On total asymptotically nonexpansive mappings in $CAT(\kappa )$ spaces

Abstract

In this article, we obtain the demiclosed principle, fixed point theorems and convergence theorems for the class of total asymptotically nonexpansive mappings on $CAT(\kappa )$ spaces with $\kappa >0$. Our results generalize the results of Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Tang et al. (Abstr. Appl. Anal. 2012:965751, 2012), Karapınar et al. (J. Appl. Math. 2014:738150, 2014) and many others.

1 Introduction

For a real number κ, a $CAT(\kappa )$ space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.

Fixed point theory in $CAT(\kappa )$ spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on $CAT(0)$ spaces (see, e.g., [3–11]). Since any $CAT(\kappa )$ space is a $CAT({\kappa }^{\prime })$ space for ${\kappa }^{\prime }\ge \kappa$, all results for $CAT(0)$ spaces immediately apply to any $CAT(\kappa )$ space with $\kappa \le 0$. However, there are only a few articles that contain fixed point results in the setting of $CAT(\kappa )$ spaces with $\kappa >0$.

The concept of total asymptotically nonexpansive mappings was first introduced in Banach spaces by Alber et al. [12]. It generalizes the concept of asymptotically nonexpansive mappings introduced by Goebel and Kirk [13] as well as the concept of nearly asymptotically nonexpansive mappings introduced by Sahu [14]. In 2012, Chang et al. [15] studied the demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in the setting of $CAT(0)$ spaces. Since then the convergence of several iteration procedures for this type of mappings has been rapidly developed and many of articles have appeared (see, e.g., [16–24]). Among other things, under some suitable assumptions, Karapınar et al. [24] obtained the demiclosed principle, fixed point theorems, and convergence theorems for the following iteration.

Let K be a nonempty closed convex subset of a $CAT(0)$ space X and $T:K\to K$ be a total asymptotically nonexpansive mapping. Given $x_1\in K$, and let $\{x_n\}\subseteq K$ be defined by

$$x_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n((1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n)),n\in \mathbb{N},$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $[0,1]$.

In this article, we extend Karapınar et al.’s results to the general setting of $CAT(\kappa )$ space with $\kappa >0$.

2 Preliminaries

Let $(X,\rho )$ 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 $\rho (c(t),c(t^{\prime }))=|t-t^{\prime }|$ for all $t,t^{\prime }\in [0,l]$. In particular, c is an isometry and $\rho (x,y)=l$. The image $c([0,l])$ of c is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by $[x,y]$. This means that $z\in [x,y]$ if and only if there exists $\alpha \in [0,1]$ such that

$$\rho (x,z)=(1-\alpha )\rho (x,y)\text{and}\rho (y,z)=\alpha \rho (x,y).$$

In this case, we write $z=\alpha x\oplus (1-\alpha )y$. The space $(X,\rho )$ 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 $\rho (x,y)<D$). 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

$$diam(K):=sup\{\rho (x,y):x,y\in K\}<\mathrm{\infty }.$$

Now we introduce the model spaces $M_{\kappa }^n$, for more details on these spaces the reader is referred to [25]. Let $n\in \mathbb{N}$. We denote by ${\mathbb{E}}^n$ the metric space ${\mathbb{R}}^n$ endowed with the usual Euclidean distance. We denote by $(\cdot |\cdot )$ the Euclidean scalar product in ${\mathbb{R}}^n$, that is,

$$(x|y)=x_1{y}_1+\cdots +x_n{y}_n\text{where }x=(x_1,\dots ,x_n),y=(y_1,\dots ,y_n).$$

Let ${\mathbb{S}}^n$ denote the n-dimensional sphere defined by

$${\mathbb{S}}^n=\{x=(x_1,\dots ,x_{n+1})\in {\mathbb{R}}^{n+1}:(x|x)=1\},$$

with metric $d_{{\mathbb{S}}^n}(x,y)=arccos(x|y)$, $x,y\in {\mathbb{S}}^n$.

Let ${\mathbb{E}}^{n,1}$ denote the vector space ${\mathbb{R}}^{n+1}$ endowed with the symmetric bilinear form which associates to vectors $u=(u_1,\dots ,u_{n+1})$ and $v=(v_1,\dots ,v_{n+1})$ the real number $〈u|v〉$ defined by

$$〈u|v〉=-u_{n+1}{v}_{n+1}+\sum _{i=1}^n{u}_i{v}_i.$$

Let ${\mathbb{H}}^n$ denote the hyperbolic n-space defined by

$${\mathbb{H}}^n=\{u=(u_1,\dots ,u_{n+1})\in {\mathbb{E}}^{n,1}:〈u|u〉=-1,u_{n+1}>0\},$$

with metric $d_{{\mathbb{H}}^n}$ such that

$$cosh{d}_{{\mathbb{H}}^n}(x,y)=-〈x|y〉,x,y\in {\mathbb{H}}^n.$$

Definition 2.1 Given $\kappa \in \mathbb{R}$, we denote by $M_{\kappa }^n$ the following metric spaces:

• (i) if $\kappa =0$, then $M_0^n$ is the Euclidean space ${\mathbb{E}}^n$;

• (ii) if $\kappa >0$, then $M_{\kappa }^n$ is obtained from the spherical space ${\mathbb{S}}^n$ by multiplying the distance function by the constant $1/\sqrt{\kappa }$;

• (iii) if $\kappa <0$, then $M_{\kappa }^n$ is obtained from the hyperbolic space ${\mathbb{H}}^n$ by multiplying the distance function by the constant $1/\sqrt{-\kappa }$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in a geodesic space $(X,\rho )$ consists of three points x, y, z in X (the vertices of △) and three geodesic segments between each pair of vertices (the edges of △). A comparison triangle for a geodesic triangle $\mathrm{△}(x,y,z)$ in $(X,\rho )$ is a triangle $\overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$ in $M_{\kappa }^2$ such that

$$\rho (x,y)=d_{M_{\kappa }^2}(\overline{x},\overline{y}),\rho (y,z)=d_{M_{\kappa }^2}(\overline{y},\overline{z}),\text{and}\rho (z,x)=d_{M_{\kappa }^2}(\overline{z},\overline{x}).$$

If $\kappa \le 0$, then such a comparison triangle always exists in $M_{\kappa }^2$. If $\kappa >0$, then such a triangle exists whenever $\rho (x,y)+\rho (y,z)+\rho (z,x)<2D_{\kappa }$, where $D_{\kappa }=\pi /\sqrt{\kappa }$. A point $\overline{p}\in [\overline{x},\overline{y}]$ is called a comparison point for $p\in [x,y]$ if $\rho (x,p)=d_{M_{\kappa }^2}(\overline{x},\overline{p})$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in X is said to satisfy the $CAT(\kappa )$ inequality if for any $p,q\in \mathrm{△}(x,y,z)$ and for their comparison points $\overline{p},\overline{q}\in \overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$, one has

$$\rho (p,q)\le d_{M_{\kappa }^2}(\overline{p},\overline{q}).$$

Definition 2.2 If $\kappa \le 0$, then X is called a $CAT(\kappa )$ space if and only if X is a geodesic space such that all of its geodesic triangles satisfy the $CAT(\kappa )$ inequality.

If $\kappa >0$, then X is called a $CAT(\kappa )$ space if and only if X is $D_{\kappa }$-geodesic and any geodesic triangle $\mathrm{△}(x,y,z)$ in X with $\rho (x,y)+\rho (y,z)+\rho (z,x)<2D_{\kappa }$ satisfies the $CAT(\kappa )$ inequality.

Notice that in a $CAT(0)$ space $(X,\rho )$ if $x,y,z\in X$, then the $CAT(0)$ inequality implies

$$(\text{CN}){\rho }^2(x,\frac{1}{2}y\oplus \frac{1}{2}z)\le \frac{1}{2}{\rho }^2(x,y)+\frac{1}{2}{\rho }^2(x,z)-\frac{1}{4}{\rho }^2(y,z).$$

This is the (CN) inequality of Bruhat and Tits [26]. This inequality is extended by Dhompongsa and Panyanak [27] as

$$({\text{CN}}^{\ast }){\rho }^2(x,(1-\alpha )y\oplus \alpha z)\le (1-\alpha ){\rho }^2(x,y)+\alpha {\rho }^2(x,z)-\alpha (1-\alpha ){\rho }^2(y,z)$$

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 $CAT(0)$ space;

• (ii) X satisfies (CN);

• (iii) X satisfies (CN^∗).

Let $R\in (0,2]$. Recall that a geodesic space $(X,\rho )$ is said to be R-convex for R (see [28]) if for any three points $x,y,z\in X$, we have

$${\rho }^2(x,(1-\alpha )y\oplus \alpha z)\le (1-\alpha ){\rho }^2(x,y)+\alpha {\rho }^2(x,z)-\frac{R}{2}\alpha (1-\alpha ){\rho }^2(y,z).$$

(1)

It follows from (CN^∗) that a geodesic space $(X,\rho )$ is a $CAT(0)$ space if and only if $(X,\rho )$ is R-convex for $R=2$. The following lemma is a consequence of Proposition 3.1 in [28].

Lemma 2.3 Let $\kappa >0$ and $(X,\rho )$ be a $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Then $(X,\rho )$ is R-convex for $R=(\pi -2\epsilon )tan(\epsilon )$.

The following lemma is also needed.

Lemma 2.4 ([[25], p.176])

Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Then

$$\rho ((1-\alpha )x\oplus \alpha y,z)\le (1-\alpha )\rho (x,z)+\alpha \rho (y,z)$$

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

We now collect some elementary facts about $CAT(\kappa )$ spaces. Most of them are proved in the setting of $CAT(1)$ spaces. For completeness, we state the results in $CAT(\kappa )$ with $\kappa >0$.

Let $\{x_n\}$ be a bounded sequence in a $CAT(\kappa )$ space $(X,\rho )$. For $x\in X$, we set

$$r(x,\{x_n\})=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (x,x_n).$$

The asymptotic radius $r(\{x_n\})$ of $\{x_n\}$ is given by

$$r(\{x_n\})=inf\{r(x,\{x_n\}):x\in X\},$$

and the asymptotic center $A(\{x_n\})$ of $\{x_n\}$ is the set

$$A(\{x_n\})=\{x\in X:r(x,\{x_n\})=r(\{x_n\})\}.$$

It is known from Proposition 4.1 of [8] that in a $CAT(\kappa )$ space X with $diam(X)<\frac{\pi }{2\sqrt{\kappa }}$, $A(\{x_n\})$ consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.

Definition 2.5 ([6, 29])

A sequence $\{x_n\}$ in X is said to Δ-converge to $x\in X$ if x is the unique asymptotic center of $\{u_n\}$ for every subsequence $\{u_n\}$ of $\{x_n\}$. In this case we write $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$ and call x the Δ-limit of $\{x_n\}$.

Lemma 2.6 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. 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 $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$, then $x\in {\bigcap }_{k=1}^{\mathrm{\infty }}\overline{conv}\{x_k,x_{k+1},\dots \}$, where $\overline{conv}(A)=\bigcap \{B:B\supseteq A\mathit{\text{ and }}B\mathit{\text{ is closed and convex}}\}$.

By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[27], Lemma 2.8]).

Lemma 2.7 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. If $\{x_n\}$ is a 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 $\{\rho (x_n,u)\}$ converges, then $x=u$.

Definition 2.8 Let K be a nonempty subset of a $CAT(\kappa )$ space $(X,\rho )$. A mapping $T:K\to K$ is called total asymptotically nonexpansive if there exist nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\psi (0)=0$ such that

$$\rho (T^n(x),T^n(y))\le \rho (x,y)+{\nu }_n\psi (\rho (x,y))+{\mu }_n\text{for all }n\in \mathbb{N},x,y\in K.$$

A point $x\in K$ is called a fixed point of T if $x=T(x)$. We denote with $F(T)$ the set of fixed points of T. A sequence $\{x_n\}$ in K is called approximate fixed point sequence for T (AFPS in short) if

$$\underset{n\to \mathrm{\infty }}{lim}\rho (x_n,T(x_n))=0.$$

Algorithm 1 The sequence $\{x_n\}$ defined by $x_1\in K$ and

$$\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\hfill \\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\hfill \end{array}$$

is called an Ishikawa iterative sequence (see [30]).

If ${\beta }_n=0$ for all $n\in \mathbb{N}$, then Algorithm 1 reduces to the following.

Algorithm 2 The sequence $\{x_n\}$ defined by $x_1\in K$ and

$$x_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(x_n),n\in \mathbb{N},$$

is called a Mann iterative sequence (see [31]).

The following lemma is also needed.

Lemma 2.9 ([[32], Lemma 1])

Let $\{s_n\}$ and $\{t_n\}$ be sequences of nonnegative real numbers satisfying

$$s_{n+1}\le s_n+t_n\mathit{\text{for all }}n\in \mathbb{N}.$$

If ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{s}_n$ exists.

3 Main results

3.1 Existence theorems

Theorem 3.1 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let K be a nonempty closed convex subset of X, and let $T:K\to K$ be a continuous total asymptotically nonexpansive mapping. Then T has a fixed point in K.

Proof Fix $x\in K$. We can consider the sequence ${\{T^n(x)\}}_{n=1}^{\mathrm{\infty }}$ as a bounded sequence in K. Let $\varphi :K\to [0,\mathrm{\infty })$ be a function defined by

$$\varphi (u):=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (T^n(x),u)\text{for all }u\in K.$$

Then there exists $w\in K$ such that $\varphi (w)=inf\{\varphi (u):u\in K\}$. Since T is total asymptotically nonexpansive, for each $n,m\in \mathbb{N}$, we have

$$\rho (T^{n+m}(x),T^m(w))\le \rho (T^n(x),w)+{\nu }_m\psi \left(\rho (T^n(x),w)\right)+{\mu }_m.$$

(2)

Let $M=diam(K)$. Taking $n\to \mathrm{\infty }$ in (2), we get that

$$\varphi (T^m(w))\le \varphi (w)+{\nu }_m\psi (M)+{\mu }_m.$$

This implies that

$$\underset{m\to \mathrm{\infty }}{lim}\varphi (T^m(w))\le \varphi (w).$$

(3)

In view of (1), we have

$$\begin{array}{rl}\rho {(T^n(x),\frac{1}{2}{T}^m(w)\oplus \frac{1}{2}{T}^h(w))}^2\le & \frac{1}{2}\rho {(T^n(x),T^m(w))}^2+\frac{1}{2}\rho {(T^n(x),T^h(w))}^2\\ -\frac{R}{8}\rho {(T^m(w),T^h(w))}^2.\end{array}$$

Taking $n\to \mathrm{\infty }$, we get that

$$\begin{array}{rl}\varphi {(w)}^2\le & \varphi {(\frac{1}{2}{T}^m(w)\oplus \frac{1}{2}{T}^h(w))}^2\le \frac{1}{2}\varphi {(T^m(w))}^2+\frac{1}{2}\varphi {(T^h(w))}^2\\ -\frac{R}{8}\rho {(T^m(w),T^h(w))}^2,\end{array}$$

yielding

$$\frac{R}{8}\rho {(T^m(w),T^h(w))}^2\le \frac{1}{2}\varphi {(T^m(w))}^2+\frac{1}{2}\varphi {(T^h(w))}^2-\varphi {(w)}^2.$$

(4)

By (3) and (4), we have ${lim}_{m,h\to \mathrm{\infty }}\rho {(T^m(w),T^h(w))}^2\le 0$. Therefore, ${\{T^n(w)\}}_{n=1}^{\mathrm{\infty }}$ is a Cauchy sequence in K and hence converges to some point $v\in K$. Since T is continuous,

$$T(v)=T(\underset{n\to \mathrm{\infty }}{lim}{T}^n(w))=\underset{n\to \mathrm{\infty }}{lim}{T}^{n+1}(w)=v.$$

 □

From Theorem 3.1 we shall now derive a result for $CAT(0)$ spaces which can also be found in [24].

Corollary 3.2 Let $(X,\rho )$ be a complete $CAT(0)$ space and K be a nonempty bounded closed convex subset of X. If $T:K\to K$ is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.

Proof It is well known that every convex subset of a $CAT(0)$ space, equipped with the induced metric, is a $CAT(0)$ space (cf. [25]). Then $(K,\rho )$ is a $CAT(0)$ space and hence it is a $CAT(\kappa )$ space for all $\kappa >0$. Notice also that K is R-convex for $R=2$. Since K is bounded, we can choose $\epsilon \in (0,\pi /2)$ and $\kappa >0$ so that $diam(K)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$. The conclusion follows from Theorem 3.1. □

3.2 Demiclosed principle

Theorem 3.3 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let K be a nonempty closed convex subset of X, and let $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping. If $\{x_n\}$ is an AFPS for T such that $\mathrm{\Delta }\text{-}{lim}_n{x}_n=w$, then $w\in K$ and $w=T(w)$.

Proof By Lemma 2.6, $w\in K$. As in Theorem 3.1, we define $\varphi (u):={lim\hspace{0.17em}sup}_n\rho (x_n,u)$ for each $u\in K$. Since ${lim}_n\rho (x_n,T(x_n))=0$, by induction we can show that ${lim}_n\rho (x_n,T^m(x_n))=0$ for all $m\in \mathbb{N}$ (cf. [16]). This implies that

$$\varphi (u)=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (T^m(x_n),u)\text{for each }u\in K\text{ and }m\in \mathbb{N}.$$

(5)

In (5), taking $u=T^m(w)$, we have

$$\begin{array}{rl}\varphi (T^m(w))& =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\rho (T^m(x_n),T^m(w))\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\rho (x_n,w)+{\nu }_m\psi (\rho (x_n,w))+{\mu }_m).\end{array}$$

Hence

$$\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (T^m(w))\le \varphi (w).$$

(6)

In view of (1), we have

$$\rho {(x_n,\frac{1}{2}w\oplus \frac{1}{2}{T}^m(w))}^2\le \frac{1}{2}\rho {(x_n,w)}^2+\frac{1}{2}\rho {(x_n,T^m(w))}^2-\frac{R}{8}\rho {(w,T^m(w))}^2,$$

where $R=(\pi -2\epsilon )tan(\epsilon )$. Since $\mathrm{\Delta }\text{-}{lim}_n{x}_n=w$, letting $n\to \mathrm{\infty }$, we get that

$$\varphi {(w)}^2\le \varphi {(\frac{1}{2}w\oplus \frac{1}{2}{T}^m(w))}^2\le \frac{1}{2}\varphi {(w)}^2+\frac{1}{2}\varphi {(T^m(w))}^2-\frac{R}{8}\rho {(w,T^m(w))}^2,$$

yielding

$$\rho {(w,T^m(w))}^2\le \frac{4}{R}[\varphi {(T^m(w))}^2-\varphi {(w)}^2].$$

(7)

By (6) and (7), we have ${lim}_{m\to \mathrm{\infty }}\rho (w,T^m(w))=0$. Since T is continuous,

$$T(w)=T(\underset{m\to \mathrm{\infty }}{lim}{T}^m(w))=\underset{m\to \mathrm{\infty }}{lim}{T}^{m+1}(w)=w.$$

 □

As we have observed in Corollary 3.2, we can derive the following result from Theorem 3.3.

Corollary 3.4 ([[24], Theorem 12])

Let $(X,\rho )$ be a complete $CAT(0)$ space, K be a nonempty bounded closed convex subset of X, and $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping. If $\{x_n\}$ is an AFPS for T such that $\mathrm{\Delta }\text{-}{lim}_n{x}_n=w$, then $w\in K$ and $w=T(w)$.

3.3 Convergence theorems

We begin this section by proving a crucial lemma.

Lemma 3.5 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let K be a nonempty closed convex subset of X, and $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$. Let $x_1\in K$ and $\{x_n\}$ be a sequence in K defined by

$$\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\hfill \\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\hfill \end{array}$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ such that ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$. Then $\{x_n\}$ is an AFPS for T and ${lim}_n\rho (x_n,p)$ exists for all $p\in F(T)$.

Proof It follows from Theorem 3.1 that $F(T)\ne \mathrm{\varnothing }$. Let $p\in F(T)$ and $M=diam(K)$. Since T is total asymptotically nonexpansive, by Lemma 2.4 we have

$$\begin{array}{rl}\rho (y_n,p)& =\rho ((1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),p)\\ \le (1-{\beta }_n)\rho (x_n,p)+{\beta }_n\rho (T^n(x_n),T^n(p))\\ \le \rho (x_n,p)+{\beta }_n{\nu }_n\psi (M)+{\beta }_n{\mu }_n.\end{array}$$

This implies that

$$\begin{array}{rl}\rho (x_{n+1},p)& =\rho ((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),p)\\ \le (1-{\alpha }_n)\rho (x_n,p)+{\alpha }_n\rho (T^n(y_n),T^n(p))\\ \le (1-{\alpha }_n)\rho (x_n,p)+{\alpha }_n[\rho (y_n,p)+{\nu }_n\psi (M)+{\mu }_n]\\ \le \rho (x_n,p)+{\alpha }_n(1+{\beta }_n)({\nu }_n\psi (M)+{\mu }_n).\end{array}$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$, by Lemma 2.9 ${lim}_n\rho (x_n,p)$ exists. Next, we show that $\{x_n\}$ is an AFPS for T. In view of (1), we have

$$\begin{array}{rl}\rho {(x_{n+1},p)}^2=& \rho {((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),p)}^2\\ \le & (1-{\alpha }_n)\rho {(x_n,p)}^2+{\alpha }_n\rho {(T^n(y_n),p)}^2\\ \le & (1-{\alpha }_n)\rho {(x_n,p)}^2+{\alpha }_n{[\rho (y_n,p)+{\nu }_n\psi (M)+{\mu }_n]}^2\\ \le & (1-{\alpha }_n)\rho {(x_n,p)}^2+{\alpha }_n\rho {(y_n,p)}^2\\ +{\alpha }_n[2\rho (y_n,p)({\nu }_n\psi (M)+{\mu }_n)+{({\nu }_n\psi (M)+{\mu }_n)}^2].\end{array}$$

This implies that

$$\rho {(x_{n+1},p)}^2\le (1-{\alpha }_n)\rho {(x_n,p)}^2+{\alpha }_n\rho {(y_n,p)}^2+A{\nu }_n+B{\mu }_n\text{for some }A,B\ge 0.$$

(8)

Again by (1), we have

$$\begin{array}{rl}\rho {(y_n,p)}^2=& \rho {((1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),p)}^2\\ \le & (1-{\beta }_n)\rho {(x_n,p)}^2+{\beta }_n\rho {(T^n(x_n),T^n(p))}^2-\frac{R}{2}{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2\\ \le & (1-{\beta }_n)\rho {(x_n,p)}^2+{\beta }_n{[\rho (x_n,p)+{\nu }_n\psi (M)+{\mu }_n]}^2\\ -\frac{R}{2}{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2\\ \le & \rho {(x_n,p)}^2+{\beta }_n[2\rho (x_n,p)({\nu }_n\psi (M)+{\mu }_n)+{({\nu }_n\psi (M)+{\mu }_n)}^2]\\ -\frac{R}{2}{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2.\end{array}$$

Substituting this into (8), we get that

$$\begin{array}{rl}\rho {(x_{n+1},p)}^2\le & \rho {(x_n,p)}^2+{\alpha }_n{\beta }_n[2\rho (x_n,p)({\nu }_n\psi (M)+{\mu }_n)+{({\nu }_n\psi (M)+{\mu }_n)}^2]\\ -\frac{R}{2}{\alpha }_n{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2+A{\nu }_n+B{\mu }_n,\end{array}$$

yielding

$$\frac{R}{2}{\alpha }_n{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2\le \rho {(x_n,p)}^2-\rho {(x_{n+1},p)}^2+C{\nu }_n+D{\mu }_n\text{for some }C,D\ge 0.$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$, we have

$$\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{\beta }_n(1-{\beta }_n)\rho {(x_n,T^n(x_n))}^2<\mathrm{\infty }.$$

This implies by ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$ that

$$\underset{n\to \mathrm{\infty }}{lim}\rho (x_n,T^n(x_n))=0.$$

(9)

By the uniform continuity of T, we have

$$\underset{n\to \mathrm{\infty }}{lim}\rho (T(x_n),T^{n+1}(x_n))=0.$$

(10)

It follows from (9) and the definitions of $x_{n+1}$ and $y_n$ that

$$\begin{array}{rl}\rho (x_n,x_{n+1})& \le \rho (x_n,T^n(y_n))\\ \le \rho (x_n,T^n(x_n))+\rho (T^n(x_n),T^n(y_n))\\ \le \rho (x_n,T^n(x_n))+\rho (x_n,y_n)+{\nu }_n\psi (M)+{\mu }_n\\ \le (1+{\beta }_n)\rho (x_n,T^n(x_n))+{\nu }_n\psi (M)+{\mu }_n⟶0\text{as }n\to \mathrm{\infty }.\end{array}$$

(11)

By (9), (10), and (11), we have

$$\begin{array}{rl}\rho (x_n,T(x_n))\le & \rho (x_n,x_{n+1})+\rho (x_{n+1},T^{n+1}(x_{n+1}))\\ +\rho (T^{n+1}(x_{n+1}),T^{n+1}(x_n))+\rho (T^{n+1}(x_n),T(x_n))\\ \le & \rho (x_n,x_{n+1})+\rho (x_{n+1},T^{n+1}(x_{n+1}))+\rho (x_{n+1},x_n)\\ +{\nu }_{n+1}\psi (M)+{\mu }_{n+1}+\rho (T^{n+1}(x_n),T(x_n))⟶0\text{as }n\to \mathrm{\infty }.\end{array}$$

 □

Now, we are ready to prove our Δ-convergence theorem.

Theorem 3.6 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let K be a nonempty closed convex subset of X, and let $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$. Let $x_1\in K$ and $\{x_n\}$ be a sequence in K defined by

$$\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\hfill \\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\hfill \end{array}$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ such that ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$. 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 ${\omega }_w(x_n)$ is contained in $F(T)$ 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 2.6, there exists a subsequence $\{v_n\}$ of $\{u_n\}$ such that $\mathrm{\Delta }\text{-}{lim}_n{v}_n=v\in K$. Hence $v\in F(T)$ by Lemma 3.5 and Theorem 3.3. Since ${lim}_n\rho (x_n,v)$ exists, $u=v$ by Lemma 2.7. This shows that ${\omega }_w(x_n)\subseteq F(T)$. Next, we show that ${\omega }_w(x_n)$ consists of exactly one point. Let $\{u_n\}$ be a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$, and let $A(\{x_n\})=\{x\}$. Since $u\in {\omega }_w(x_n)\subseteq F(T)$, by Lemma 3.5 ${lim}_n\rho (x_n,u)$ exists. Again, by Lemma 2.7, $x=u$. This completes the proof. □

As a consequence of Theorem 3.6, we obtain the following.

Corollary 3.7 ([[24], Theorem 17])

Let $(X,\rho )$ be a complete $CAT(0)$ space, K be a nonempty bounded closed convex subset of X, and $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$. Let $x_1\in K$ and $\{x_n\}$ be a sequence in K defined by

$$\begin{array}{r}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\end{array}$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ such that ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$. Then $\{x_n\}$ Δ-converges to a fixed point of T.

Recall that a mapping $T:K\to K$ is said to be semi-compact if K is closed and each bounded AFPS for T in K has a convergent subsequence. Now, we prove a strong convergence theorem for uniformly continuous total asymptotically nonexpansive semi-compact mappings.

Theorem 3.8 Let $\kappa >0$ and $(X,\rho )$ be a complete $CAT(\kappa )$ space with $diam(X)\le \frac{\pi /2-\epsilon }{\sqrt{\kappa }}$ for some $\epsilon \in (0,\pi /2)$. Let K be a nonempty closed convex subset of X, and let $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$. Let $x_1\in K$ and $\{x_n\}$ be a sequence in K defined by

$$\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\hfill \\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\hfill \end{array}$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ such that ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$. Suppose that $T^m$ is semi-compact for some $m\in \mathbb{N}$. Then $\{x_n\}$ converges strongly to a fixed point of T.

Proof By Lemma 3.5, ${lim}_n\rho (x_n,T(x_n))=0$. Since T is uniformly continuous, we have

$$\rho (x_n,T^m(x_n))\le \rho (x_n,T(x_n))+\rho (T(x_n),T^2(x_n))+\cdots +\rho (T^{m-1}(x_n),T^m(x_n))⟶0$$

as $n\to \mathrm{\infty }$. That is, $\{x_n\}$ is an AFPS for $T^m$. By the semi-compactness of $T^m$, there exist a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ and $p\in K$ such that ${lim}_{j\to \mathrm{\infty }}{x}_{n_j}=p$. Again, by the uniform continuity of T, we have

$$\rho (T(p),p)\le \rho (T(p),T(x_{n_j}))+\rho (T(x_{n_j}),x_{n_j})+\rho (x_{n_j},p)⟶0\text{as }j\to \mathrm{\infty }.$$

That is, $p\in F(T)$. By Lemma 3.5, ${lim}_n\rho (x_n,p)$ exists, thus p is the strong limit of the sequence $\{x_n\}$ itself. □

Corollary 3.9 ([[24], Theorem 22])

Let $(X,\rho )$ be a complete $CAT(0)$ space, K be a nonempty bounded closed convex subset of X, and $T:K\to K$ be a uniformly continuous total asymptotically nonexpansive mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\nu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$. Let $x_1\in K$ and $\{x_n\}$ be a sequence in K defined by

$$\begin{array}{r}{x}_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n(y_n),\\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n(x_n),n\in \mathbb{N},\end{array}$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ such that ${lim\hspace{0.17em}inf}_n{\alpha }_n{\beta }_n(1-{\beta }_n)>0$. Suppose that $T^m$ is semi-compact for some $m\in \mathbb{N}$. Then $\{x_n\}$ converges strongly to a fixed point of T.

Remark 3.10 The results in this article also hold for the class of weakly total asymptotically nonexpansive mappings in the following sense. A mapping $T:K\to K$ is called weakly total asymptotically nonexpansive if there exist nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a nondecreasing function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

$$\rho (T^n(x),T^n(y))\le \rho (x,y)+{\nu }_n\psi (\rho (x,y))+{\mu }_n\text{for all }n\in \mathbb{N},x,y\in K.$$

Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.