Convergence of three-step iterations for nearly asymptotically nonexpansive mappings in \(\operatorname{CAT}(k)\) spaces
- Gurucharan S Saluja^{1},
- Mihai Postolache^{2}Email author and
- Alia Kurdi^{2}
https://doi.org/10.1186/s13660-015-0670-z
© Saluja et al.; licensee Springer. 2015
Received: 24 February 2015
Accepted: 20 April 2015
Published: 9 May 2015
Abstract
In this paper, we study strong and △-convergence of a newly defined three-step iteration process for nearly asymptotically nonexpansive mappings in the setting of \(\operatorname{CAT}(k)\) spaces. Our results generalize, unify and extend many known results from the existing literature.
Keywords
MSC
1 Introduction
For a real number k, a \(\operatorname{CAT}(k)\) space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature k. The precise definition is given below. The term ‘\(\operatorname{CAT}(k)\)’ was coined by Gromov ([1], p.119). The initials are in honor of Cartan, Alexandrov and Toponogov, each of whom considered similar conditions in varying degrees of generality.
Fixed point theory in \(\operatorname{CAT}(k)\) spaces was first studied by Kirk (see [2, 3]). His works were followed by a series of new works by many authors, mainly focusing on \(\operatorname{CAT}(0)\) spaces (see, e.g., [4–11]). 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 [12], ‘Metric spaces of non-positive curvature’).
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [13] in 1972, as an important generalization of the class of nonexpansive mappings, and they proved that if C is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping of C has a fixed point.
There are many papers dealing with the approximation of fixed points of asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces, using modified Mann, Ishikawa and three-step iteration processes (see, e.g., [14–22]; see also [23–27]).
The concept of Δ-convergence in a general metric space was introduced by Lim [28]. In 2008, Kirk and Panyanak [29] used the notion of Δ-convergence introduced by Lim [28] to prove in the CAT(0) space and analogous of some Banach space results which involve weak convergence. Further, Dhompongsa and Panyanak [30] obtained Δ-convergence theorems for the Picard, Mann and Ishikawa iterations in a \(\operatorname{CAT}(0)\) space. Since then, the existence problem and the Δ-convergence problem of iterative sequences to a fixed point for nonexpansive mapping, asymptotically nonexpansive mapping, nearly asymptotically nonexpansive mapping, asymptotically nonexpansive mapping in the intermediate sense, asymptotically quasi-nonexpansive mapping in the intermediate sense, total asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping through Picard, Mann [31], Ishikawa [32], modified Agarwal et al. [33] have been rapidly developed in the framework of \(\operatorname{CAT}(0)\) spaces and many papers have appeared in this direction (see, e.g., [5, 30, 34–39]).
The aim of this article is to establish Δ-convergence and strong convergence of a modified three-step iteration process which contains a modified S-iteration process for a class of mappings which is wider than that of asymptotically nonexpansive mappings in \(\operatorname{CAT}(k)\) spaces. Our results extend and improve the corresponding results of Abbas et al. [34], Dhompongsa and Panyanak [30], Khan and Abbas [35] and many other results of this direction.
2 Preliminaries
Let \(F(T)=\{x\in K: Tx=x\}\) denote the set of fixed points of the mapping T. We begin with the following definitions.
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 \(n\geq1\);
- (3)
asymptotically quasi-nonexpansive if \(F(T)\neq\emptyset\), and there exists a sequence \(\{u_{n}\}\subset[0,\infty)\), with \(\lim_{n\to\infty}u_{n}=0\), such that \(d(T^{n}x,p)\leq(1+u_{n})d(x,p)\) for all \(x\in K\), \(p\in F(T)\) and \(n\geq1\);
- (4)
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 \(n\geq1\);
- (5)
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 \(x_{n_{k}}\to p\in K\) as \(k\to \infty\);
- (6)
a sequence \(\{x_{n}\}\) in K is called approximate fixed point sequence for T (AFPS, in short) if \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\).
The class of nearly Lipschitzian mappings is an important generalization of the class of Lipschitzian mappings and was introduced by Sahu [40].
Definition 2.2
The infimum of the constants \(k_{n}\), for which the above inequality holds, is denoted by \(\eta(T^{n})\) and is called nearly Lipschitz constant of \(T^{n}\).
- (i)
nearly nonexpansive if \(\eta(T^{n})=1\) for all \(n\geq1\);
- (ii)
nearly asymptotically nonexpansive if \(\eta(T^{n})\geq1\) for all \(n\geq1\) and \(\lim_{n\to\infty}\eta(T^{n})=1\);
- (iii)
nearly uniformly k-Lipschitzian if \(\eta(T^{n})\leq k\) for all \(n\geq1\).
Let \((X,d)\) be a metric space. A geodesic path joining \(x\in X\) to \(y\in X\) (or, more briefly, a geodesic from x to y) is a map c from a closed interval \([0,l]\subset\mathbb{R}\) to X such that \(c(0)=x\), \(c(l)=y\) and \(d(c(t),c(t'))=|t-t'|\) for all \(t,t'\in[0,l]\). In particular, c is an isometry, and \(d(x,y)=l\). The image α of c is called a geodesic (or metric) segment joining x and y. We say that X is (i) a geodesic space if any two points of X are joined by a geodesic, and (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. This means that \(z\in[x, y]\) if and only if there exists \(\alpha\in[0,1]\) such that \(d(x,z)=(1-\alpha)d(x,y)\) and \(d(y,z)=\alpha d(x,y)\).
In this case, we write \(z=\alpha x\oplus(1-\alpha)y\). The space \((X,d)\) is said to be a geodesic space (D-geodesic space) if every two points of X (every two points of distance smaller than D) are joined by a geodesic, and X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic joining x and y for each \(x,y\in X\) (for \(x,y\in X\) with \(d(x,y)< D\)). A subset K of X is said to be convex if K includes every geodesic segment joining any two of its points. The set K is said to be bounded if \(\operatorname{diam}(K):= \sup\{d(x,y):x,y\in K\}< \infty\).
The model spaces \(M_{k}^{2}\) are defined as follows.
- (i)
if \(k=0\), then \(M_{k}^{2}\) is an Euclidean space \(\mathbb{E}^{n}\);
- (ii)
if \(k>0\), then \(M_{k}^{2}\) is obtained from the sphere \(\mathbb{S}^{n}\) by multiplying the distance function by \(\frac{1}{\sqrt{k}}\);
- (iii)
if \(k<0\), then \(M_{k}^{2}\) is obtained from a hyperbolic space \(\mathbb{H}^{n}\) by multiplying the distance function by \(\frac{1}{\sqrt{-k}}\).
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 \(M_{k}^{2}\) such that \(d(x_{1},x_{2})=d_{M_{k}^{2}}(\overline{x_{1}},\overline{x_{2}})\), \(d(x_{2},x_{3})=d_{M_{k}^{2}}(\overline{x_{2}},\overline{x_{3}})\) and \(d(x_{3},x_{1})=d_{M_{k}^{2}}(\overline{x_{3}},\overline{x_{1}})\). If \(k\leq 0\), then such a comparison triangle always exists in \(M_{k}^{2}\). If \(k>0\), then such a triangle exists whenever \(d(x_{1},x_{2})+d(x_{2},x_{3})+d(x_{3},x_{1})< 2D_{k}\), where \(D_{k}=\pi/\sqrt{k}\). A point \(\bar{p}\in[\bar{x},\bar{y}]\) is called a comparison point for \(p\in[x,y]\) if \(d(x,p)=d_{M_{k}^{2}}(\bar{x},\bar{p})\).
A geodesic triangle \(\triangle(x_{1},x_{2},x_{3})\) in X is said to satisfy the \(\operatorname{CAT}(k)\) inequality if for any \(p,q\in \triangle(x_{1},x_{2},x_{3})\) and for their comparison points \(\bar{p},\bar{q}\in \overline{\triangle}(\bar{x_{1}},\bar{x_{2}},\bar{x_{3}})\), one has \(d(p,q)=d_{M_{k}^{2}}(\overline{p},\overline{q})\).
Definition 2.3
If \(k\leq0\), then X is called a \(\operatorname{CAT}(k)\) space if and only if X is a geodesic space such that all of its geodesic triangles satisfy the \(\operatorname{CAT}(k)\) inequality.
If \(k>0\), then X is called a \(\operatorname{CAT}(k)\) space if and only if X is \(D_{k}\)-geodesic and any geodesic triangle \(\triangle(x_{1},x_{2},x_{3})\) in X with \(d(x_{1},x_{2})+d(x_{2},x_{3})+d(x_{3},x_{1})< 2D_{k}\) satisfies the \(\operatorname{CAT}(k)\) inequality.
- (i)
X is a \(\operatorname{CAT}(0)\) space;
- (ii)
X satisfies the (CN) inequality;
- (iii)
X satisfies the (CN^{∗}) inequality.
In the sequel we need the following lemma.
Lemma 2.1
([12], p.176)
We now recall some elementary facts about \(\operatorname{CAT}(k)\) spaces. Most of them are proved in the framework of \(\operatorname{CAT}(1)\) spaces. For completeness, we state the results in a \(\operatorname{CAT}(k)\) space with \(k>0\).
It is known from Proposition 4.1 of [8] that a \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi}{2\sqrt{k}}\), \(A(\{x_{n}\})\) consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.
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 \(\{x_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case we write \(\Delta\mbox{-}\!\lim_{n}x_{n}=x\) and call x the Δ-limit of \(\{x_{n}\}\).
Lemma 2.2
- (i)
([8], Corollary 4.4) Every sequence in X has a Δ-convergent subsequence.
- (ii)
([8], Proposition 4.5) If \(\{x_{n}\}\subseteq X\) and \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=x\), then \(x\in \bigcap_{k=1}^{\infty}\overline{\operatorname{conv}}\{x_{k},x_{k+1},\dots\}\),
By the uniqueness of asymptotic center, we can obtain the following lemma in [30].
Lemma 2.3
([30], Lemma 2.8)
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). If \(\{x_{n}\}\) is a bounded sequence in 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 2.4
(see [20])
Proposition 2.1
([37], Proposition 3.12)
- (i)
\(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=x\) implies that \(\{x_{n}\}\rightharpoonup x\),
- (ii)
the converse is true if \(\{x_{n}\}\) is regular.
Algorithm 1
If \(T^{n}=T\) for all \(n\geq1\), then Algorithm 1 reduces to the following.
Algorithm 2
Algorithm 3
If \(\beta_{n}=0\) for all \(n\geq1\), then Algorithm 3 reduces to the following.
Algorithm 4
Motivated and inspired by [33] and some others, we modify iteration scheme (2.2) as follows.
Algorithm 5
If \(\gamma_{n}=0\) for all \(n\geq1\), then Algorithm 5 reduces to Algorithm 1.
Iteration procedures in fixed point theory are led by considerations in summability theory. For example, if a given sequence converges, then we do not look for the convergence of the sequence of its arithmetic means. Similarly, if the sequence of Picard iterates of any mapping T converges, then we do not look for the convergence of other iteration procedures.
The three-step iterative approximation problems were studied extensively by Noor [43, 44], Glowinski and Le Tallec [45], and Haubruge et al. [46]. The three-step iterations lead to highly parallelized algorithms under certain conditions. They are also a natural generalization of the splitting methods for solving partial differential equations. It has been shown [45] that a three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations. Thus we conclude that a three-step scheme plays an important and significant role in solving various problems which arise in pure and applied sciences. These facts motivated us to study a class of three-step iterative schemes in the setting of \(\operatorname{CAT}(k)\) spaces with \(k>0\).
In this paper, we study a newly defined modified three-step iteration scheme to approximate a fixed point for nearly asymptotically nonexpansive mappings in the setting of a \(\operatorname{CAT}(k)\) space with \(k>0\) and also establish Δ-convergence and strong convergence results for the above mentioned iteration scheme and mappings.
3 Main results
Now, we shall introduce existence theorems.
Theorem 3.1
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed convex subset of X, and let \(T\colon K\to K\) be a continuous nearly asymptotically nonexpansive mapping. Then T has a fixed point.
Proof
From Theorem 3.1 we shall now derive a result for a \(\operatorname{CAT}(0)\) space as follows.
Corollary 3.1
Let \((X,d)\) be a complete \(\operatorname{CAT}(0)\) space and K be a nonempty bounded, closed convex subset of X. If \(T\colon K\to K\) is a continuous nearly asymptotically nonexpansive mapping, then T has a fixed point.
Proof
It is well known that every convex subset of a \(\operatorname{CAT}(0)\) space, equipped with the induced metric, is a \(\operatorname{CAT}(k)\) space (see [12]). Then \((K,d)\) is a \(\operatorname{CAT}(0)\) space and hence it is a \(\operatorname{CAT}(k)\) space for all \(k>0\). Also note that K is R-convex for \(R=2\). Since K is bounded, we can chose \(\varepsilon\in(0,\pi/2)\) and \(k>0\) so that \(\operatorname{diam}(K)\leq\frac{\pi/2-\varepsilon}{\sqrt{k}}\). The conclusion follows from Theorem 3.1. This completes the proof. □
Theorem 3.2
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed convex subset of X, and let \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping. If \(\{x_{n}\}\) is an AFPS for T such that \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=z\), then \(z\in K\) and \(z=Tz\).
Proof
From Theorem 3.2 we can derive the following result as follows.
Corollary 3.2
Let \((X,d)\) be a complete \(\operatorname{CAT}(0)\) space, K be a nonempty bounded, closed convex subset of X and \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping. If \(\{x_{n}\}\) is an AFPS for T such that \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=z\), then \(z\in K\) and \(z=Tz\).
Now, we prove the following lemma using iteration scheme (2.6) needed in the sequel.
Lemma 3.1
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed and convex subset of X, and let \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping with sequence \(\{(a_{n},\eta(T^{n}))\}\) such that \(\sum_{n=1}^{\infty}a_{n}< \infty\) and \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\). Let \(\{x_{n}\}\) be a sequence in K defined by (2.6). Then \(\lim_{n\to\infty}d(x_{n},p)\) exists for each \(p\in F(T)\).
Proof
Lemma 3.2
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed convex subset of X, and let \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping with sequence \(\{(a_{n},\eta(T^{n}))\}\) such that \(\sum_{n=1}^{\infty}a_{n}< \infty\) and \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\). Let \(\{x_{n}\}\) be a sequence in K defined by (2.6). Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be sequences in \((0,1)\) such that \(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\), \(\liminf_{n\to\infty}\beta_{n}(1-\beta_{n})>0\) and \(\liminf_{n\to\infty}\gamma_{n}(1-\gamma_{n})>0\). Then \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\).
Proof
It follows from Theorem 3.1 that \(F(T)\neq\emptyset\). Let \(p\in F(T)\). From Lemma 3.1, we obtain that \(\lim_{n\to\infty}d(x_{n},p)\) exists for each \(p\in F(T)\). We claim that \(\lim_{n\to\infty}d(Tx_{n},x_{n})=0\).
Now, we are in a position to prove the Δ-convergence and strong convergence theorems.
Theorem 3.3
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space, with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\), for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed convex subset of X, and let \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping with sequence \(\{(a_{n},\eta(T^{n}))\}\) such that \(\sum_{n=1}^{\infty}a_{n}< \infty\) and \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\). Let \(\{x_{n}\}\) be a sequence in K defined by (2.6). Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be sequences in \((0,1)\) such that \(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\), \(\liminf_{n\to\infty}\beta_{n}(1-\beta_{n})>0\) and \(\liminf_{n\to\infty}\gamma_{n}(1-\gamma_{n})>0\). Then \(\{x_{n}\}\) Δ-converges to a fixed point of T.
Proof
Let \(\omega_{w}(x_{n}):=\bigcup A(\{u_{n}\})\) where the union is taken over all subsequences \(\{u_{n}\}\) of \(\{x_{n}\}\). We can complete the proof by showing that \(\omega_{w}(x_{n})\subseteq F(T)\) and \(\omega_{w}(x_{n})\) consists of exactly one point. Let \(u\in \omega_{w}(x_{n})\), then there exists a subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{u_{n}\})=\{u\}\). By Lemma 2.2, there exists a subsequence \(\{v_{n}\}\) of \(\{u_{n}\}\) such that \(\Delta\mbox{-}\!\lim_{n}v_{n}=v\in K\). Hence \(v\in F(T)\) by Lemma 3.1 and Lemma 3.2. Since \(\lim_{n\to\infty}d(x_{n},v)\) exists, so by Lemma 2.3, \(v=u\), i.e., \(\omega_{w}(x_{n})\subseteq F(T)\).
To show that \(\{x_{n}\}\) Δ-converges to a fixed point of T, it is sufficient to show that \(\omega_{w}(x_{n})\) consists of exactly one point.
Let \(\{w_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{w_{n}\})=\{w\}\) and let \(A(\{x_{n}\})=\{x\}\). Since \(w\in\omega_{w}(x_{n})\subseteq F(T)\) and by Lemma 3.1, \(\lim_{n\to\infty}d(x_{n},w)\) exists. Again by Lemma 3.1, we have \(x=w\in F(T)\). Thus \(\omega_{w}(x_{n})=\{x\}\). This shows that \(\{x_{n}\}\) Δ-converges to a fixed point of T. This completes the proof. □
Theorem 3.4
Let \(k>0\) and \((X,d)\) be a complete \(\operatorname{CAT}(k)\) space with \(\operatorname{diam}(X)=\frac{\pi/2-\varepsilon}{\sqrt{k}}\) for some \(\varepsilon\in(0,\pi/2)\). Let K be a nonempty closed convex subset of X, and let \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping with sequence \(\{(a_{n},\eta(T^{n}))\}\) such that \(\sum_{n=1}^{\infty}a_{n}< \infty\) and \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\). Let \(\{x_{n}\}\) be a sequence in K defined by (2.6). Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be sequences in \((0,1)\) such that \(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\), \(\liminf_{n\to\infty}\beta_{n}(1-\beta_{n})>0\) and \(\liminf_{n\to\infty}\gamma_{n}(1-\gamma_{n})>0\). Suppose that \(T^{m}\) is semi-compact for some \(m\in\mathbb{N}\). Then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
Proof
That is, \(p\in F(T)\). By Lemma 3.1, \(d(x_{n},p)\) exists, thus p is the strong limit of the sequence \(\{x_{n}\}\) itself. This shows that the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T. This completes the proof. □
Remark 3.1
Since T is completely continuous, the image of \(T^{m}\), for some \(m\in\mathbb{N}\), is semi-compact, \(\{x_{n}\}\) is a bounded sequence and \(d(x_{n},T^{m}x_{n})\to0\) as \(n\to \infty\). Thus \(T^{m}\), for some \(m\in\mathbb{N}\), is semi-compact, that is, the continuous image of a semi-compact space is semi-compact.
Example 3.1
([47])
The following example shows that there is a semi-compact mapping that is not compact.
Example 3.2
([47])
From Theorem 3.4 we can derive the following result as a corollary.
Corollary 3.3
Let \((X,d)\) be a complete \(\operatorname{CAT}(0)\) space, K be a nonempty bounded, closed convex subset of X and \(T\colon K\to K\) be a uniformly continuous nearly asymptotically nonexpansive mapping with sequence \(\{(a_{n},\eta(T^{n}))\}\) such that \(\sum_{n=1}^{\infty}a_{n}< \infty\) and \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\). Let \(\{x_{n}\}\) be a sequence in K defined by (2.6). Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be sequences in \((0,1)\) such that \(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\), \(\liminf_{n\to\infty}\beta_{n}(1-\beta_{n})>0\) and \(\liminf_{n\to\infty}\gamma_{n}(1-\gamma_{n})>0\). Suppose that \(T^{m}\) is semi-compact for some \(m\in\mathbb{N}\). Then the sequence \(\{x_{n}\}\) converges strongly to a fixed point of T.
Example 3.3
([40])
Example 3.4
The following example shows that, if the sequence \(\{x_{n}\}\) is weakly convergent, then it is not Δ-convergent.
Example 3.5
([37])
Let \(X=\mathbb{R}\), d be the usual metric on X, \(K=[-1,1]\), \(\{x_{n}\}=\{1,-1,1,-1, \ldots\}\), \(\{u_{n}\}=\{-1,-1,-1,\dots\}\) and \(\{v_{n}\}=\{1,1,1,\dots\}\). Then \(A(\{x_{n}\})=A_{K}(\{x_{n}\})=\{0\}\), \(A(\{u_{n}\})=\{-1\}\) and \(A(\{v_{n}\})=\{1\}\). This shows that \(\{x_{n}\}\rightharpoonup0\) but it does not have a Δ-limit.
4 Conclusions
- 1.
We proved strong and Δ convergence theorems of a modified three-step iteration process which contains a modified S-iteration process in the framework of \(\operatorname{CAT}(k)\) spaces.
- 2.
Theorem 3.1 extends Theorem 3.3 of Dhompongsa and Panyanak [30] to the case of a more general class of nonexpansive mappings which are not necessarily Lipschitzian, a modified three-step iteration scheme and from a \(\operatorname{CAT}(0)\) space to a \(\operatorname{CAT}(k)\) space considered in this paper.
- 3.
Theorem 3.1 also extends Theorem 3.5 of Niwongsa and Panyanak [48] to the case of a more general class of asymptotically nonexpansive mappings which are not necessarily Lipschitzian, a modified three-step iteration scheme and from a \(\operatorname{CAT}(0)\) space to a \(\operatorname{CAT}(k)\) space considered in this paper.
- 4.
Our results extend the corresponding results of Xu and Noor [22] to the case of a more general class of asymptotically nonexpansive mappings, a modified three-step iteration scheme and from a Banach space to a \(\operatorname{CAT}(k)\) space considered in this paper.
- 5.
Our results also extend and generalize the corresponding results of [35, 38, 49–52] for a more general class of non-Lipschitzian mappings, a modified three-step iteration scheme and from a uniformly convex metric space, a \(\operatorname{CAT}(0)\) space to a \(\operatorname{CAT}(k)\) space considered in this paper.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions and comments that helped to improve the manuscript.
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
References
- Gromov, M: Hyperbolic groups. In: Gersten, SM (ed.) Essays in Group Theory, vol. 8, pp. 75-263. Springer, Berlin (1987) View ArticleGoogle Scholar
- Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Coleccion Abierta, vol. 64, pp. 195-225. University of Seville Secretary of Publications, Seville (2003) Google Scholar
- Kirk, WA: Geodesic geometry and fixed point theory. II. In: International Conference on Fixed Point Theory and Applications, pp. 113-142. Yokohama Publishers, Yokohama (2004) Google Scholar
- Abkar, A, Eslamian, M: Common fixed point results in \(\operatorname{CAT}(0)\) spaces. Nonlinear Anal. TMA 74(5), 1835-1840 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Chang, SS, Wang, L, Joesph Lee, HW, Chan, CK, Yang, L: Demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Appl. Math. Comput. 219(5), 2611-2617 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Chaoha, P, Phon-on, A: A note on fixed point sets in \(\operatorname{CAT}(0)\) spaces. J. Math. Anal. Appl. 320(2), 983-987 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Dhompongsa, S, Kaewkho, A, Panyanak, B: Lim’s theorems for multivalued mappings in \(\operatorname{CAT}(0)\) spaces. J. Math. Anal. Appl. 312(2), 478-487 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Espinola, R, Fernandez-Leon, A: \(\operatorname{CAT}(k)\)-Spaces, weak convergence and fixed point. J. Math. Anal. Appl. 353(1), 410-427 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Laowang, W, Panyanak, B: Strong and △ convergence theorems for multivalued mappings in CAT(0) spaces. J. Inequal. Appl. 2009, Article ID 730132 (2009) View ArticleMathSciNetGoogle Scholar
- Leustean, L: A quadratic rate of asymptotic regularity for \(\operatorname{CAT}(0)\)-spaces. J. Math. Anal. Appl. 325(1), 386-399 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Saluja, GS: Fixed point theorems in CAT(0) spaces using a generalized Z-type condition. J. Adv. Math. Stud. 7(1), 89-96 (2014) MATHMathSciNetGoogle Scholar
- Bridson, MR, Haefliger, A: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999) MATHGoogle Scholar
- Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171-174 (1972) View ArticleMATHMathSciNetGoogle Scholar
- Liu, QH: Iterative sequences for asymptotically quasi-nonexpansive mappings. J. Math. Anal. Appl. 259, 1-7 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Liu, QH: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. J. Math. Anal. Appl. 259, 18-24 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Saluja, GS: Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space. Tamkang J. Math. 38(1), 85-92 (2007) MATHMathSciNetGoogle Scholar
- Saluja, GS: Convergence result of \((L,\alpha)\)-uniformly Lipschitz asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces. Jñānābha 38, 41-48 (2008) MATHMathSciNetGoogle Scholar
- Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43(1), 153-159 (1991) View ArticleMATHMathSciNetGoogle Scholar
- Shahzad, N, Udomene, A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2006, Article ID 18909 (2006) View ArticleMathSciNetGoogle Scholar
- Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Tan, KK, Xu, HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 122, 733-739 (1994) View ArticleMATHMathSciNetGoogle Scholar
- Xu, BL, Noor, MA: Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 267(2), 444-453 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Ceng, LC, Sahu, DR, Yao, JC: Implicit iterative algorithm for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings. J. Comput. Appl. Math. 233(11), 2902-2915 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Ceng, LC, Yao, JC: Strong convergence theorems for variational inequalities and fixed point problem of asymptotically strict pseudocontractive mappings in the intermediate sense. Acta Appl. Math. 115, 167-191 (2011) View ArticleMATHMathSciNetGoogle Scholar
- He, JS, Fang, DH, Li, C, López, G: Mann’s algorithm for nonexpansive mappings in \(\operatorname{CAT}(k)\) spaces. Nonlinear Anal. 75, 445-452 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Li, C, López, G, Martin-Máquez, V: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwan. J. Math. 14(2), 541-559 (2010) MATHGoogle Scholar
- Naraghirad, E, Wong, NC, Yao, JC: Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces. J. Inequal. Appl. 2011, Article ID 119 (2011) View ArticleMathSciNetGoogle Scholar
- Lim, TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, 179-182 (1976) View ArticleGoogle Scholar
- Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689-3696 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Dhompongsa, S, Panyanak, B: On △-convergence theorem in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56(10), 2572-2579 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953) View ArticleMATHGoogle Scholar
- Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974) View ArticleMATHMathSciNetGoogle Scholar
- Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61-79 (2007) MATHMathSciNetGoogle Scholar
- Abbas, M, Kadelburg, Z, Sahu, DR: Fixed point theorems for Lipschitzian type mappings in \(\operatorname{CAT}(0)\) spaces. Math. Comput. Model. 55, 1418-1427 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Khan, SH, Abbas, M: Strong and △-convergence of some iterative schemes in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 61(1), 109-116 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Kumam, P, Saluja, GS, Nashine, HK: Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces. J. Inequal. Appl. 2014, Article ID 368 (2014) View ArticleGoogle Scholar
- Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2010, Article ID 268780 (2010) View ArticleMathSciNetGoogle Scholar
- Şahin, A, Başarir, M: On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a CAT(0) space. Fixed Point Theory Appl. 2013, Article ID 12 (2013) View ArticleGoogle Scholar
- Saluja, GS: Modified S-iteration process for two asymptotically quasi-nonexpansive mappings in a CAT(0) space. J. Adv. Math. Stud. 7(2), 151-159 (2014) MATHMathSciNetGoogle Scholar
- Sahu, DR: Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces. Comment. Math. Univ. Carol. 46(4), 653-666 (2005) MATHMathSciNetGoogle Scholar
- Bruhat, F, Tits, J: Groupes réductifs sur un corps local. Publ. Math. IHES 41, 5-251 (1972) View ArticleMATHMathSciNetGoogle Scholar
- Ohta, S: Convexities of metric spaces. Geom. Dedic. 125, 225-250 (2007) View ArticleMATHGoogle Scholar
- Noor, MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251(1), 217-229 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Noor, MA: Three-step iterative algorithms for multivalued quasi variational inclusions. J. Math. Anal. Appl. 255, 589-604 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Glowinski, R, Le Tallec, P: Augmented Lagrangian and operator-splitting methods. In: Nonlinear Mechanics. SIAM, Philadelphia (1989) Google Scholar
- Haubruge, S, Nguyen, VH, Strodiot, JJ: Convergence analysis and applications of the Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 97, 645-673 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Agarwal, RP, O’Regan, D, Sahu, DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, Berlin (2009). doi:10.1007/978-0-387-75818-3 MATHGoogle Scholar
- Niwongsa, Y, Panyanak, B: Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces. Int. J. Math. Anal. 4(13), 645-656 (2010) MATHMathSciNetGoogle Scholar
- Beg, I: An iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces. Nonlinear Anal. Forum 6, 27-34 (2001) MATHMathSciNetGoogle Scholar
- Osilike, MO, Aniagbosor, SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 32, 1181-1191 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Şahin, A, Başarir, M: On the strong convergence of SP-iteration on \(\operatorname{CAT}(0)\) space. J. Inequal. Appl. 2013, Article ID 311 (2013) View ArticleGoogle Scholar
- Saluja, GS: Strong and Δ-convergence of new three-step iteration process in CAT(0) spaces. Nonlinear Anal. Forum 20, 43-52 (2015) Google Scholar