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
nearly asymptotically nonexpansive mapping three-step iteration scheme fixed point strong convergence Δ-convergence \(\operatorname{CAT}(k)\) spaceMSC
54H25 54E401 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.
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