAT}(0)\) space X. A bounded sequence \(\{x_{n}\}\) in K is said to converge weakly to \(q\in K\) if and only if \(\Phi(q)=\inf_{x\in K}\Phi(x)\), where \(\Phi(x)=\limsup_{n\to\infty}d(x_{n},x)\).
Note that \(\{x_{n}\}\rightharpoonup q\) if and only if \(A_{K}\{x_{n}\}=\{q\}\).
Nanjaras and Panyanak [28] established the following relation between △-convergence and weak convergence in a \(\operatorname{CAT}(0)\) space.
Lemma 2.3
([28, Proposition 3.12])
- (i)
\(\triangle\mbox{-}\!\lim_{x_{n}}=x\) implies \(x_{n}\rightharpoonup x\).
- (ii)
The converse of (i) is true if \(\{x_{n}\}\) is regular.
Lemma 2.4
([22, Lemma 2.8])
If \(\{x_{n}\}\) is a bounded sequence in a \(\operatorname{CAT}(0)\) space X, with \(A(\{x_{n}\})=\{x\}\), and \(\{u_{n}\}\) is a subsequence of \(\{x_{n}\}\), with \(A(\{u_{n}\})=\{u\}\), and the sequence \(\{d(x_{n},u)\}\) converges, then \(x=u\).
Lemma 2.5
([29, Proposition 2.1])
If K is a closed convex subset of a \(\operatorname{CAT}(0)\) space X and if \(\{x_{n}\}\) is a bounded sequence in K, then the asymptotic center of \(\{x_{n}\}\) is in K.
Lemma 2.6
([30])
Suppose that \(\{a_{n}\}\) and \(\{b_{n}\}\) are two sequences of nonnegative numbers such that \(a_{n+1}\leq a_{n} + b_{n}\) for all \(n\geq1\). If \(\sum_{n=1}^{\infty} b_{n}\) converges, then \(\lim_{n\to\infty}a_{n}\) exists.
Lemma 2.7
([26, Theorem 3.1])
Let X be a complete \(\operatorname{CAT}(0)\) space, K be a nonempty closed convex subset of X. If \(T\colon K\to K\) is an asymptotically nonexpansive mapping in the intermediate sense, then T has a fixed point.
Lemma 2.8
([26, Theorem 3.2])
Let X be a complete \(\operatorname{CAT}(0)\) space, K be a nonempty closed convex subset of X. If \(T\colon K\to K\) is an asymptotically nonexpansive mapping in the intermediate sense, then \(\operatorname{Fix}(T)\) is closed and convex.
Lemma 2.9
(Demiclosed principle [26, Proposition 3.3])
Let K be a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X and \(T\colon K\to K\) be an asymptotically nonexpansive mapping in the intermediate sense. If \(\{x_{n}\}\) is a bounded sequence in K such that \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\) and \(\{x_{n}\}\rightharpoonup w\), then \(Tw=w\).
Lemma 2.10
([26, Corollary 3.4])
Let K be a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X and \(T\colon K\to K\) be an asymptotically nonexpansive mapping in the intermediate sense. If \(\{x_{n}\}\) is a bounded sequence in K △-converging to x and \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\), then \(x\in K\) and \(Tx=x\).
3 The main results
Now, we prove the following lemmas using the newly defined two-step iteration scheme (1.8) needed in the sequel.
Lemma 3.1
- (i)
\(\lim_{n\to\infty}d(x_{n},p)\) exists for all \(p\in F\).
- (ii)
\(\lim_{n\to\infty}d(x_{n},F)\) exists.
Proof
Lemma 3.2
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two asymptotically nonexpansive mappings in the intermediate sense with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(c_{n}\) and \(d_{n}\) are taken as in Lemma 3.1. Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). If \(d(x,Sx)\leq d(Tx,Sx)\) for all \(x\in K\), then \(\lim_{n\to\infty}d(x_{n},Sx_{n})=0\) and \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\).
Proof
Now we prove the △-convergence and strong convergence results.
Theorem 3.1
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two asymptotically nonexpansive mappings in the intermediate sense with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(c_{n}\) and \(d_{n}\) are taken as in Lemma 3.1. Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). Then the sequence \(\{x_{n}\}\) is △-convergent to a point of F.
Proof
We first show that \(w_{w}(\{x_{n}\})\subseteq F\). Let \(v\in w_{w}(\{x_{n}\})\), then there exists a subsequence \(\{v_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{x_{n}\})=\{v\}\). By Lemma 2.5, there exists a subsequence \(\{w_{n}\}\) of \(\{v_{n}\}\) such that \(\triangle\mbox{-}\!\lim_{n}w_{n}=w\in K\). By Lemma 2.10, \(w\in F(T)\) and \(w\in F(S)\) and so \(w\in F\). By Lemma 3.1 \(\lim_{n\to\infty}d(x_{n},F)\) exists, so by Lemma 2.4 we have \(v=w\), i.e., \(w_{w}(\{x_{n}\})\subseteq F\).
To show that \(\{x_{n}\}\) △-converges to a point in F, it is sufficient to show that \(w_{w}(\{x_{n}\})\) consists of exactly one point.
Let \(\{v_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{v_{n}\})=\{v\}\), and let \(A(\{x_{n}\})=\{x\}\) for some \(v\in w_{w}(\{x_{n}\})\subseteq F\) and \(\{d(x_{n},w)\}\) converges. By Lemma 2.4, we have \(x=w\in F\). Thus \(w_{w}(\{x_{n}\})=\{x\}\). This shows that \(\{x_{n}\}\) is △-convergent to a point of F. This completes the proof. □
Theorem 3.2
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two asymptotically nonexpansive mappings in the intermediate sense with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(c_{n}\) and \(d_{n}\) are taken as in Lemma 3.1. Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). If \(\liminf_{n\to\infty}d(x_{n},F)=0\) or \(\limsup_{n\to\infty}d(x_{n},F)=0\), where \(d(x,F)=\inf_{p\in F}d(x,p)\), then the sequence \(\{x_{n}\}\) converges strongly to a point in F.
Proof
Theorem 3.3
- (i)
\(\lim_{n\to\infty}d(x_{n},Sx_{n})=0\) and \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\).
- (ii)
If the sequence \(\{z_{n}\}\) in K satisfies \(\lim_{n\to\infty}d(z_{n},Sz_{n})=0\) and \(\lim_{n\to\infty}d(z_{n},Tz_{n})=0\), then \(\liminf_{n\to\infty}d(z_{n},F)=0\) or \(\limsup_{n\to\infty}d(z_{n},F)=0\).
Then the sequence \(\{x_{n}\}\) converges strongly to a point of F.
Proof
It follows from the hypothesis that \(\lim_{n\to\infty}d(x_{n},Sx_{n})=0\) and \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\). From (ii), \(\liminf_{n\to\infty}d(x_{n},F)=0\) or \(\limsup_{n\to\infty}d(x_{n},F)=0\). Therefore, the sequence \(\{x_{n}\}\) must converge strongly to a point in F by Theorem 3.2. This completes the proof. □
Theorem 3.4
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two asymptotically nonexpansive mappings in the intermediate sense with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(c_{n}\) and \(d_{n}\) are taken as in Lemma 3.1. Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). If either S or T is semi-compact, then the sequence \(\{x_{n}\}\) converges strongly to a point of F.
Proof
We recall the following definition.
A mapping \(T\colon K\to K\), where K is a subset of a normed linear space E, is said to satisfy Condition (A) [31] if there exists a nondecreasing function \(f\colon[0,\infty)\to[0,\infty)\) with \(f(0)=0\) and \(f(t)>0\) for all \(t\in(0,\infty)\) such that \(\Vert x-Tx\Vert \geq f(d(x,F(T)))\) for all \(x\in K\), where \(d(x,F(T))=\inf\{\|x-p\|:p\in F(T)\neq\emptyset \}\).
Now, we modify this definition for two mappings.
Two mappings \(S,T\colon K\to K\), where K is a subset of a normed linear space E, are said to satisfy Condition (B) if there exists a nondecreasing function \(f\colon[0,\infty)\to[0,\infty)\) with \(f(0)=0\) and \(f(t)>0\) for all \(t\in(0,\infty)\) such that \(a_{1}\|x-Sx\|+a_{2}\|x-Tx\|\geq f(d(x,F))\) for all \(x\in K\), where \(d(x,F)=\inf\{\|x-p\|:p\in F\neq\emptyset \}\) and \(a_{1}\) and \(a_{2}\) are two nonnegative real numbers such that \(a_{1}+a_{2}=1\). It is to be noted that Condition (B) is weaker than the compactness of the domain K.
Remark 3.1
Condition (B) reduces to Condition (A) when \(S=T\).
As an application of Theorem 3.2, we establish some strong convergence results as follows.
Theorem 3.5
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two asymptotically nonexpansive mappings in the intermediate sense with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(c_{n}\) and \(d_{n}\) are taken as in Lemma 3.1. Suppose that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). If S and T satisfy Condition (B), then the sequence \(\{x_{n}\}\) converges strongly to a point of F.
Proof
Theorem 3.6
Let K be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X, and let \(S,T\colon K\to K\) be two uniformly continuous asymptotically nonexpansive mappings with \(F\neq\emptyset \). Suppose that \(\{x_{n}\}\) is defined by the iteration process (1.8) and \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([l,m]\) for some \(l,m\in(0,1)\). If \(\liminf_{n\to\infty}d(x_{n},F)=0\) or \(\limsup_{n\to\infty}d(x_{n},F)=0\), where \(d(x,F)=\inf_{p\in F}d(x,p)\), then the sequence \(\{x_{n}\}\) converges strongly to a point in F.
Proof
Example 3.1
4 Concluding remarks
This work contains our dedicated study to develop and improve methods for solving equations by means of iteration methods. We have introduced our results by using as basic framework the research of Kirk; please see [1, 2], and the inspired work of Şahin and Başarir [7] and some others. This study is motivated by relevant applications for solving many real-world problems which give rise to mathematical models in the sphere of functional analysis.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their careful reading and suggestions on the manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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