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., [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

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) 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

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:

• (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.

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

• (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.

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:

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

• (ii) X satisfies the (CN) inequality;

• (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:

• (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 $x\in \bigcap_{k=1}^{\infty}\overline{\operatorname{conv}}\{x_{k},x_{k+1},\dots\}$,

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

• (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

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. 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.