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Convergence of three-step iterations for nearly asymptotically nonexpansive mappings in \(\operatorname{CAT}(k)\) spaces

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.

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., [411]). 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., [1422]; see also [2327]).

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, 3439]).

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

Let \((X,d)\) be a metric space and K be its nonempty subset. Then the mapping \(T\colon K\to K\) is said to be:

  1. (1)

    nonexpansive if \(d(Tx,Ty)\leq d(x,y)\) for all \(x,y\in K\);

  2. (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. (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. (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. (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. (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

Let K be a nonempty subset of a metric space \((X,d)\) and fix a sequence \(\{a_{n}\}\subset[0,\infty)\) with \(\lim_{n\to\infty}a_{n}=0\). A mapping \(T\colon K\to K\) is said to be nearly Lipschitzian with respect to \(\{a_{n}\}\) if, for all \(n\geq1\), there exists a constant \(k_{n}\geq0\) such that

$$d\bigl(T^{n}x,T^{n}y\bigr)\leq k_{n} \bigl[d(x,y)+a_{n}\bigr] \quad\mbox{for all } x, y\in K. $$

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 \(T^{n}\).

A nearly Lipschitzian mapping T with sequence \(\{a_{n},\eta(T^{n})\}\) is said to be:

  1. (i)

    nearly nonexpansive if \(\eta(T^{n})=1\) for all \(n\geq1\);

  2. (ii)

    nearly asymptotically nonexpansive if \(\eta(T^{n})\geq1\) for all \(n\geq1\) and \(\lim_{n\to\infty}\eta(T^{n})=1\);

  3. (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.

Given a real number k, we denote by \(M_{k}^{2}\) the following metric spaces:

  1. (i)

    if \(k=0\), then \(M_{k}^{2}\) is an Euclidean space \(\mathbb{E}^{n}\);

  2. (ii)

    if \(k>0\), then \(M_{k}^{2}\) is obtained from the sphere \(\mathbb{S}^{n}\) by multiplying the distance function by \(\frac{1}{\sqrt{k}}\);

  3. (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.

Notice that in a \(\operatorname{CAT}(0)\) space \((X,d)\) if \(x,y,z\in X\), then the \(\operatorname{CAT}(0)\) inequality implies

$$\begin{aligned} (\mathrm{CN})\quad d^{2} \biggl(x,\frac{y\oplus z}{2} \biggr) \leq& \frac{1}{2} d^{2}(x,y)+\frac{1}{2} d^{2}(x,z)- \frac{1}{4} d^{2}(y,z). \end{aligned}$$

This is the (CN) inequality of Bruhat and Tits [41]. This inequality is extended by Dhompongsa and Panyanak in [30] as

$$\begin{aligned} \bigl(\mathrm{CN}^{*}\bigr)\quad d^{2}\bigl(z,\alpha x\oplus(1-\alpha)y \bigr)\leq \alpha d^{2}(z,x)+(1-\alpha)d^{2}(z,y)-\alpha(1- \alpha)d^{2}(x,y) \end{aligned}$$

for all \(\alpha\in[0,1]\) and \(x,y,z\in X\). In fact, if X is a geodesic space, then the following statements are equivalent:

  1. (i)

    X is a \(\operatorname{CAT}(0)\) space;

  2. (ii)

    X satisfies the (CN) inequality;

  3. (iii)

    X satisfies the (CN) inequality.

Let \(R\in(0,2]\). Recall that a geodesic space \((X,d)\) is said to be R-convex for R (see [42]) if for any three points \(x,y,z\in X\), we have

$$\begin{aligned} d^{2}\bigl(z,\alpha x\oplus(1-\alpha)y\bigr) \leq& \alpha d^{2}(z,x)+(1-\alpha)d^{2}(z,y) \\ &{} -\frac{R}{2}\alpha(1-\alpha)d^{2}(x,y). \end{aligned}$$
(2.1)

It follows from (CN) that a geodesic space \((X,d)\) is a \(\operatorname{CAT}(0)\) space if and only if \((X,d)\) is R-convex for R=2.

In the sequel we need the following lemma.

Lemma 2.1

([12], p.176)

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 \(\varepsilon\in(0,\pi/2)\). Then

$$d\bigl((1-\alpha) x\oplus\alpha y,z\bigr)\leq (1-\alpha)d(x,z)+\alpha d(y,z) $$

for all \(x,y,z\in X\) and \(\alpha\in[0,1]\).

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 \(k>0\).

Let \(\{x_{n}\}\) be a bounded sequence in a \(\operatorname{CAT}(k)\) space \((X,d)\). For \(x\in X\), set

$$r\bigl(x,\{x_{n}\}\bigr)=\limsup_{n\to\infty}d(x,x_{n}). $$

The asymptotic radius \(r(\{x_{n}\})\) of \(\{x_{n}\}\) is given by

$$r\bigl(\{x_{n}\}\bigr)=\inf\bigl\{ r\bigl(x,\{x_{n}\}\bigr):x \in X\bigr\} , $$

and the asymptotic center \(A(\{x_{n}\})\) of \(\{x_{n}\}\) is the set

$$A\bigl(\{x_{n}\}\bigr)= \bigl\{ x\in X:r\bigl(\{x_{n}\} \bigr)=r\bigl(x,\{x_{n}\}\bigr) \bigr\} . $$

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

([28, 29])

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

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 \(\varepsilon\in(0,\pi/2)\). Then the following statements hold:

  1. (i)

    ([8], Corollary 4.4) Every sequence in X has a Δ-convergent subsequence.

  2. (ii)

    ([8], Proposition 4.5) If \(\{x_{n}\}\subseteq X\) and \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=x\), then \(x\in \bigcap_{k=1}^{\infty}\overline{\operatorname{conv}}\{x_{k},x_{k+1},\dots\}\),

where \(\overline{\operatorname{conv}}(A)=\bigcap \{B:B\supseteq A\textit{ and } B \textit{ is closed and convex}\}\).

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])

Let \(\{p_{n}\}_{n=1}^{\infty}\), \(\{q_{n}\}_{n=1}^{\infty}\) and \(\{r_{n}\}_{n=1}^{\infty}\) be sequences of nonnegative numbers satisfying the inequality

$$p_{n+1} \leq(1+q_{n})p_{n} + r_{n},\quad \forall n\geq1. $$

If \(\sum_{n=1}^{\infty}q_{n} < \infty\) and \(\sum_{n=1}^{\infty}r_{n} < \infty\), then \(\lim_{n\to\infty}p_{n}\) exists.

Proposition 2.1

([37], Proposition 3.12)

Let \(\{x_{n}\} \) be a bounded sequence in a \(\operatorname{CAT}(0)\) space X, and let C be a closed convex subset of X which contains \(\{x_{n}\}\). Then

  1. (i)

    \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=x\) implies that \(\{x_{n}\}\rightharpoonup x\),

  2. (ii)

    the converse is true if \(\{x_{n}\}\) is regular.

Algorithm 1

The sequence \(\{x_{n}\}\) defined by \(x_{1}\in K\) and

$$ \begin{aligned} &y_{n}= (1-\beta_{n})x_{n} \oplus\beta_{n}T^{n}x_{n}, \\ &x_{n+1}= (1-\alpha_{n})T^{n}x_{n}\oplus \alpha_{n}T^{n}y_{n},\quad n\geq 1, \end{aligned} $$
(2.2)

where \(\{\alpha_{n}\}_{n=1}^{\infty}\) and \(\{\beta_{n}\}_{n=1}^{\infty}\) are appropriate sequences in \((0,1)\), is called a modified S-iterative sequence (see [33]).

If \(T^{n}=T\) for all \(n\geq1\), then Algorithm 1 reduces to the following.

Algorithm 2

The sequence \(\{x_{n}\}\) defined by \(x_{1}\in K\) and

$$ \begin{aligned} &y_{n}= (1-\beta_{n})x_{n}\oplus \beta_{n}Tx_{n}, \\ &x_{n+1}= (1-\alpha_{n})Tx_{n}\oplus \alpha_{n}Ty_{n}, \quad n\geq 1, \end{aligned} $$
(2.3)

where \(\{\alpha_{n}\}_{n=1}^{\infty}\) and \(\{\beta_{n}\}_{n=1}^{\infty}\) are appropriate sequences in \((0,1)\), is called an S-iterative sequence (see [33]).

Algorithm 3

The sequence \(\{x_{n}\}\) defined by \(x_{1}\in K\) and

$$ \begin{aligned} &y_{n}= (1-\beta_{n})x_{n} \oplus\beta_{n}T^{n}x_{n}, \\ &x_{n+1}= (1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n},\quad n\geq 1, \end{aligned} $$
(2.4)

where \(\{\alpha_{n}\}_{n=1}^{\infty}\) and \(\{\beta_{n}\}_{n=1}^{\infty}\) are appropriate sequences in \([0,1]\), is called an Ishikawa iterative sequence (see [32]).

If \(\beta_{n}=0\) for all \(n\geq1\), then Algorithm 3 reduces to the following.

Algorithm 4

The sequence \(\{x_{n}\}\) defined by \(x_{1}\in K\) and

$$\begin{aligned} x_{n+1}= (1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}x_{n},\quad n\geq 1, \end{aligned}$$
(2.5)

where \(\{\alpha_{n}\}_{n=1}^{\infty}\) is a sequence in \((0,1)\), is called a Mann iterative sequence (see [31]).

Motivated and inspired by [33] and some others, we modify iteration scheme (2.2) as follows.

Algorithm 5

The sequence \(\{x_{n}\}\) defined by \(x_{1}\in K\) and

$$ \begin{aligned} &z_{n}= (1-\gamma_{n})x_{n}\oplus \gamma_{n} T^{n}x_{n}, \\ &y_{n}= (1-\beta_{n})x_{n}\oplus \beta_{n}T^{n}z_{n}, \\ &x_{n+1}= (1-\alpha_{n})T^{n}x_{n}\oplus \alpha_{n}T^{n}y_{n}, \quad n\geq 1, \end{aligned} $$
(2.6)

where \(\{\alpha_{n}\}_{n=1}^{\infty}\), \(\{\beta_{n}\}_{n=1}^{\infty}\), \(\{\gamma_{n}\}_{n=1}^{\infty}\) are appropriate sequences in \((0,1)\), is called a modified three-step iterative sequence. Iteration scheme (2.6) is independent of modified Noor iteration, modified Ishikawa iteration and modified Mann iteration schemes.

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

Fix \(x\in K\). We can consider the sequence \(\{T^{n}x\}_{n=1}^{\infty}\) as a bounded sequence in K. Let ϕ be a function defined by

$$\phi\colon K\to[0,\infty) , \quad\phi(u)=\limsup_{n\to\infty }d \bigl(T^{n}x,u\bigr) \quad\mbox{for all } u\in K. $$

Then there exists \(z\in K\) such that \(\phi(z)=\inf\{\Phi(u):u\in K\}\). Since T is a nearly asymptotically nonexpansive mapping, for each \(n,m\in\mathbb{N}\), we have

$$d\bigl(T^{n+m}x,T^{m}z\bigr)\leq\eta\bigl(T^{m} \bigr) \bigl(d\bigl(T^{n}x,z\bigr)+a_{m}\bigr). $$

On taking limit as \(n\to\infty\), we obtain

$$\begin{aligned} \phi\bigl(T^{m}z\bigr)\leq \eta\bigl(T^{m}\bigr) \phi(z)+ \eta\bigl(T^{m}\bigr)a_{m} \end{aligned}$$
(3.1)

for any \(m\in\mathbb{N}\). This implies that

$$\begin{aligned} \lim_{m\to\infty}\phi\bigl(T^{m}z\bigr)\leq \phi(z). \end{aligned}$$
(3.2)

In view of inequality (2.1), we obtain

$$\begin{aligned} d \biggl(T^{n}x,\frac{T^{m}z\oplus T^{h}z}{2} \biggr)^{2} \leq& \frac{1}{2}d\bigl(T^{n}x,T^{m}z\bigr)^{2}+ \frac{1}{2}d\bigl(T^{n}x,T^{h}z\bigr)^{2} \\ &{} -\frac{R}{8}d\bigl(T^{m}z,T^{h}z \bigr)^{2}, \end{aligned}$$

which, on taking limit as \(n\to\infty\), gives

$$\begin{aligned} \phi(z)^{2} \leq& \Phi \biggl(\frac{T^{m}z\oplus T^{h}z}{2} \biggr)^{2} \\ \leq& \frac{1}{2} \phi\bigl(T^{m}z\bigr)^{2}+ \frac{1}{2} \phi\bigl(T^{h}z\bigr)^{2}- \frac{R}{8} d\bigl(T^{m}z,T^{h}z\bigr)^{2}. \end{aligned}$$
(3.3)

The above inequality yields

$$\begin{aligned} \frac{R}{8} d\bigl(T^{m}z,T^{h}z\bigr)^{2} \leq& \frac{1}{2} \phi\bigl(T^{m}z\bigr)^{2}+ \frac{1}{2} \phi\bigl(T^{h}z\bigr)^{2}- \phi(z)^{2}. \end{aligned}$$
(3.4)

By (3.2) and (3.4), we have \(\limsup_{m,h\to\infty}d(T^{m}z,T^{h}z)\leq0\). Therefore, \(\{T^{n}z\}_{n=1}^{\infty}\) is a Cauchy sequence in K and hence converges to some point \(v\in K\). Since T is continuous,

$$Tv=T \Bigl(\lim_{n\to\infty}T^{n}z \Bigr)=\lim _{n\to\infty}T^{n+1}z=v. $$

This shows that T has a fixed point in K. This completes the 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

By Lemma 2.2, we get that \(z\in K\). As in Theorem 3.1, we define

$$\phi(u)=\limsup_{n\to\infty}d(x_{n},u) $$

for each \(u\in K\). Since \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\), by induction we can show that

$$\lim_{n\to\infty}d\bigl(x_{n},T^{m}x_{n}\bigr)=0 $$

for some \(m\in \mathbb{N}\). This implies that

$$\begin{aligned} \phi(u)=\limsup_{n\to\infty}d\bigl(T^{m}x_{n},u \bigr) \quad\mbox{for each } u\in K \mbox{ and } m\in\mathbb{N}. \end{aligned}$$
(3.5)

Taking \(u=T^{m}z\) in (3.5), we have

$$\begin{aligned} \phi\bigl(T^{m}z\bigr) =&\limsup_{n\to\infty}d \bigl(T^{m}x_{n},T^{m}z\bigr) \\ \leq& \limsup_{n\to\infty}\bigl[\eta\bigl(T^{m}\bigr) \bigl(d(x_{n},z)+a_{m}\bigr)\bigr]. \end{aligned}$$
(3.6)

Hence

$$\begin{aligned} \limsup_{m\to\infty}\phi\bigl(T^{m}z\bigr) \leq& \phi(z). \end{aligned}$$
(3.7)

In view of inequality (2.1), we have

$$\begin{aligned} d \biggl(x_{n},\frac{z\oplus T^{m}z}{2} \biggr)^{2} \leq& \frac{1}{2} d(x_{n},z)^{2}+\frac{1}{2} d \bigl(x_{n},T^{m}z\bigr)^{2} -\frac{R}{8}d\bigl(z,T^{m}z\bigr)^{2}, \end{aligned}$$

where \(R=(\pi-2\varepsilon)\tan(\varepsilon)\). Since \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=z\), letting \(n\to\infty\), we get

$$\begin{aligned} \phi(z)^{2} \leq& \Phi \biggl(\frac{z\oplus T^{m}z}{2} \biggr)^{2} \\ \leq& \frac{1}{2} \phi(z)^{2}+\frac{1}{2} \phi \bigl(T^{m}z\bigr)^{2}-\frac{R}{8} d \bigl(z,T^{m}z\bigr)^{2}. \end{aligned}$$
(3.8)

This yields

$$\begin{aligned} d\bigl(z,T^{m}z\bigr)^{2} \leq& \frac{4}{R} \bigl[ \phi\bigl(T^{m}z\bigr)^{2}-\phi(z)^{2} \bigr]. \end{aligned}$$
(3.9)

By (3.7) and (3.9), we have \(\lim_{m\to\infty}d(z,T^{m}z)=0\). Since T is continuous,

$$Tz=T \Bigl(\lim_{m\to\infty}T^{m}z \Bigr)=\lim _{n\to\infty}T^{m+1}z=z. $$

This shows that T has a fixed point in K. This completes the 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

It follows from Theorem 3.1 that \(F(T)\neq\emptyset\). Let \(p\in F(T)\) and since T is nearly asymptotically nonexpansive, by (2.6) and Lemma 2.1, we have

$$\begin{aligned} d(z_{n},p) =& d\bigl((1-\gamma_{n})x_{n} \oplus\gamma_{n} T^{n}x_{n},p\bigr) \\ \leq& (1-\gamma_{n})d(x_{n},p)+\gamma_{n}d \bigl(T^{n}x_{n},p\bigr) \\ \leq& (1-\gamma_{n})d(x_{n},p)+\gamma_{n}\bigl[ \eta\bigl(T^{n}\bigr) \bigl(d(x_{n},p)+a_{n}\bigr) \bigr] \\ \leq& \eta\bigl(T^{n}\bigr)\bigl[(1-\gamma_{n})d(x_{n},p)+ \gamma_{n}d(x_{n},p)\bigr]+\gamma_{n}\eta \bigl(T^{n}\bigr)a_{n} \\ \leq& \eta\bigl(T^{n}\bigr) d(x_{n},p)+\eta \bigl(T^{n}\bigr)a_{n}. \end{aligned}$$
(3.10)

Again using (2.6), (3.10) and Lemma 2.1, we have

$$\begin{aligned} d(y_{n},p) =& d\bigl((1-\beta_{n})x_{n} \oplus\beta_{n}T^{n}z_{n},p\bigr) \\ \leq& (1-\beta_{n})d(x_{n},p)+\beta_{n}d \bigl(T^{n}z_{n},p\bigr) \\ \leq& (1-\beta_{n})d(x_{n},p)+\beta_{n}\bigl[\eta \bigl(T^{n}\bigr) \bigl(d(z_{n},p)+a_{n}\bigr)\bigr] \\ \leq& (1-\beta_{n})d(x_{n},p)+\beta_{n}\eta \bigl(T^{n}\bigr)d(z_{n},p)+\eta\bigl(T^{n} \bigr)a_{n} \\ \leq& (1-\beta_{n})d(x_{n},p)+\beta_{n}\eta \bigl(T^{n}\bigr)\bigl[\eta\bigl(T^{n}\bigr) \bigl(d(x_{n},p)+\eta \bigl(T^{n}\bigr)a_{n}\bigr) \bigr]+\eta\bigl(T^{n}\bigr)a_{n} \\ \leq& \eta\bigl(T^{n}\bigr)^{2}\bigl[(1- \beta_{n})d(x_{n},p)+\beta_{n}d(x_{n},p) \bigr] + \bigl(\eta\bigl(T^{n}\bigr)+\eta\bigl(T^{n} \bigr)^{2} \bigr)a_{n} \\ \leq& \eta\bigl(T^{n}\bigr)^{2} d(x_{n},p)+ \bigl(\eta\bigl(T^{n}\bigr)+\eta\bigl(T^{n} \bigr)^{2} \bigr)a_{n}. \end{aligned}$$
(3.11)

Finally, using (2.6), (3.11) and Lemma 2.1, we get

$$\begin{aligned} d(x_{n+1},p) =& d\bigl((1-\alpha_{n})T^{n}x_{n} \oplus\alpha_{n}T^{n}y_{n},p\bigr) \\ \leq&(1-\alpha_{n})d\bigl(T^{n}x_{n},p\bigr)+ \alpha_{n}d\bigl(T^{n}y_{n},p\bigr) \\ \leq&(1-\alpha_{n})\bigl[\eta\bigl(T^{n}\bigr) \bigl(d(x_{n},p)+a_{n}\bigr)\bigr]+\alpha_{n}\bigl[ \eta \bigl(T^{n}\bigr) \bigl(d(y_{n},p)+a_{n}\bigr) \bigr] \\ =& (1-\alpha_{n})\eta\bigl(T^{n}\bigr) d(x_{n},p)+ \alpha_{n}\eta\bigl(T^{n}\bigr) d(y_{n},p)+\eta \bigl(T^{n}\bigr)a_{n} \\ \leq& (1-\alpha_{n})\eta\bigl(T^{n}\bigr) d(x_{n},p)+\alpha_{n}\eta\bigl(T^{n}\bigr) \\ &{}\times \bigl[\eta\bigl(T^{n}\bigr)^{2} d(x_{n},p)+ \bigl(\eta\bigl(T^{n}\bigr)+\eta \bigl(T^{n}\bigr)^{2} \bigr)a_{n} \bigr]+\eta \bigl(T^{n}\bigr)a_{n} \\ \leq& \eta\bigl(T^{n}\bigr)^{3}\bigl[(1- \alpha_{n})d(x_{n},p)+\alpha_{n} d(x_{n},p)\bigr] \\ &{} +\eta\bigl(T^{n}\bigr) \bigl(\eta\bigl(T^{n}\bigr)+\eta \bigl(T^{n}\bigr)^{2} \bigr)a_{n}+\eta \bigl(T^{n}\bigr)a_{n} \\ =&\eta\bigl(T^{n}\bigr)^{3} d(x_{n},p)+ \bigl( \eta\bigl(T^{n}\bigr)+\eta\bigl(T^{n}\bigr)^{2}+\eta \bigl(T^{n}\bigr)^{3} \bigr)a_{n} \\ =& (1+w_{n})d(x_{n},p)+v_{n}, \end{aligned}$$
(3.12)

where \(w_{n}= (\eta(T^{n})^{3}-1 )= (\eta(T^{n})^{2}+\eta(T^{n})+1 ) (\eta (T^{n})-1 )\) and \(v_{n}= (\eta(T^{n})+\eta(T^{n})^{2}+\eta(T^{n})^{3} )a_{n}\). Since \(\sum_{n=1}^{\infty} (\eta(T^{n})-1 )< \infty\) and \(\sum_{n=1}^{\infty}a_{n}< \infty\), it follows that \(\sum_{n=1}^{\infty}w_{n}< \infty\) and \(\sum_{n=1}^{\infty}v_{n}< \infty\). Hence, by Lemma 2.4, we get that \(\lim_{n\to\infty}d(x_{n},p)\) exists. This completes the 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 \(\lim_{n\to\infty}d(Tx_{n},x_{n})=0\).

Since \(\{x_{n}\}\) is bounded, there exists \(R>0\) such that \(\{x_{n}\},\{y_{n}\},\{z_{n}\}\subset B_{R}'(p)\) for all \(n\geq1\) with \(R'< D_{k}/2\). In view of (2.1), we have

$$\begin{aligned} d(z_{n},p)^{2} =& d\bigl((1- \gamma_{n})x_{n}\oplus \gamma_{n}T^{n}x_{n},p \bigr)^{2} \\ \leq& \gamma_{n}d\bigl(T^{n}x_{n},p \bigr)^{2}+(1-\gamma_{n})d(x_{n},p)^{2} -\frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr) \\ \leq& \gamma_{n}\bigl[\eta\bigl(T^{n}\bigr) \bigl(d(x_{n},p)+a_{n}\bigr)\bigr]^{2}+(1-\gamma _{n})d(x_{n},p)^{2} -\frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr) \\ \leq& \eta\bigl(T^{n}\bigr)^{2} d(x_{n},p)^{2}+P a_{n} -\frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr) \end{aligned}$$
(3.13)

for some \(P>0\). This implies that

$$\begin{aligned} d(z_{n},p)^{2} \leq& \eta\bigl(T^{n} \bigr)^{2} d(x_{n},p)^{2}+P a_{n}. \end{aligned}$$
(3.14)

Again from (2.1) and using (3.14), we have

$$\begin{aligned} d(y_{n},p)^{2} =& d^{2}\bigl((1- \beta_{n})x_{n}\oplus \beta_{n}T^{n}z_{n},p \bigr)^{2} \\ \leq& \beta_{n}d\bigl(T^{n}z_{n},p \bigr)^{2}+(1-\beta_{n})d^{2}(x_{n},p)^{2} \\ &{}-\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \\ \leq& \beta_{n}\bigl[\eta\bigl(T^{n}\bigr) \bigl(d(z_{n},p)+a_{n}\bigr)\bigr]^{2}+(1- \beta_{n})d(x_{n},p)^{2} \\ &{}-\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \\ \leq& \eta\bigl(T^{n}\bigr)^{2}\beta_{n}d(z_{n},p)^{2}+Q a_{n}+(1-\beta _{n})d(x_{n},p)^{2} \\ &{}-\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \\ \leq& \eta\bigl(T^{n}\bigr)^{2}\beta_{n}\bigl[ \eta\bigl(T^{n}\bigr)^{2} d^{2}(x_{n},p)+P a_{n}\bigr]+Q a_{n} \\ &{} +(1-\beta_{n})d(x_{n},p)^{2}- \beta_{n}(1-\beta_{n})d\bigl(T^{n}z_{n},x_{n} \bigr)^{2} \\ \leq& \eta\bigl(T^{n}\bigr)^{4}d(x_{n},p)^{2}+(L+Q)a_{n} \\ &{} -\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \end{aligned}$$
(3.15)

for some \(L, Q>0\).

This implies that

$$\begin{aligned} d(y_{n},p)^{2} \leq& \eta\bigl(T^{n} \bigr)^{4} d(x_{n},p)^{2}+(L+Q) a_{n}. \end{aligned}$$
(3.16)

Finally, from (2.1) and using (3.16), we have

$$\begin{aligned} d(x_{n+1},p)^{2} =& d\bigl((1- \alpha_{n})T^{n}x_{n}\oplus \alpha_{n}T^{n}y_{n},p \bigr)^{2} \\ \leq& \alpha_{n}d\bigl(T^{n}y_{n},p \bigr)^{2}+(1-\alpha_{n})d\bigl(T^{n}x_{n},p \bigr)^{2} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ \leq& \alpha_{n}\bigl[\eta\bigl(T^{n}\bigr) \bigl(d(y_{n},p)+a_{n}\bigr)\bigr]^{2}+(1- \alpha_{n})\bigl[\eta \bigl(T^{n}\bigr)d(x_{n},p)+a_{n} \bigr]^{2} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ \leq& \alpha_{n}\eta\bigl(T^{n}\bigr)^{2}d(y_{n},p)^{2}+M a_{n}+(1-\alpha_{n})\eta \bigl(T^{n} \bigr)^{2}d(x_{n},p)^{2} \\ &{} +N a_{n}-\frac{R}{2}\alpha_{n}(1- \alpha_{n})d\bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ \leq& \alpha_{n}\eta\bigl(T^{n}\bigr)^{2}\bigl[ \eta\bigl(T^{n}\bigr)^{4} d(x_{n},p)^{2}+(L+Q) a_{n}\bigr] \\ &{} +(M+N) a_{n}+(1-\alpha_{n})\eta\bigl(T^{n} \bigr)^{2}d(x_{n},p)^{2} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ \leq& \eta\bigl(T^{n}\bigr)^{6} d(x_{n},p)^{2}+(L+Q+M+N)a_{n} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ =& \bigl[1+\bigl(\eta\bigl(T^{n}\bigr)^{6}-1\bigr)\bigr] d(x_{n},p)^{2}+(L+Q+M+N)a_{n} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \\ =& \bigl[1+\bigl(\eta\bigl(T^{n}\bigr)-1\bigr)\rho\bigr] d(x_{n},p)^{2}+(L+Q+M+N)a_{n} \\ &{}-\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \end{aligned}$$
(3.17)

for some \(M, N, \rho>0\).

This implies that

$$\begin{aligned} \alpha_{n}(1-\alpha_{n})d\bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2} \leq& d(x_{n},p)^{2}-d(x_{n+1},p)^{2}+ \bigl(\eta\bigl(T^{n}\bigr)-1\bigr)\rho d(x_{n},p)^{2} \\ &{} +(L+Q+M+N)a_{n}. \end{aligned}$$

Since \(\sum_{n=1}^{\infty}a_{n}< \infty\), \(\sum_{n=1}^{\infty}(\eta(T^{n})-1)< \infty\) and \(d(x_{n},p)< R'\), we have

$$\frac{R}{2}\alpha_{n}(1-\alpha_{n})d \bigl(T^{n}x_{n},T^{n}y_{n} \bigr)^{2}< \infty. $$

Hence by the fact that \(\liminf_{n\to\infty}\alpha_{n}(1-\alpha_{n})>0\), we have

$$\begin{aligned} \lim_{n\to\infty}d\bigl(T^{n}x_{n},T^{n}y_{n} \bigr)= 0. \end{aligned}$$
(3.18)

Now, consider (3.15), we have

$$\begin{aligned} d(y_{n},p)^{2} \leq& \bigl[1+\bigl(\eta \bigl(T^{n}\bigr)^{4}-1\bigr)\bigr]d(x_{n},p)^{2}+(L+Q)a_{n} \\ &{} -\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \\ \leq& \bigl[1+\bigl(\eta\bigl(T^{n}\bigr)-1\bigr)\mu \bigr]d(x_{n},p)^{2}+(L+Q)a_{n} \\ &{} -\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \end{aligned}$$
(3.19)

for some \(\mu>0\).

Equation (3.19) yields

$$\begin{aligned} \frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2} \leq& d(x_{n},p)^{2}-d(y_{n},p)^{2}+\bigl(\eta \bigl(T^{n}\bigr)-1\bigr)\mu d(x_{n},p)^{2} +(L+Q)a_{n}. \end{aligned}$$

Since \(\sum_{n=1}^{\infty}a_{n}< \infty\), \(\sum_{n=1}^{\infty}(\eta(T^{n})-1)< \infty\), \(d(x_{n},p)< R'\) and \(d(y_{n},p)< R'\), we have

$$\frac{R}{2}\beta_{n}(1-\beta_{n})d \bigl(T^{n}z_{n},x_{n}\bigr)^{2}< \infty. $$

Thus by the fact that \(\liminf_{n\to\infty}\beta_{n}(1-\beta_{n})>0\), we have

$$\begin{aligned} \lim_{n\to\infty}d\bigl(T^{n}z_{n},x_{n} \bigr) =& 0. \end{aligned}$$
(3.20)

Next, consider (3.13), we have

$$\begin{aligned} d(z_{n},p)^{2} \leq& \eta\bigl(T^{n} \bigr)^{2} d(x_{n},p)^{2}+P a_{n}- \frac{R}{2}\gamma _{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr) \\ \leq& \bigl[1+\bigl(\eta\bigl(T^{n}\bigr)-1\bigr)\nu \bigr]d(x_{n},p)^{2}+P a_{n} -\frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr) \end{aligned}$$
(3.21)

for some \(\nu>0\).

Equation (3.21) yields

$$\begin{aligned} \frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr)^{2} \leq& d(x_{n},p)^{2}-d(z_{n},p)^{2} +\bigl(\eta\bigl(T^{n}\bigr)-1\bigr)\nu d(x_{n},p)^{2}+P a_{n}. \end{aligned}$$

Since \(\sum_{n=1}^{\infty}a_{n}< \infty\), \(\sum_{n=1}^{\infty}(\eta(T^{n})-1)< \infty\), \(d(x_{n},p)< R'\) and \(d(z_{n},p)< R'\), we have

$$\frac{R}{2}\gamma_{n}(1-\gamma_{n})d \bigl(T^{n}x_{n},x_{n}\bigr)^{2}< \infty. $$

Hence by the fact that \(\liminf_{n\to\infty}\gamma_{n}(1-\gamma_{n})>0\), we have

$$\begin{aligned} \lim_{n\to\infty}d\bigl(T^{n}x_{n},x_{n} \bigr) =& 0. \end{aligned}$$
(3.22)

Now, we have

$$\begin{aligned} d\bigl(T^{n}y_{n},x_{n}\bigr) \leq& d\bigl(T^{n}y_{n},T^{n}x_{n}\bigr)+d \bigl(T^{n}x_{n},x_{n}\bigr) \\ & \to0 \quad\mbox{as } n\to\infty. \end{aligned}$$
(3.23)

Again, note that

$$\begin{aligned} d(x_{n},y_{n}) \leq& \beta_{n}d \bigl(x_{n},T^{n}z_{n}\bigr)\to0 \quad\mbox{as } n \to\infty. \end{aligned}$$
(3.24)

By the definitions of \(x_{n+1}\) and \(y_{n}\), we have

$$\begin{aligned} d(x_{n},x_{n+1}) \leq& d\bigl(x_{n},T^{n}y_{n} \bigr) \\ \leq& d\bigl(x_{n},T^{n}x_{n}\bigr)+d \bigl(T^{n}x_{n},T^{n}y_{n}\bigr) \\ \leq& d\bigl(x_{n},T^{n}x_{n}\bigr)+\eta \bigl(T^{n}\bigr) \bigl(d(x_{n},y_{n})+a_{n} \bigr) \\ \to&0 \quad\mbox{as } n\to\infty. \end{aligned}$$
(3.25)

By (3.22), (3.24) and the uniform continuity of T, we have

$$\begin{aligned} d(x_{n},Tx_{n}) \leq& d(x_{n},x_{n+1})+d \bigl(x_{n+1},T^{n+1}x_{n+1}\bigr) \\ &{} +d\bigl(T^{n+1}x_{n+1},T^{n+1}x_{n} \bigr)+d\bigl(T^{n+1}x_{n},Tx_{n}\bigr) \\ \leq& d(x_{n},x_{n+1})+d\bigl(x_{n+1},T^{n+1}x_{n+1} \bigr) \\ &{} + \eta\bigl(T^{n+1}\bigr) d(x_{n+1},x_{n})+a_{n+1}+d \bigl(T^{n+1}x_{n},Tx_{n}\bigr) \\ =& \bigl(1+\eta\bigl(T^{n+1}\bigr) \bigr)d(x_{n},x_{n+1})+d \bigl(x_{n+1},T^{n+1}x_{n+1}\bigr) \\ &{} +d\bigl(T^{n+1}x_{n},Tx_{n} \bigr)+a_{n+1}\to0 \quad\mbox{as } n\to\infty. \end{aligned}$$
(3.26)

This completes the proof. □

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

By Lemma 3.2, \(\lim_{n\to\infty}d(x_{n},Tx_{n})=0\). Since T is uniformly continuous, we have

$$d\bigl(x_{n},T^{m}x_{n}\bigr)\leq d(x_{n},Tx_{n})+d\bigl(Tx_{n},T^{2}x_{n} \bigr)+\cdots+d\bigl(T^{m-1}x_{n},T^{m}x_{n} \bigr)\to0, $$

as \(n\to\infty\). That is, \(\{x_{n}\}\) is an AFPS for \(T^{m}\). By the semi-compactness of \(T^{m}\), there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) and \(p\in K\) such that \(\lim_{j\to\infty}x_{n_{j}}=p\). Again, by the uniform continuity of T, we have

$$d(Tp,p)\leq d(Tp,Tx_{n_{j}})+d(Tx_{n_{j}},x_{n_{j}})+d(x_{n_{j}},p) \to 0 \quad\mbox{as } j\to\infty. $$

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])

Let \(X=K=[0,1]\), with the usual metric, and

$$T\colon K\to K,\quad T(x)= \left \{ \begin{array}{@{}l@{\quad}l} \frac{x}{2} &\mbox{if }x\neq0,\\ 1 &\mbox{if }x=0. \end{array} \right . $$

Then T is not continuous. However, T is semi-compact. In fact, if \(\{x_{n}\}\) is a bounded sequence in K such that \(|x_{n}-Tx_{n}|\to0\) as \(n\to\infty\), then by Balzano-Weierstrass theorem, it follows that \(\{x_{n}\}\) has a convergent subsequence.

The following example shows that there is a semi-compact mapping that is not compact.

Example 3.2

([47])

Let \(X=\ell_{2}\) and \(K=\{e_{1},e_{2},\ldots,e_{n},\ldots\}\) be the usual orthonormal basis for \(\ell_{2}\). Define

$$T\colon K\to K, \quad T(e_{i})=e_{i+1}, \quad i\in \mathbb{N}. $$

Then T is continuous (in fact, an isometry) but not compact. However, T is semi-compact. Indeed, if \(\{e_{i}\}_{i\in\mathbb{N}}\) is a bounded sequence in K such that \(e_{i}-Te_{i}\) converges, \(\{e_{i}\}_{i\in\mathbb{N}}\) must be finite.

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])

Let \(E=\mathbb{R}\), \(C=[0,1]\) and T be a mapping defined by

$$T\colon C\to C, \quad T(x)=\left \{ \begin{array}{@{}l@{\quad}l} \frac{1}{2} &\mbox{if }x\in[0,\frac{1}{2}],\\ 0 &\mbox{if }x\in(\frac{1}{2},1]. \end{array} \right . $$

Here \(F(T)=\{\frac{1}{2}\}\). Clearly, T is a discontinuous and non-Lipschitzian mapping. However, it is a nearly nonexpansive mapping and hence a nearly asymptotically nonexpansive mapping with sequence \(\{a_{n},\eta(T^{n})\}=\{\frac{1}{2^{n}},1\}\). Indeed, for a sequence \(\{a_{n}\}\) with \(a_{1}=\frac{1}{2}\) and \(a_{n}\to0\), we have

$$d(Tx,Ty)\leq d(x,y)+a_{1} \quad\mbox{for all } x, y\in C $$

and

$$d\bigl(T^{n}x,T^{n}y\bigr)\leq d(x,y)+a_{n} \quad \mbox{for all } x, y\in C \mbox{ and } n\geq2, $$

since

$$T^{n}x=\frac{1}{2} \quad\mbox{for all } x\in[0,1] \mbox{ and } n \geq 2. $$

Example 3.4

Let \(X=K=[0,1]\) with the usual metric d, \(\{x_{n}\}=\{\frac{1}{n}\}\), \(\{u_{n_{k}}\}=\{\frac{1}{kn}\}\), for all \(n, k\in\mathbb{N}\) are sequences in K. Then \(A(\{x_{n}\})=\{0\}\) and \(A(\{u_{n_{k}}\})=\{0\}\). This shows that \(\{x_{n}\}\) Δ-converges to 0, that is, \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n}=0\). The sequence \(\{x_{n}\}\) also converges strongly to 0, that is, \(|x_{n}-0|\to0\) as \(n\to\infty\). Also it is weakly convergent to 0, that is, \(x_{n}\rightharpoonup0\) as \(n\to\infty\), by Proposition 2.1. Thus, we conclude that

$$\mbox{strong convergence}\quad \Rightarrow \quad \Delta\mbox{-convergence} \quad \Rightarrow \quad \mbox{weak convergence}, $$

but the converse is not true in general.

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. 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. 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. 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. 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. 5.

    Our results also extend and generalize the corresponding results of [35, 38, 4952] 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.

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Saluja, G.S., Postolache, M. & Kurdi, A. Convergence of three-step iterations for nearly asymptotically nonexpansive mappings in \(\operatorname{CAT}(k)\) spaces. J Inequal Appl 2015, 156 (2015). https://doi.org/10.1186/s13660-015-0670-z

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