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# Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces

- Liang-cai Zhao
^{1}, - Shih-sen Chang
^{2}Email author and - Jong Kyu Kim
^{3}

**2013**:353

https://doi.org/10.1186/1687-1812-2013-353

© Zhao et al.; licensee Springer. 2013

**Received:**24 July 2013**Accepted:**3 December 2013**Published:**31 December 2013

## Abstract

The purpose of this paper is to introduce the concept of *total asymptotically nonexpansive mappings* and to prove some Δ-convergence theorems of the mixed type iteration process to approximating a common fixed point for two *asymptotically nonexpansive mappings* and two *total asymptotically nonexpansive mappings* in *hyperbolic spaces*. The results presented in the paper extend and improve some recent results announced in the current literature.

**MSC:**47H09, 47H10.

## Keywords

- asymptotically nonexpansive mapping
- total asymptotically nonexpansive mapping
- common fixed point
- hyperbolic space

## 1 Introduction and preliminaries

Most of the problems in various disciplines of science are nonlinear in nature, whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the linear structure of the underlying spaces. A nonlinear framework for fixed point theory is a metric space embedded with a ‘convex structure’. The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting with rich geometrical structure for metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory.

Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1], defined below, which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in [3].

- (i)
$d(u,W(x,y,\alpha ))\le \alpha d(u,x)+(1-\alpha )d(u,y)$;

- (ii)
$d(W(x,y,\alpha ),W(x,y,\beta ))=|\alpha -\beta |d(x,y)$;

- (iii)
$W(x,y,\alpha )=W(y,x,(1-\alpha ))$;

- (iv)
$d(W(x,z,\alpha ),W(y,w,\alpha ))\le (1-\alpha )d(x,y)+\alpha d(z,w)$

for all $x,y,z,w\in X$ and $\alpha ,\beta \in [0,1]$. A nonempty subset *K* of a hyperbolic space *X* is convex if $W(x,y,\alpha )\in K$ for all $x,y\in K$ and $\alpha \in [0,1]$. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [4], Hadamard manifolds as well as $CAT(0)$ spaces in the sense of Gromov (see [5]).

provided $d(x,u)\le r$, $d(y,u)\le r$ and $d(x,y)\ge \u03f5r$.

A map $\eta :(0,\mathrm{\infty})\times (0,2]\to (0,1]$, which provides such $\delta =\eta (r,\u03f5)$ for given $r>0$ and $\u03f5\in (0,2]$, is known as a modulus of uniform convexity of *X*. We call *η* monotone if it decreases with *r* (for fixed *ϵ*), *i.e.*, $\mathrm{\forall}\u03f5>0$, $\mathrm{\forall}{r}_{2}\ge {r}_{1}>0$ ($\eta ({r}_{2},\u03f5)\le \eta ({r}_{1},\u03f5)$).

In the sequel, let $(X,d)$ be a metric space, and let *K* be a nonempty subset of *X*. We shall denote the fixed point set of a mapping *T* by $F(T)=\{x\in K:Tx=x\}$.

*nonexpansive*if

*asymptotically nonexpansive*if there exists a sequence $\{{k}_{n}\}\subset [0,\mathrm{\infty})$ with ${k}_{n}\to 0$ such that

*uniformly L-Lipschitzian*if there exists a constant $L>0$ such that

**Definition 1.1**A mapping $T:K\to K$ is said to be $(\{{\mu}_{n}\},\{{\xi}_{n}\},\rho )$

*-total asymptotically nonexpansive*if there exist nonnegative sequences $\{{\mu}_{n}\}$, $\{{\xi}_{n}\}$ with ${\mu}_{n}\to 0$, ${\xi}_{n}\to 0$ and a strictly increasing continuous function $\rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\rho (0)=0$ such that

**Remark 1.1** From the definitions, it is to know that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence $\{{k}_{n}=0\}$, and each asymptotically nonexpansive mapping is a $(\{{\mu}_{n}\},\{{\xi}_{n}\},\rho )$-total asymptotically nonexpansive mapping with ${\xi}_{n}=0$, and $\rho (t)=t$, $t\ge 0$.

The existence of fixed points of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is difficult to derive conditions for the existence of fixed points for certain types of nonlinear mappings. It is worth to mention that fixed point theory for nonexpansive mappings, a limit case of a contraction mapping when the Lipschitz constant is allowed to be 1, requires tools far beyond metric fixed point theory. Iteration schemas are the only main tool for analysis of generalized nonexpansive mappings. Fixed point theory has a computational flavor as one can define effective iteration schemas for the computation of fixed points of various nonlinear mappings. The problem of finding a common fixed point of some nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics.

The purpose of this paper is to introduce the concepts of total asymptotically nonexpansive mappings and to prove some Δ-convergence theorems of the mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive mappings and two total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6–19].

In order to define the concept of Δ-convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.

*X*. For $x\in X$, we define a continuous functional $r(\cdot ,\{{x}_{n}\}):X\to [0,\mathrm{\infty})$ by

*asymptotic radius*$r(\{{x}_{n}\})$ of $\{{x}_{n}\}$ is given by

*asymptotic center*${A}_{k}(\{{x}_{n}\})$ of a bounded sequence $\{{x}_{n}\}$ with respect to $K\subset X$ is the set

This is the set of minimizers of the functional $r(\cdot ,\{{x}_{n}\})$. If the asymptotic center is taken with respect to *X*, then it is simply denoted by $A(\{{x}_{n}\})$. It is known that uniformly convex Banach spaces and $CAT(0)$ spaces enjoy the property that ‘bounded sequences have unique asymptotic centers with respect to closed convex subsets’. The following lemma is due to Leustean [20] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

**Lemma 1.1** [20]

*Let* $(X,d,W)$ *be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity*. *Then every bounded sequence* $\{{x}_{n}\}$ *in* *X* *has a unique asymptotic center with respect to any nonempty closed convex subset* *K* *of* *X*.

Recall that 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 $\mathrm{\Delta}\text{-}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ and call *x* the $\mathrm{\Delta}\text{-}\text{limit}$ of $\{{x}_{n}\}$.

A mapping $T:K\to K$ is semi-compact if every bounded sequence $\{{x}_{n}\}\subset K$ satisfying $d({x}_{n},T{x}_{n})\to 0$ has a convergent subsequence.

**Lemma 1.2** [9]

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$

*and*$\{{\delta}_{n}\}$

*be sequences of nonnegative real numbers satisfying*

*If* ${\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$, *then the limit* ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ *exists*. *If there exists a subsequence* $\{{a}_{{n}_{i}}\}\subset \{{a}_{n}\}$ *such that* ${a}_{{n}_{i}}\to 0$, *then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.

**Lemma 1.3** [12]

*Let*$(X,d,W)$

*be a uniformly convex hyperbolic space with monotone modulus of uniform convexity*

*η*.

*Let*$x\in X$

*and*$\{{\alpha}_{n}\}$

*be a sequence in*$[a,b]$

*for some*$a,b\in (0,1)$.

*If*$\{{x}_{n}\}$

*and*$\{{y}_{n}\}$

*are sequences in*

*X*

*such that*

*for some* $c\ge 0$, *then* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{y}_{n})=0$.

**Lemma 1.4** [12]

*Let* *K* *be a nonempty closed convex subset of uniformly convex hyperbolic space and* $\{{x}_{n}\}$ *be a bounded sequence in* *K* *such that* $A(\{{x}_{n}\})=\{y\}$ *and* $r(\{{x}_{n}\})=\zeta $. *If* $\{{y}_{m}\}$ *is another sequence in* *K* *such that* ${lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\zeta $, *then* ${lim}_{m\to \mathrm{\infty}}{y}_{m}=y$.

## 2 Main results

**Theorem 2.1**

*Let*

*K*

*be a nonempty closed convex subset of a complete uniformly convex hyperbolic space*

*X*

*with monotone modulus of uniform convexity*

*η*.

*Let*${T}_{i}:K\to K$, $i=1,2$,

*be a uniformly*${L}_{i}$-

*Lipschitzian and*$(\{{\mu}_{n}^{i}\},\{{\xi}_{n}^{i}\},{\rho}^{i})$-

*total asymptotically nonexpansive mapping with sequence*$\{{\mu}_{n}^{i}\}$

*and*$\{{\xi}_{n}^{i}\}$

*satisfying*${lim}_{n\to \mathrm{\infty}}{\mu}_{n}^{i}=0$, ${lim}_{n\to \mathrm{\infty}}{\xi}_{n}^{i}=0$

*and a strictly increasing function*${\rho}^{i}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*with*${\rho}^{i}(0)=0$, $i=1,2$,

*let*${S}_{i}:K\to K$, $i=1,2$,

*be a uniformly*$\tilde{{L}_{i}}$-

*Lipschitzian and asymptotically nonexpansive mapping with sequence*$\{{k}_{n}^{i}\}$

*satisfying*${lim}_{n\to \mathrm{\infty}}{k}_{n}^{i}=0$.

*Assume that*$\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})\ne \mathrm{\varnothing}$,

*and for arbitrarily chosen*${x}_{1}\in K$, $\{{x}_{n}\}$

*is defined as follows*:

*where*$\{{\mu}_{n}^{i}\}$, $\{{\xi}_{n}^{i}\}$, ${\rho}^{i}$, ${k}_{n}^{i}$, $i=1,2$, $\{{\alpha}_{n}\}$

*and*$\{{\beta}_{n}\}$

*satisfy the following conditions*:

- (1)
${\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}^{i}<\mathrm{\infty}$, ${\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}^{i}<\mathrm{\infty}$, ${\sum}_{n=1}^{\mathrm{\infty}}{k}_{n}^{i}<\mathrm{\infty}$, $i=1,2$;

- (2)
*There exist constants*$a,b\in (0,1)$*with*$0<b(1-a)\le \frac{1}{2}$*such that*$\{{\alpha}_{n}\}\subset [a,b]$*and*$\{{\beta}_{n}\}\subset [a,b]$; - (3)
*There exists a constant*${M}^{\ast}>0$*such that*${\rho}^{i}(r)\le {M}^{\ast}r$, $r>0$, $i=1,2$; - (4)
$d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y)$

*for all*$x,y\in K$*and*$i=1,2$.

*Then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) Δ-*converges to a common fixed point of* $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$.

*Proof* Set $L=max\{{L}_{i},\tilde{{L}_{i}},i=1,2\}$, ${\mu}_{n}=max\{{\mu}_{n}^{i},{k}_{n}^{i},i=1,2\}$ and ${\xi}_{n}=max\{{\xi}_{n}^{i},i=1,2\}$, $\rho =max\{{\rho}^{i},i=1,2\}$. By condition (1), we know that ${\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}$, ${\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty}$. The proof of Theorem 2.1 is divided into four steps.

Step 1. First we prove that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists for each $p\in \mathcal{F}$.

where ${\delta}_{n}={\mu}_{n}(1+{\alpha}_{n}{M}^{\ast}(1+{\beta}_{n}+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}))$, ${b}_{n}=(1+{\beta}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}){\alpha}_{n}{\xi}_{n}$. Since ${\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}$, ${\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty}$ and condition (2), it follows from Lemma 1.2 that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists for $p\in \mathcal{F}$.

Step 3. Now we prove that the sequence $\{{x}_{n}\}$ Δ-converges to a common fixed point of $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$.

In fact, for each $p\in F$, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists. This implies that the sequence $\{d({x}_{n},p)\}$ is bounded, so is the sequence $\{{x}_{n}\}$. Hence, by virtue of Lemma 1.1, $\{{x}_{n}\}$ has a unique asymptotic center ${A}_{K}(\{{x}_{n}\})=\{x\}$.

*K*by ${z}_{j}={T}_{i}^{j}u$. So we calculate

*L*-Lipschitzian, from (2.22) we have

It follows from Lemma 1.4 that ${lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u=u$. As ${T}_{i}$ is uniformly continuous, so that ${T}_{i}u={T}_{i}({lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u)={lim}_{j\to \mathrm{\infty}}{T}_{i}^{j+1}u=u$. That is, $u\in F({T}_{i})$. Similarly, we also can show that $u\in F({S}_{i})$. Hence, *u* is the common fixed point of ${T}_{i}$ and ${S}_{i}$. Reasoning as above by utilizing the uniqueness of asymptotic centers, we get that $x=u$. Since $\{{u}_{n}\}$ is an arbitrary subsequence of $\{{x}_{n}\}$, therefore $A(\{{u}_{n}\})=\{u\}$ for all subsequence $\{{u}_{n}\}$ of $\{{x}_{n}\}$. This proves that $\{{x}_{n}\}$ Δ-converges to a common fixed point of $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$. This completes the proof. □

The following theorem can be obtained from Theorem 2.1 immediately.

**Theorem 2.2**

*Let*

*K*

*be a nonempty closed convex subset of a complete uniformly convex hyperbolic space*

*X*

*with monotone modulus of uniform convexity*

*η*.

*Let*${T}_{i}:K\to K$, $i=1,2$,

*be a uniformly*${L}_{i}$-

*Lipschitzian and asymptotically nonexpansive mapping with sequence*$\{{t}_{n}^{i}\}\subset [1,\mathrm{\infty})$

*satisfying*${lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1$,

*and*${S}_{i}:K\to K$, $i=1,2$,

*be a nonexpansive mapping*.

*Assume that*$\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$,

*for arbitrarily chosen*${x}_{1}\in K$, $\{{x}_{n}\}$

*is defined as follows*:

*where*$\{{t}_{n}^{i}\}$, $i=1,2$, $\{{\alpha}_{n}\}$

*and*${\beta}_{n}$

*satisfy the following conditions*:

- (1)
${\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}-1)<\mathrm{\infty}$, $i=1,2$;

- (2)
*There exist constants*$a,b\in (0,1)$*with*$0<b(1-a)\le \frac{1}{2}$*such that*$\{{\alpha}_{n}\}\subset [a,b]$*and*$\{{\beta}_{n}\}\subset [a,b]$; - (3)
$d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y)$

*for all*$x,y\in K$*and*$i=1,2$.

*Then the sequence* $\{{x}_{n}\}$ *defined in* (2.23) Δ-*converges to a common fixed point of* $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$.

*Proof* Take ${\rho}^{i}(t)=t$, $t\ge 0$, ${\xi}_{n}^{i}=0$, ${\mu}_{n}^{i}={t}_{n}^{i}-1$, ${k}_{n}^{i}=0$, $i=1,2$, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence $\{{x}_{n}\}$ Δ-converges to a common fixed point of $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})$.

This completes the proof of Theorem 2.2. □

**Theorem 2.3**

*Let*

*K*

*be a nonempty closed convex subset of a complete uniformly convex hyperbolic space*

*X*

*with monotone modulus of uniform convexity*

*η*.

*Let*${T}_{i}:K\to K$, $i=1,2$,

*be a uniformly*${L}_{i}$-

*Lipschitzian and asymptotically nonexpansive mapping with sequence*$\{{t}_{n}^{i}\}\subset [1,\mathrm{\infty})$

*satisfying*${lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1$.

*Assume that*$\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})$,

*for arbitrarily chosen*${x}_{1}\in K$, $\{{x}_{n}\}$

*is defined as follows*:

*where*$\{{t}_{n}^{i}\}$, $i=1,2$, $\{{\alpha}_{n}\}$

*and*${\beta}_{n}$

*satisfy the following conditions*:

- (1)
${\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}-1)<\mathrm{\infty}$, $i=1,2$;

- (2)
*There exist constants*$a,b\in (0,1)$*with*$0<b(1-a)\le \frac{1}{2}$*such that*$\{{\alpha}_{n}\}\subset [a,b]$*and*$\{{\beta}_{n}\}\subset [a,b]$.

*Then the sequence* $\{{x}_{n}\}$ *defined in* (2.24) Δ-*converges to a common fixed point of* $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})$.

*Proof* Take ${\rho}^{i}(t)=t$, $t\ge 0$, ${\xi}_{n}^{i}=0$, ${\mu}_{n}^{i}={t}_{n}^{i}-1$, ${S}_{i}=I$, $i=1,2$, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence $\{{x}_{n}\}$ Δ-converges to a common fixed point of $\mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})$.

This completes the proof of Theorem 2.3. □

## Declarations

### Acknowledgements

The authors would like to express their thanks to the editors and referees for their useful comments and suggestions. This work is supported by Scientific Research Fund of SiChuan Provincial Education Department (No. 11ZA222) and the Natural Science Foundation of Yibin University (No. 2012S07) and the National Natural Science foundation of China (Grant No. 11361070).

## Authors’ Affiliations

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