# On the convergence of fixed points for Lipschitz type mappings in hyperbolic spaces

- Shin Min Kang
^{1}Email author, - Samir Dashputre
^{2}, - Bhuwan Lal Malagar
^{2}and - Arif Rafiq
^{3}

**2014**:229

https://doi.org/10.1186/1687-1812-2014-229

© Kang et al.; licensee Springer. 2014

**Received: **17 April 2014

**Accepted: **31 October 2014

**Published: **12 November 2014

## Abstract

In this paper, we prove strong and Δ-convergence theorems for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings on hyperbolic space through the *S*-iteration process introduced by Agarwal *et al.* (J. Nonlinear Convex Anal. 8:61-79, 2007) which is faster and independent of the Mann (Proc. Am. Math. Soc. 4:506-510, 1953) and Ishikawa (Proc. Am. Math. Soc. 44:147-150, 1974) iteration processes. Our results generalize, extend, and unify the corresponding results of Abbas *et al.* (Math. Comput. Model. 55:1418-1427, 2012), Agarwal *et al.* (J. Nonlinear Convex Anal. 8:61-79, 2007), Dhompongsa and Panyanak (Comput. Math. Appl. 56:2572-2579, 2008), and Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011).

**MSC:**47H10.

## Keywords

*S*-iteration processuniformly convex hyperbolic spacenearly asymptotically nonexpansive mapping

## 1 Introduction

The class of asymptotically nonexpansive mappings, introduced by Goebel and Kirk [1] in 1972, is an important generalization of the class of nonexpansive mapping 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 numerous papers dealing with the approximation of fixed points of nonexpansive and asymptotically nonexpansive mappings in uniformly convex Banach spaces through modified Mann and Ishikawa iteration processes (see, *e.g.*, [2–9] and references therein). The class of Lipschitz mappings is larger than the classes of nonexpansive and asymptotically nonexpansive mappings. However, the theory of the computation of fixed points of non-Lipschitz mappings is equally important and interesting. There are few a results in this direction (see, *e.g.*, [10–13]).

In 1976, Lim [14] introduced a concept of convergence in a general metric space setting which he called ‘Δ-convergence’. In 2008, Kirk and Panyanak [15] specialized Lim’s concept to $CAT(0)$ spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. 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, asymptotically nonexpansive mapping in intermediate sense, asymptotically nonexpansive nonself-mapping via Picard, Mann [16], Ishikawa [17], Agarwal *et al.* [18] in the framework of $CAT(0)$ space have been rapidly developed and many papers have appeared in this direction (see, *e.g.*, [19–23]).

The purpose of the paper is to establish Δ-convergence as well as strong convergence through the *S*-iteration process for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings on a nonlinear domain, uniformly convex hyperbolic space which includes both uniformly convex Banach spaces and $CAT(0)$ spaces. Therefore, our results extend and improve the corresponding ones proved by Abbas *et al.* [19], Dhompongsa and Panyanak [22], Khan and Abbas [23] and many other results in this direction.

## 2 Preliminaries

Let $F(T)=\{Tx=x:x\in C\}$ denotes the set of fixed point. We begin with the following definitions.

**Definition 2.1**Let

*C*be a nonempty subset of metric space

*X*and $T:C\to C$ a mapping. A sequence $\{{x}_{n}\}$ in

*C*is said to be an

*approximating fixed point sequence*of

*T*if

**Definition 2.2**Let

*C*be a nonempty subset of a metric space

*X*. The mapping $T:C\to C$ is said to be

- (1)
*uniformly**L*-*Lipschitzian*if for each $n\in \mathbb{N}$, there exists a positive number $L>0$ such that$d({T}^{n}x,{T}^{n}y)\le Ld(x,y)\phantom{\rule{1em}{0ex}}\text{for all}x,y\in C;$ - (2)
*asymptotically nonexpansive*if there exists a sequence $\{{k}_{n}\}$ in $[0,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=0$ such that$d({T}^{n}x,{T}^{n}y)\le (1+{k}_{n})d(x,y)\phantom{\rule{1em}{0ex}}\text{for all}x,y\in C\text{and}n\in \mathbb{N};$ - (3)
*asymptotically quasi*-*nonexpansive*if $F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{k}_{n}\}$ in $[0,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=0$ such that$d({T}^{n}{x}_{n},p)\le (1+{k}_{n})d({x}_{n},p)\phantom{\rule{1em}{0ex}}\text{for all}x\in C,p\in F(T)\text{and}n\in \mathbb{N}.$

The class of nearly Lipschitzian mappings is an important generalization of the class of Lipschitzian mappings and was introduced by Sahu [11].

*C*be a nonempty subset of a metric space

*X*and fix a sequence $\{{a}_{n}\}$ in $[0,\mathrm{\infty})$ with ${a}_{n}\to 0$. A mapping $T:C\to C$ is said to be

*nearly Lipschitzian*with respect to $\{{a}_{n}\}$ if for each $n\in \mathbb{N}$, there exists a constant ${k}_{n}\ge 0$ such that

The infimum of the constants ${k}_{n}$ for which (2.1) holds is denoted by $\eta ({T}^{n})$ and is called the *nearly Lipschitz constant* of ${T}^{n}$.

*T*with the sequence $\{({a}_{n},\eta ({T}^{n}))\}$ is said to be

- (4)
*nearly nonexpansive*if $\eta ({T}^{n})=1$ for all $n\in \mathbb{N}$; - (5)
*nearly asymptotically nonexpansive*if $\eta ({T}^{n})\ge 1$ for all $n\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty}}\eta ({T}^{n})=1$; - (6)
*nearly uniformly**k*-*Lipschitzian*if $\eta ({T}^{n})\le k$ for all $n\in \mathbb{N}$.

**Definition 2.3**Let

*C*be a nonempty subset of a metric space

*X*and fix a sequence $\{{a}_{n}\}$ in $[0,\mathrm{\infty})$ with ${a}_{n}\to 0$. A mapping $T:C\to C$ is said to be

*nearly asymptotically quasi*-

*nonexpansive*with respect to $\{{a}_{n}\}$ if $F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{u}_{n}\}$ in $[0,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$ such that

for all $x\in C$, $p\in F(T)$ and $n\in \mathbb{N}$.

In fact, if *T* is a nearly asymptotically nonexpansive mapping and $F(T)$ is nonempty, then *T* is a nearly asymptotically quasi-nonexpansive mapping. The following is an example of a nearly asymptotically quasi-nonexpansive mapping with $F(T)\ne \varphi $.

**Example 2.4** [19]

Here, $F(T)=\{0\}$ and also, *T* is nearly asymptotically quasi-nonexpansive mapping with $\{{u}_{n}\}=\{1,\frac{1}{2},\frac{1}{{2}^{2}},\frac{1}{{2}^{3}},\dots \}$ and $\{{a}_{n}\}=\{1,\frac{1}{2},\frac{1}{{2}^{2}},\frac{1}{{2}^{3}},\dots \}$.

*C*is a bounded subset of a metric space and $T:C\to C$ a nearly asymptotically quasi-nonexpansive mapping with sequence $\{({a}_{n},{u}_{n})\}$, then

for all $x\in C$, $p\in F(T)$ and $n\in \mathbb{N}$.

The following example shows that *T* is a nearly quasi-nonexpansive mapping but not Lipschitzian and quasi-nonexpansive.

**Example 2.5** [19]

*T*is not Lipschitzian. For each fixed $n\in \mathbb{N}$, define

Clearly, *T* is a nearly quasi-nonexpansive mapping with respect to $\{{a}_{n}\}$ and it is not Lipschitz and not quasi-nonexpansive.

**Lemma 2.6** [[19], Lemma 2.11]

*Let*

*C*

*be a nonempty subset of a metric space*$(X,d)$

*and*$T:C\to C$

*a quasi*-

*L*-

*Lipschitzian*,

*i*.

*e*., $F(T)\ne \mathrm{\varnothing}$

*and there exists a constant*$L>0$

*such that*

*If* $\{{x}_{n}\}$ *is a sequence in* *C* *such that* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$ *and* ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x\in C$, *where* $d(x,F(T))=inf\{d(x,p):p\in F(T)\}$, *then* *x* *is a fixed point of* *T*.

Throughout this paper we consider the following definition of a hyperbolic space introduced by Kohlenbach [24]. It is worth noting that they are different from the Gromov hyperbolic space [25] or from other notions of hyperbolic space that can be found in the literature (see, *e.g.*, [26–28]).

**Definition 2.7**A metric space $(X,d)$ is a hyperbolic space if there exists a map $W:{X}^{2}\times [0,1]\to X$ satisfying

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

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

An important example of a hyperbolic space is a $CAT(0)$ space. It is nonlinear in nature and its brief introduction is as follows.

A metric space $(X,d)$ is a *length space* if any two points of *X* are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points of *X* is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, *d* is known as a *length metric* (otherwise an *inner metric* or *intrinsic metric*). In the case that no rectifiable path joins two points of the space, the distance between them is taken to be ∞.

A *geodesic path* joining $x\in X$ to $y\in X$ is a map *c* from a closed interval $[0,l]\subset \mathbb{R}$ to *X* such that $c(0)=x$, $c(l)=y$, and $d(c(t),c({t}^{\prime}))=|t-{t}^{\prime}|$ for all $t,{t}^{\prime}\in [0,l]$. 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*. The space $(X,d)$ is said to be a *geodesic space* if any two points of *X* are joined by a geodesic path and *X* is said to be *uniquely geodesic* if there is exactly one geodesic path denoted by $\alpha x\oplus (1-\alpha )y$ joining *x* and *y* for each $x,y\in X$. The set $\{\alpha x\oplus (1-\alpha )y:\alpha \in [0,1]\}$ will be denoted by $[x,y]$, called the segment joining *x* to *y*. A subset *C* of a geodesic space *X* is convex if for any $x,y\in C$, we have $[x,y]\subset C$.

A geodesic triangle $\mathrm{\Delta}({x}_{1},{x}_{2},{x}_{3})$ in a geodesic metric space $(X,d)$ is defined to be a collection of three points in *X* (the *vertices* of Δ) and three geodesic segments between each pair of vertices (the *edges* of Δ). A *comparison triangle* for geodesic triangle $\mathrm{\Delta}({x}_{1},{x}_{2},{x}_{3})$ in $(X,d)$ is a triangle $\overline{\mathrm{\Delta}}({x}_{1},{x}_{2},{x}_{3}):=\mathrm{\Delta}({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3})$ in ${\mathbb{R}}^{2}$ such that ${d}_{{\mathbb{R}}^{2}}({\overline{x}}_{i},{\overline{x}}_{j})=d({x}_{i},{x}_{j})$ for $i,j\in \{1,2,3\}$ and such a triangle always exists (see [25]).

*X*with a comparison triangle $\overline{\mathrm{\Delta}}\subset {\mathbb{R}}^{2}$ satisfy the $CAT(0)$ inequality

for all $x,y\in \mathrm{\Delta}$ and for all comparison points $\overline{x},\overline{y}\in \overline{\mathrm{\Delta}}$. Let *X* be a $CAT(0)$ space. Define $W:{X}^{2}\times [0,1]\to X$ by $W(x,y,\alpha )=\alpha x\oplus (1-\alpha )y$. Then *W* satisfies the four properties of a hyperbolic space. Also if *X* is a Banach space and $W(x,y,\alpha )=\alpha x+(1-\alpha )y$, then *X* is a hyperbolic space. Therefore, our hyperbolic space represents a unified approach for both linear and nonlinear structures simultaneously.

To elaborate that there are hyperbolic spaces which are not imbedded in any Banach space, we give the following example.

**Example 2.8**Let

*B*be the open unit ball in complex Hilbert space with respect to the Poincaré metric (also called ‘Poincaré distance’)

Then *B* is a hyperbolic space which is not imbedded in any Banach space.

A metric space $(X,d)$ is called a convex metric space introduced by Takahashi [29] if it satisfies only (i). A subset *C* of a hyperbolic space *X* is *convex* if $W(x,y,\alpha )\in C$ for all $x,y\in C$ and $\alpha \in [0,1]$.

A hyperbolic space $(X,d,W)$ is *uniformly convex* [30] if for any $u,x,y\in X$, $r>0$ and $\u03f5\in (0,2]$, there exists a $\delta \in (0,1]$ such that $d(W(x,y,\frac{1}{2}),u)\le (1-\delta )r$ whenever $d(x,u)\le r$, $d(y,u)\le r$ and $d(x,y)\ge \u03f5r$.

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

The hyperbolic space introduced by Kohlenbach [24] is slightly restrictive than the space of hyperbolic type [26] but general than hyperbolic space of [28]. Moreover, this class of hyperbolic spaces also contains Hadamard manifolds, Hilbert balls equipped with the hyperbolic metric [31], ℝ-trees and Cartesian products of Hilbert balls as special cases.

*C*be a nonempty subset of hyperbolic space

*X*. Let $\{{x}_{n}\}$ be a bounded sequence in a hyperbolic space

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

*C*of

*X*is the set

This is the set of minimizers of the functional ${r}_{a}(\cdot ,\{{x}_{n}\})$. If the asymptotic center is taken with respect to *X*, then it is simply denoted by $A(\{{x}_{n}\})$.

It is well known that uniformly convex Banach spaces and even $CAT(0)$ spaces enjoy the property that bounded sequences have unique asymptotic centers with respect to closed convex subsets. The following lemma is due to Leustean [32] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

**Lemma 2.9** [32]

*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* *C* *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 $\mathrm{\Delta}\text{-}{lim}_{n}{x}_{n}=x$ and call *x* the Δ-*limit* of $\{{x}_{n}\}$.

**Lemma 2.10** [33]

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

**Lemma 2.11** [33]

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

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

*η*.

*Let*$x\in X$

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

*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 2.12** [5]

*Let*$\{{\delta}_{n}\}$, $\{{\beta}_{n}\}$,

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

*be three sequences of nonnegative numbers such that*

*If* ${\beta}_{n}\ge 1$ *for all* $n\in \mathbb{N}$, ${\sum}_{n=1}^{\mathrm{\infty}}({\beta}_{n}-1)<\mathrm{\infty}$ *and* ${\gamma}_{n}<\mathrm{\infty}$, *then* ${lim}_{n\to \mathrm{\infty}}{\delta}_{n}$ *exists*.

## 3 Strong and Δ-convergence theorems in hyperbolic space

In this section, we approximate fixed point for nearly asymptotically nonexpansive mappings in a hyperbolic space. More briefly, we established Δ-convergence and strong convergence theorems for iteration scheme (3.1).

First, we define the *S*-iteration process in hyperbolic space as follows.

*C*be a nonempty closed convex subset of a hyperbolic space

*X*and $T:C\to C$ be a nearly asymptotically nonexpansive mapping. Then, for arbitrarily chosen ${x}_{1}\in C$, we construct the sequence $\{{x}_{n}\}$ in

*C*such that

where $\{{\alpha}_{n}\}$ and $\{{\beta}_{n}\}$ are sequences in $(0,1)$ is called an *S*-*iteration process*.

**Lemma 3.1** *Let* *C* *be a nonempty convex subset of a hyperbolic space* *X* *and* $T:C\to C$ *a nearly asymptotically quasi*-*nonexpansive mapping with sequence* $\{({a}_{n},{u}_{n})\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. *Let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *Then* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ *exists for each* $p\in F(T)$.

*Proof*First, we show that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists for each $p\in F(T)$, we have

for some $M,{M}_{1}\ge 0$. $\{{u}_{n}\}$ is bounded. By Lemma 2.12, we find that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists. □

**Lemma 3.2** *Let* *C* *be a nonempty and closed convex subset of a uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and let* $T:C\to C$ *be a nearly asymptotically quasi*-*nonexpansive mapping with sequences* $\{({a}_{n},{u}_{n})\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. *Let* $F(T)\ne \mathrm{\varnothing}$, *then for the sequence* $\{{x}_{n}\}$ *in* *C* *defined by* (3.1), *we have* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}^{n}{x}_{n})=0$.

*Proof*From Lemma 3.1, we find that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists for each $p\in F(T)$. We suppose that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=c\ge 0$. Since

□

**Theorem 3.3** *Let* *C* *be a nonempty closed convex subset of a complete uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and let* $T:C\to C$ *be a uniformly continuous nearly asymptotically nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$ *and sequence* $\{({a}_{n},\eta ({T}^{n}))\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}(\eta ({T}^{n})-1)<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty}$. *For arbitrary* ${x}_{1}\in C$, *let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *Then* $\{{x}_{n}\}$ *is* Δ-*convergent to an element of* $F(T)$.

*Proof*By Lemma 3.2, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}^{n}{x}_{n})=0$. By uniform continuity of

*T*, $d({x}_{n},{T}^{n}{x}_{n})\to 0$ implies that $d(T{x}_{n},{T}^{n+1}{x}_{n})\to 0$, observe that

Next, we have to show that $\{{x}_{n}\}$ is Δ-convergent to an element of $F(T)$.

*T*is a nearly asymptotically nonexpansive mapping with sequence $\{({a}_{n},\eta ({T}^{n}))\}$. By uniform continuity of

*T*

*v*is a fixed point of

*T*. For this, we define a sequence $\{{z}_{n}\}$ in

*C*by ${z}_{m}={T}^{m}v$, $m\in \mathbb{N}$. For integers $m,n\in \mathbb{N}$, we have

*T*, we have

which implies that *v* is a fixed point of *T*, *i.e.*, $v\in F(T)$.

*v*is the unique asymptotic center for each subsequence $\{{y}_{n}\}$ of $\{{x}_{n}\}$. Assume contrarily, that is, $x\ne v$. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},v)$ exists by Lemma 3.1, therefore, by the uniqueness of asymptotic centers, we have

a contradiction and hence $x=v$. Since $\{{y}_{n}\}$ is an arbitrary subsequence of $\{{x}_{n}\}$, therefore, ${A}_{C}(\{{y}_{n}\})=\{v\}$ for all subsequence of $\{{y}_{n}\}$ of $\{{x}_{n}\}$. This proves that $\{{x}_{n}\}$ Δ-converges to a fixed point of *T*. □

We now discuss the strong convergence for the *S*-iteration process defined by (3.1) for Lipschitzian type mappings in a uniformly convex hyperbolic space setting.

**Theorem 3.4** *Let* *C* *be a nonempty closed convex subset of a complete uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and let* $T:C\to C$ *be a nearly asymptotically quasi*-*nonexpansive mapping with sequence* $\{({a}_{n},{u}_{n})\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. *Assume that* $F(T)$ *is a closed set*. *Let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *Then* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T* *if and only if* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$.

*Proof* Necessity is obvious.

Hence $\{{x}_{n}\}$ is a Cauchy sequence in closed subset *C* of a complete hyperbolic space and so it must converge strongly to a point *q* in *C*. Now, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$ gives $d(q,F(T))=0$. Since $F(T)$ is closed, we have $q\in F(T)$. □

In the next result, the closedness assumption on $F(T)$ is not required.

**Theorem 3.5** *Let* *C* *be a nonempty closed convex subset of a complete uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and* $T:C\to C$ *an asymptotically quasi*-*nonexpansive mapping with sequence* $\{{u}_{n}\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. *Let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *Then* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T* *if* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$.

*Proof* Following an argument similar to those of Theorem 3.4, we see that $\{{x}_{n}\}$ is a Cauchy sequence in *C*. Let ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$. Since an asymptotically quasi-nonexpansive mapping is quasi-*L*-Lipschitzian, it follows from Lemma 2.6 that *x* is a fixed point of *T*. □

**Theorem 3.6** *Let* *C* *be a nonempty closed convex subset of a complete uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and* $T:C\to C$ *a uniformly continuous nearly asymptotically nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$ *and sequence* $\{{a}_{n},\eta ({T}^{n})\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}(\eta ({T}^{n})-1)<\mathrm{\infty}$ *and* ${a}_{n}<\mathrm{\infty}$. *For arbitrary* ${x}_{1}\in C$, *let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *If* *T* *is uniformly continuous and* ${T}^{m}$ *is demicompact for some* $m\in N$, *it follows that* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*By (3.13), we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. By the uniformly continuous of

*T*, we have

Since $d({x}_{n},{T}^{m}{x}_{n})\to 0$, and ${T}^{m}$ is demicompact, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that ${lim}_{j\to \mathrm{\infty}}{T}^{m}{x}_{{n}_{j}}=x\in C$.

Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, we get $x\in F(T)$. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},x)$ exists by Lemma 3.1, and ${lim}_{j\to \mathrm{\infty}}d({x}_{{n}_{j}},x)=0$, we conclude that ${x}_{n}\to x$. □

*T*from a subset of a metric space $(X,d)$ into itself with $F(T)\ne \mathrm{\varnothing}$ is said to

*satisfy condition*(A) (see [35]) if there exists a nondecreasing function $f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $f(0)=0$, $f(t)>0$ for $t\in (0,\mathrm{\infty})$ such that

**Theorem 3.7** *Let* *C* *be a nonempty closed convex subset of a complete uniformly convex hyperbolic space* *X* *with monotone modulus of uniform convexity* *η* *and* $T:C\to C$ *a uniformly continuous nearly asymptotically nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$ *and sequence* $\{({a}_{n},\eta ({T}^{n}))\}$ *such that* ${\sum}_{n=1}^{\mathrm{\infty}}\eta ({T}^{n}-1)<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty}$. *For arbitrary* ${x}_{1}\in C$, *let* $\{{x}_{n}\}$ *be a sequence in* *C* *defined by* (3.1), *where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$. *Suppose that* *T* *satisfies the condition* (A). *Then* $\{{x}_{n}\}$ *converges strongly to a fixed point of T*.

*Proof*By (3.13), we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$ Further, by condition (A),

It follows that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$. Therefore, the result follows from Theorem 3.4. □

## 4 Conclusion

- 1.
We prove strong and Δ-convergence of the

*S*-iteration process, which is faster than the iteration processes used by Abbas*et al.*[19], Dhompongsa and Panyanak [22], and Khan and Abbas [23]. - 2.
Theorem 3.3 extends Agarwal

*et al.*[[18], Theorem 3.8] from a uniformly convex Banach space to a uniformly convex hyperbolic space. - 3.
Theorem 3.3 extends Dhompongsa and Panyanak [[22], Theorem 3.3] from the class of nonexpansive mappings to the class of mappings which are not necessarily Lipschitzian.

- 4.
Theorem 3.6, extends corresponding results of Beg [36], Chang [37], Khan and Takahashi [4] and Osilike and Aniagbosor [5] for a more general class of non-Lipschitzian mappings in the framework of a uniformly convex hyperbolic space. It also extends the corresponding results of Dhomponsga and Panyanak [22] from the class of nonexpansive mappings to a more general class of non-Lipschitzian mappings in the same space setting.

- 5.
Theorem 3.7 extends Sahu and Beg [[12], Theorem 4.4] from a Banach to a uniformly convex hyperbolic space.

## Declarations

### Acknowledgements

The authors would like to thank the editor and all referees for their valuable comments and suggestions for improving the paper.

## Authors’ Affiliations

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