Existence and higher arity iteration for total asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces
- Muhammad Aqeel Ahmad Khan^{1}Email author,
- Hafiz Fukhar-ud-din^{2, 3} and
- Amna Kalsoom^{3}
https://doi.org/10.1186/s13663-015-0483-2
© Khan et al. 2016
Received: 24 February 2015
Accepted: 6 December 2015
Published: 6 January 2016
Abstract
This paper provides a fixed point theorem and iterative construction of a common fixed point for a general class of nonlinear mappings in the setup of uniformly convex hyperbolic spaces. We translate a multi-step iteration, essentially due to Chidume and Ofoedu (J. Math. Anal. Appl. 333:128-141, 2007) in such a setting for the approximation of common fixed points of a finite family of total asymptotically nonexpansive mappings. As a consequence, we establish strong and △-convergence results which extend and generalize various corresponding results established in the current literature.
Keywords
MSC
1 Introduction
The above characteristics is referred to as a constant speed of θ, the parametrization of θ with respect to the arc length or distance preservation of θ. The points \(x=\theta(0)\) and \(y=\theta(l)\) are called the end points or the extreme (maximal or minimal) points of the segment. The metric space \((X,d)\) is called a geodesic space if for every pair of points \(x,y\in X\), there is a geodesic segment from x to y. Moreover, \((X,d)\) is uniquely geodesic if for all \(x,y\in X\) there is exactly one geodesic from x to y.
The class of hyperbolic spaces introduced by Kohlenbach [2] is an important example of a uniquely geodesic space. It is worth to mention that this class is prominent among various other notions of hyperbolic spaces in the current literature, for convenience of the reader; see [3–6]. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory. We remark that the non-positively curved spaces, such as hyperbolic spaces, play a significant role in many branches of applied mathematics.
Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [2].
The class of hyperbolic spaces in the sense of Kohlenbach [2] contains all normed linear spaces and convex subsets thereof as well as Hadamard manifolds and \(\operatorname{CAT}(0)\) spaces in the sense of Gromov. An important example of a hyperbolic space due to Goebel and Reich [4] is stated as follows.
It is worth to mention that the fixed point theory of nonexpansive mappings (i.e., \(d(Tx,Ty)\leq d(x,y) \)for \(x,y\in K\)) and its various generalizations majorly depends on the geometrical characteristics of the underlying space. The class of nonexpansive mappings enjoys the fixed point property (FPP) and the approximate fixed point property (AFPP) in various settings of spaces, see for example [8] for the later property for the class of nonexpansive mappings. Moreover, it is natural to extend such powerful results to generalized nonexpansive mappings as a mean of testing the limit of the theory of nonexpansive mappings. It is remarked that the FPP and even AFPP, in a nonlinear domain, of various generalizations of nonexpansive mappings are still developing. The class of hyperbolic spaces is endowed with rich geometric structures for different results with applications in topology, graph theory, multivalued analysis and metric fixed point theory. An important ingredient for metric fixed point theory of nonexpansive mappings is uniform convexity.
A mapping \(\eta:(0,\infty)\times(0,2]\rightarrow(0,1]\) providing such \(\delta=\eta(r,\varepsilon)\) for given \(r>0\) and \(\varepsilon\in (0,2]\) is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε), i.e., \(\forall \varepsilon >0\), \(\forall r_{2}\geq r_{1}>0\) (\(\eta ( r_{2},\varepsilon ) \leq \eta ( r_{1},\varepsilon ) \)). The \(\operatorname{CAT}(0)\) spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity \(\eta (r,\varepsilon)=\frac{\varepsilon^{2}}{8}\) [9]. Therefore, the class of uniformly convex hyperbolic spaces includes \(\operatorname{CAT}(0)\) spaces as a special case.
Metric fixed point theory of nonlinear mappings in a general setup of hyperbolic spaces is a fascinating field of research in nonlinear functional analysis. Moreover, iteration schemas are the only main tool to study fixed point problems of nonexpansive mappings and its various generalizations in spaces of non-positive sectional curvature. In 2006, Alber et al. [10] introduced a unified and generalized notion of a class of nonlinear mappings in Banach spaces, which can be introduced in the general setup of hyperbolic spaces as follows.
Example 1.1
- (i)
Let \(X=\mathbb{R}\), \(K=[0,\infty)\) and \(T:K\rightarrow K\) be defined by \(Tx=\sin x\). Then T is a total asymptotically nonexpansive.
- (ii)
Let \(X= \mathbb{R}\), \(K= [ -\frac{1}{\pi},\frac{1}{\pi} ] \) and \(T:K\rightarrow K\) be defined by \(Tx=kx\sin\frac{1}{x}\), where \(k\in(0,1)\). Then T is a total asymptotically nonexpansive.
- (iii)Let \(K=\{x:=(x_{1},x_{2},\ldots,x_{n},\ldots) \mid x_{1}\leq 0,x_{i}\in \mathbb{R}, i\geq2\}\) be a nonempty subset of \(X=l^{2}\) with the norm \(\Vert \cdot \Vert \) defined asIf \(T:K\rightarrow K\) is defined by$$ \Vert x\Vert =\sqrt{\sum_{i=1}^{\infty}x_{i}^{2}}. $$then T is an asymptotically nonexpansive.$$ T(x)=(0,4x_{2},0,0,0,\ldots), $$
- (iv)Let \(X= \mathbb{R}\) and \(K=[0,2]\). Let \(T:K\rightarrow K\) be a mapping defined by$$ T(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1, &x\in[0,1], \\ \frac{1}{\sqrt{3}}\sqrt{4-x^{2}}, &x\in[1,2].\end{array}\displaystyle \right . $$
The class of total asymptotically nonexpansive mappings and asymptotically nonexpansive mappings have been studied extensively in the literature [11–16] and the references cited therein. It is worth mentioning that the results established for total asymptotically nonexpansive mappings are applicable to the mappings associated with the class of asymptotically nonexpansive mappings and which are extensions of nonexpansive mappings.
The purpose of this paper is to establish a fixed point result for a total asymptotically nonexpansive mapping along with the iterative construction of common fixed point of a finite family of these mappings in uniformly convex hyperbolic spaces. We therefore establish results concerning strong convergence and △-convergence results of iteration (1.1). Our convergence results can be viewed not only as an analog of various existing results but also improve and generalize various results in the current literature.
2 Preliminaries and some auxiliary lemmas
We start this section with the notion of asymptotic center - essentially due to Edelstein [21] - of a sequence which is not only useful in proving a fixed point result but also plays a key role to define the concept of △-convergence in hyperbolic spaces. In 1976, Lim [22] introduced the concept of △-convergence in the general setting of metric spaces. In 2008, Kirk and Panyanak [23] further analyzed this concept in geodesic spaces. They showed that many Banach space results involving weak convergence have a precise analog version of △-convergence in geodesic spaces.
Recall that a sequence \(\{x_{n}\}\) in X is said to △-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 \(\triangle\mbox{-}\!\lim_{n}x_{n}=x\) and call x the △-limit of \(\{x_{n}\}\).
A mapping \(T:K\rightarrow K\) is semi-compact if every bounded sequence \(\{x_{n}\}\subset K\) satisfying \(\lim_{n\rightarrow \infty}d(x_{n},Tx_{n})= 0\), has a convergent subsequence.
We now list some useful lemmas as well as establish some auxiliary results required in the sequel.
Lemma 2.1
([24])
Let \((X,d,W)\) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Then every bounded sequence \(\{x_{n}\}\) in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Proposition 2.2
([25])
Let \((X,d,W)\) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η. The intersection of any decreasing sequence of nonempty bounded closed convex subsets of X is nonempty.
Lemma 2.3
([26])
Let \(\{ a_{n} \} \), \(\{ b_{n} \}\), and \(\{ c_{n} \} \) be sequences of nonnegative real numbers such that \(\sum_{n=1}^{\infty }b_{n}<\infty\) and \(\sum_{n=1}^{\infty}c_{n}<\infty\). If \(a_{n+1} \leq(1+b_{n})a_{n}+c_{n}\), \(n\geq1\), then \(\lim_{n\rightarrow\infty}a_{n}\) exists.
Lemma 2.4
([27])
Let \((X,d,W)\) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let \(x\in X\) and \(\{\alpha_{n}\}\) be a sequence in \([a,b]\) for some \(a,b\in(0,1)\). If \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in X such that \(\limsup_{n\rightarrow\infty}d(x_{n},x)\leq c\), \(\limsup_{n\rightarrow\infty}d(y_{n},x)\leq c\) and \(\lim_{n\rightarrow\infty}d(W(x_{n},y_{n},\alpha _{n}),x)=c\) for some \(c\geq0\), then \(\lim_{n\rightarrow\infty }d(x_{n},y_{n})=0\).
Lemma 2.5
([27])
Let K be a nonempty, closed, and convex subset of a uniformly convex hyperbolic space X and \(\{x_{n}\}\) a bounded sequence in K such that \(A_{K}(\{x_{n}\})=\{y\}\) and \(r_{K}(\{x_{n}\})=\rho\). If \(\{y_{m}\}\) is another sequence in K such that \(\lim_{m\rightarrow\infty}r(y_{m},\{x_{n}\})=\rho\), then \(\lim_{m\rightarrow\infty}y_{m}=y\).
Lemma 2.6
- (C1)
\(\sum_{n=1}^{\infty}k_{in}<\infty\) and \(\sum_{n=1}^{\infty}\varphi_{in}<\infty\);
- (C2)
there exist constants \(M_{i}, M_{i}^{\ast}>0\) such that \(\xi_{i} ( \lambda_{i} ) \leq M_{i}^{\ast}\lambda _{i}\) for all \(\lambda_{i}\geq M_{i}\), \(i=1,2,3,\ldots,m\),
Proof
Appealing to Lemma 2.3, the above inequality implies that \(\lim_{n\rightarrow\infty}d ( x_{n},p ) \) exists and hence the sequence \(\{ x_{n} \} \) is bounded. □
Lemma 2.7
- (C1)
\(\sum_{n=1}^{\infty}k_{in}<\infty\) and \(\sum_{n=1}^{\infty}\varphi_{in}<\infty\);
- (C2)
there exist constants \(M_{i}, M_{i}^{\ast}>0\) such that \(\xi_{i} ( \lambda_{i} ) \leq M_{i}^{\ast}\lambda _{i}\) for all \(\lambda_{i}\geq M_{i}\),
Proof
It follows from Lemma 2.6 that \(\lim_{n\rightarrow \infty}d ( x_{n},p ) \) exists. Without loss of generality, we can assume that \(\lim_{n\rightarrow\infty}d ( x_{n},p ) =r>0\). We first distinguish two cases to show that \(\lim_{n\rightarrow \infty}d ( x_{n},T_{i}^{n}x_{n} ) =0\), \(i=1,2,\ldots ,m\).
3 Existence and convergence results
In this section, we first establish the existence of fixed point of total asymptotically nonexpansive mapping. Our proof closely follows Theorem 3.3 in [25] for the existence of fixed point of asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces.
Theorem 3.1
Let \((X,d,W)\) be a complete uniformly convex hyperbolic space having a monotone modulus of uniform convexity η. Let K be a nonempty, bounded, closed, and convex subset of X. Then a continuous and total asymptotically nonexpansive mapping \(T:K\rightarrow K\) has a fixed point.
Proof
We now use the concept of asymptotic center of a bounded sequence to strengthen the above existence result. The proof, in fact, follows closely the proof of Lemma 2.6 in [27].
Theorem 3.2
Let \((X,d,W)\) be a complete uniformly convex hyperbolic space having a monotone modulus of uniform convexity η. Let K be a nonempty, closed, and convex subset of X and let \(T:K\rightarrow K\) be a continuous and total asymptotically nonexpansive mapping. If \(\{T^{n}x\}\) is bounded for some \(x\in K\) and \(z\in A_{K}(\{T^{n}x\} )\), then z is a fixed point of T.
Proof
Remark 3.3
It is remarked that Theorems 3.1 and 3.2 can also be adopted to prove the existence of a common fixed point for a total asymptotically nonexpansive semigroup. A different approach to prove the existence of a common fixed point for such semigroups can be found in a recent paper due to Suantai and Phuengrattana [28].
The rest of the paper deals with the convergence analysis of iteration process (1.1) for the approximation of common fixed points of a finite family of total asymptotically nonexpansive mappings.
Theorem 3.4
- (C1)
\(\sum_{n=1}^{\infty}k_{in}<\infty\) and \(\sum_{n=1}^{\infty}\varphi_{in}<\infty\);
- (C2)
there exist constants \(M_{i}, M_{i}^{\ast}>0\) such that \(\xi_{i} ( \lambda_{i} ) \leq M_{i}^{\ast}\lambda _{i}\) for all \(\lambda_{i}\geq M_{i}\),
Proof
The strong convergence of iteration process (1.1) can easily be established under the compactness condition of K or \(T(K)\). As a consequence, we can get a generalized version of Theorem 12 in [1] and Theorem 3.3 in [17] to the general setup of uniformly convex hyperbolic spaces, respectively. Next, we give a necessary and sufficient condition for the strong convergence of iteration process (1.1).
Theorem 3.5
- (C1)
\(\sum_{n=1}^{\infty}k_{in}<\infty\) and \(\sum_{n=1}^{\infty}\varphi_{in}<\infty\);
- (C2)
there exist constants \(M_{i}, M_{i}^{\ast}>0\) such that \(\xi_{i} ( \lambda_{i} ) \leq M_{i}^{\ast}\lambda _{i}\) for all \(\lambda_{i}\geq M_{i}\),
Proof
The necessity of the conditions is obvious. Thus, we only prove the sufficiency. It follows from Lemma 2.6 that \(\{ d(x_{n},p) \} _{n=1}^{\infty}\) converges. Moreover, \(\lim \inf_{n\rightarrow\infty}d(x_{n},F)=0\) implies that \(\lim_{n\rightarrow \infty}d(x_{n},F)=0\). This completes the proof. □
It is remarked that there are certain situations when the domain \(D(T)\) of a nonlinear mapping T is a proper subset of the underlying space X. In such situations, the iterative schema for the approximation of fixed points of T may fail to be well defined. It is therefore natural to study the non-self behavior of the nonlinear mappings.
Remark 3.6
- (i)
It is worth mentioning that Theorems 3.4 and 3.5 can easily be extended to the class of mappings with bounded error terms as well as to approximate common fixed points of total asymptotically nonexpansive semigroup;
- (ii)
Lemma 2.6 improves and generalizes Theorem 7 in [1] and Lemma 3.1(i) in [17] to the general setup of uniformly convex hyperbolic spaces, respectively;
- (iii)
Lemma 2.7 improves and generalizes Theorem 11 in [1] and Lemma 3.1(iii) in [17] to the setting as mentioned in (ii);
- (iv)
Theorem 3.4 improves Theorem 3.5 in [11] and Theorem 3.1 in [14] for a finite family of total asymptotically nonexpansive mappings to the setup of spaces as mentioned in (ii);
- (v)
- (vi)
Declarations
Acknowledgements
The authors are very grateful to the editor and anonymous referees for their helpful comments. The author H Fukhar-ud-din is grateful to King Fahd University of Petroleum and Minerals for supporting this research.
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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