Some stability and strong convergence results for the algorithm with perturbations for a T-Ciric quasicontraction in CAT(0) spaces

Abstract

In this paper, we establish the stability and strong convergence theorems, for the three-step iteration with perturbations for a T-Ciric quasicontraction, in the environment of the CAT(0) space. Finally, an application to the integral-type contraction and an example are shown.

1 Introduction

Below, we state some basic concepts of the metric environment in which we will work, although several of these definitions have been extensively studied and are easily accessible, it is worth highlighting their importance, at least in some fundamental aspects. One aspect is given by the next concept. 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 f from a closed interval $[0,\ell ]\subset \mathbb{R}$ to X such that $f(0) = x$; $f(\ell ) = y$, and $d (f(s),f(s^{\prime}) )=|s-s^{\prime}|$ for all $s,s^{\prime}\in [0,1]$. In particular, f is an isometry and $d(x,y)=\ell$. The image γ of f is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment is denoted by $[x,y]$. The space $(X, d)$ is said to be a geodesic space if every two points of X are joined by a geodesic, and is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each $x, y \in X$. Let $x_{1}$, $x_{2}$, and $x_{3}$ be points not necessarily distinct in a metric space X and $[x_{1}, x_{2} ]$, $[x_{2}, x_{3} ]$, and $[x_{3},x_{1}]$ geodesic segments in X, not necessarily determined by its ends. We say that the union of the three segments, which we denote △ and give the induced metric, is a geodesic triangle in X with vertices $x_{1}$, $x_{2}$, and $x_{3}$. A comparison triangle for the geodesic triangle $\Delta ( x_{1},x_{2},x_{3} )$ in $( X,d )$ is a triangle $\overline{\Delta} ( x_{1},x_{2},x_{3} ) :=\Delta ( \bar{x}_{1},\bar{x}_{2},\bar{x}_{3} )$ in $\mathbb{R}^{2}$ such that $d_{\mathbb{R}^{2}} ( \bar{x}_{i},\bar {x}_{j} )=d ( x_{i},x_{j} )$ for $i,j\in \{ 1,2,3 \}$. Such a triangle always exists [1]. The study of spaces of nonpositive curvature, also known as CAT(0) spaces, has its origins in the discovery of hyperbolic spaces, work carried out by Hadamard at the beginning of the last century, and in the work of Cartan in the 1920s. The idea of what it means for a geodesic metric space to have a nonpositive curvature (or, more generally, a curvature bounded by a real number k) dates back to the work of Busemann and Alexandrov in the 1950s. Later, Gromov (see [1, 2]) contributed to a better understanding of these spaces and named them CAT(0) spaces.

A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.

CAT(0): Let △ be a geodesic triangle in X and let $\overline{\bigtriangleup}$ be a comparison triangle for △. Then, △ is said to satisfy the CAT(0) inequality if for all $x,y \in \bigtriangleup$ and all comparison points $\bar{x},\bar{y} \in \overline{\bigtriangleup}$,

$$ d(x, y) \le d_{\mathbb{R}^{2}} (\bar{x}, \bar{y}).$$

If x, $y_{1}$, $y_{2}$ are points in a $\operatorname {CAT}(0)$ space and if $y_{0}$ is the midpoint of the segment $[y_{1},y_{2}]$, which we will denote by $\frac{y_{1}\oplus y_{2}}{2}$, then the $\operatorname {CAT}(0)$ inequality implies

$$ d(x,y_{0})^{2}\leq \frac{1}{2}d(x,y_{1})^{2}+ \frac{1}{2}d(x,y_{2})^{2}- \frac{1}{4}d(y_{1},y_{2})^{2}. $$

This is the so-called (CN) inequality of Bruhat and Tits [3]. In fact, Bridson and Haefliger [1], a geodesic is a $\operatorname {CAT}(0)$ space if and only if it satisfies the (CN) inequality. Some additional properties of $\operatorname {CAT}(0)$ spaces are the following: Let $(X,d)$ be a $\operatorname {CAT}(0)$ space. Then,

• (i) $(X,d)$ is uniquely geodesic.

• (ii) Let p, x, y be points of X, let $\alpha \in [0,1]$, and let $m_{1}$ and $m_{2}$ denote, respectively, the points of $[p,x]$ and $[p,y]$ satisfying $d(p,m_{1})=\alpha d(p,x)$ and $d(p,m_{2})=\alpha d(p,y)$. Then,

  $$ d(m_{1},m_{2})\leq \alpha d(x,y). $$

• (iii) Let $x,y\in X$, $x\neq y$ and $z,w \in [x,y]$ such that $d(x,z)=d(x,w)$. Then, $z=w$.

• (iv) Let $x,y\in X$. For each $t\in [0,1]$, there exists a unique point $z\in [x,y]$ such that,

  $$ d(x,z)=td(x,y) \quad \text{and} \quad d(y,z)=(1-t)d(x,y). $$

  (1)

For convenience, from now on we will use the notation $(1-t)x\oplus ty$ for a unique point z satisfying (1). For $x_{i}\in X$ and $t_{i}\in [0,1]$, $i=1,\dots ,n$ such that $\sum_{i=1}^{n}t_{i}=1$, by induction we write

$$ \bigoplus_{i=1}^{n}t_{i}x_{i}=(1-t_{n}) \biggl(\frac{t_{1}}{1-t_{n}}x_{1} \oplus \frac{t_{2}}{1-t_{n}}x_{2} \oplus \cdots \oplus \frac{t_{n-1}}{1-t_{n}}x_{n} \biggr)\oplus t_{n}x_{n}. $$

Lemma 1.1

([4])

Let X be a $\operatorname{CAT}(0)$ space, with $x,x_{i}\in X$ and $t_{i}\in [0,1]$ for $i = 1, 2, \ldots, n$ ($n\ge 2$) such that $\sum_{i=1}^{n} t_{i}=1$. Then,

$$ d \Biggl(\bigoplus_{i=1}^{n} t_{i}x_{i},x \Biggr)\leq \sum_{i=1}^{n} t_{i}d(x_{i},x).$$

On the other hand, after Banach published the principle of contraction and his famous theorem, there have been several researchers who have proposed applications of the contractive type, among them are the one that studied by Beiranvand et al. [5] who introduced the concept of T-Banach contraction and T-contractive mappings, in such a way that the Banach contraction principle (BCP) was extended. Following this, Moradi [6] introduced T-Kannan contractive mappings, extending in this way, the well-known Kannan fixed-point theorem. Morales and Rojas [7] obtained conditions for the existence of a unique fixed point of T-Zamfirescu and T-weak contraction mappings in the framework of complete cone metric spaces.

Definition 1.1

([7])

Let $(X,d)$ be a metric space and $S, T : X \to X$ be two mappings. A mapping S is said be a T-Zamfirescu operator (TZ-operator), if there are real numbers $0\leq a < 1$, $0 \leq b < \frac{1}{2}$, $0 \leq c < \frac{1}{2}$ such that for all $x, y \in X$ at least one of the following conditions holds:

$TZ_{1}$:

$d (TSx,TSy )\leq ad (Tx,Ty )$;

$TZ_{2}$:

$d (TSx,TSy )\leq b [d (Tx,TSx )+d (Ty,TSy ) ]$;

$TZ_{3}$:

$d (TSx,TSy )\leq c [d (Tx,TSy )+d (Ty,TSx ) ]$.

If we take $T = I$, the identity map, in Definition 1.1, then we obtain the definition of a Zamfirescu operator

Theorem 1.1

([7])

Let $(M, d)$ be a complete metric space and $T, S : M \to M$ be two mappings such that T is continuous, one-to-one, and subsequently convergent. If S is a TZ operator, S has a unique fixed point. Moreover, if T is sequentially convergent, then for every $x_{0} \in M$ the T-Picard iteration associated to S, $TS^{n}x_{0}$ converges to $Tx^{*}$, where $x^{*}$ is the fixed point of S.

Definition 1.2

([8])

Let X be a $CAT(0)$ space and $S, T : X \to X$ be two mappings. Then, S is called a T-Ciric quasicontraction mapping if it satisfies the following condition:

$$ \begin{aligned} &d(TSx,TSy)\\ &\quad \leq h \max \biggl\lbrace d(Tx,Ty), \frac{d(Tx,TSx)+d(Ty,TSy)}{2}, \frac{d(Tx,TSy)+d(Ty,TSx)}{2} \biggr\rbrace \end{aligned} $$

(2)

for all $x, y \in X$ and $0 < h < 1$.

If we take $T = I$, then (2) reduces to a quasicontraction mapping introduced by Ciric [9].

The Man iteration process is defined by the sequence $\lbrace x_{n} \rbrace$,

$$ x_{n+1}= (1-\alpha _{n} )x_{n}+\alpha _{n}T(x_{n}), \quad n=1,2,3\ldots , $$

(3)

where $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=o}$ is a sequence in $(0,1)$. The Ishikawa iteration process is defined by the sequence $\lbrace x_{n} \rbrace$,

$$ \textstyle\begin{cases} x_{n+1}=(1-\alpha _{n})x_{n}+\alpha _{n}Ty_{n}, \\ y_{n}=(1-\beta _{n})x_{n}+\beta _{n}Tx_{n}, \quad n=1,2,3,\ldots, \end{cases} $$

(4)

where $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=o}$ and $\lbrace \beta _{n} \rbrace ^{\infty}_{n=o}$ is a sequence in $(0,1)$.

In [10] they investigated the best proximity points of multivalued mappings through Mann and Ishikawa iteration schemes. Within the framework of CAT(0) spaces, several convergence theorems of best proximity points have been established.

During previous years, much attention has been given to the following iteration processes. Among them is the one made by Agarwal, O’Regan, and Sahu [11] For a nonempty convex subset K of a normed space E and $S : K\to K$,

$$ \textstyle\begin{cases} x_{n+1}=(1-\alpha _{n})Sx_{n}+\alpha _{n}Sy_{n}, \\ y_{n}=(1-\beta _{n})x_{n}+\beta _{n}Sx_{n}, \end{cases}\displaystyle \quad n\geq 1, $$

(5)

where $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=o}$ and $\lbrace \beta _{n} \rbrace ^{\infty}_{n=o}$ are appropiate sequences in $[0,1]$. In 2013, Sahin and Basarir [12] modified the iteration process proposed by Agarwal et al. [11] introducing a version in a $\operatorname {CAT}(0)$ space as follows:

$$ \textstyle\begin{cases} x_{n+1}=(1-\alpha _{n})Sx_{n}\oplus \alpha _{n}Sy_{n}, \\ y_{n}=(1-\beta _{n})x_{n}\oplus \beta _{n}Sx_{n}, \end{cases}\displaystyle \quad n\geq 1, $$

(6)

where $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=o}$ and $\lbrace \beta _{n} \rbrace ^{\infty}_{n=o}$ are appropriate sequences in $[0,1]$. Note that (6) is independent of an Ishikawa-type iterative process (and hence a Mann-type iterative process).

Recently, in [13] they modified the algorithm proposed by [14] for the problem of approximating common fixed points of a pair of mappings by properly including T in its definition:

$$ \textstyle\begin{cases} x_{n+1}=(1-\eta _{n}-\delta _{n})Sx_{n}+\eta _{n}Sy_{n}+\delta _{n}Sz_{n}, \\ y_{n}=S ((1-\zeta _{n})Tx_{n}+\zeta _{n}Sz_{n} ), \\ z_{n}=(1-\overline{\zeta _{n}})x_{n}+\overline{\zeta _{n}}Tx_{n} , \end{cases} $$

(7)

where $\lbrace \eta _{n} \rbrace ^{\infty}_{n=o}$, $\lbrace \delta _{n} \rbrace ^{\infty}_{n=o}$, $\lbrace \zeta _{n} \rbrace ^{\infty}_{n=o}$, and $\lbrace \overline{\zeta _{n}} \rbrace ^{\infty}_{n=o}$ are sequences of real numbers in $(0,1)$. Additionally, we assume further that the parametric sequence $\lbrace \overline{\zeta _{n}} \rbrace ^{\infty}_{n=o}$ satisfies $0< p\leq \overline{\zeta _{n}}\leq q<1$ and $\lbrace \zeta _{n} \rbrace ^{\infty}_{n=o}$ is a convergent sequence to some $\zeta \in (0,1)$.

Several authors have directed their work in the search for the approximation of fixed points. In [15] convergence results for approximation of common solutions for a finite family of generalized demimetric mappings and monotone inclusion problems were presented. Other works presenting interesting iterative processes include the so-called K iteration process, see [16], which proved some strong and convergence theorems for two different classes of generalized, nonexpansive mapppings in CAT(0) spaces. Also, in [17] the M-iteration process in hyperbolic spaces was studied, obtaining convergence results for nonexpansive, generalized mappings and establishing a type of weak stability.

Motivated and inspired by the iterative processes of Agarwal, O’Regan, and Sahu [11] and Sahin and Basarir [12], we propose a new process, which includes bounded sequences in K, which we call perturbations. This is in order to study the possible causes of such disturbances, since, as is well known, perturbations are phenomena that appear quite frequently in nature and are of great interest in the scientific community.

Let K be a nonempty closed convex subset of a complete $\operatorname {CAT}(0)$ space X and $S, T : K \to K$ be two mappings where the operator S is a T-Ciric quasicontraction mapping. Then, for a given $x_{1}=x_{0}\in K$, compute the sequence $\lbrace x_{n} \rbrace$ by the scheme as follows:

$$ \textstyle\begin{cases} Tx_{n+1}=(1-\alpha _{n}-\alpha _{n}^{\prime})TSx_{n}\oplus \alpha _{n}TSy_{n} \oplus \alpha _{n}^{\prime}\epsilon _{n}, \\ Ty_{n}=(1-\beta _{n}-\beta _{n}^{\prime})Tx_{n}\oplus \beta _{n}TSz_{n} \oplus \beta _{n}^{\prime}\epsilon _{n}^{\prime }, \\ Tz_{n}=(1-\gamma _{n}-\gamma _{n}^{\prime})Tx_{n}\oplus \gamma _{n}TSx_{n} \oplus \gamma _{n}^{\prime}\epsilon _{n}^{\prime \prime}, \end{cases}\displaystyle \quad n\geq 1, $$

(8)

where $\lbrace \epsilon _{n} \rbrace$, $\lbrace \epsilon _{n}^{ \prime} \rbrace$, $\lbrace \epsilon _{n}^{\prime \prime} \rbrace$ are bounded sequences in K and $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \gamma _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \alpha _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, and $\lbrace \gamma _{n}^{\prime } \rbrace ^{\infty}_{n=1}$ are appropriate sequences in $[0,1]$,

Remark 1.1

If $T=I_{d}$ and $\lbrace \gamma _{n} \rbrace ^{\infty}_{n=o}= \lbrace \alpha _{n}^{\prime } \rbrace ^{\infty}_{n=o}= \lbrace \beta _{n}^{\prime } \rbrace ^{\infty}_{n=o}= \lbrace \gamma _{n}^{\prime } \rbrace ^{\infty}_{n=o}=0$, in (8) we obtain the iterative process scheme (6).

Definition 1.3

Let $T,S:X\longrightarrow X$ such that T is a continuous function and $x^{*}$ is the fixed point of S. Let $\lbrace Tx_{n} \rbrace _{n=0}^{\infty}$ sequence be generated by an iterative procedure,

$$ Tx_{n+1}=H(g,f,x_{n}), \quad n=0,1,2\ldots, $$

(9)

where $x_{0}\in X$ is the initial approximation and H is some function. Suppose that $\lbrace Tx_{n} \rbrace _{n=0}^{\infty}$ converges to $Tx^{*}$. Let $\lbrace Ty_{n} \rbrace _{n=0}^{\infty}\subset X$ be an arbitrary sequence and set $\varepsilon _{n}=d(Ty_{n+1},H(S,y_{n}))$, $n=1,2,3\ldots$ . Then, the iterative procedure (9) is said to be $(T,S)$-stable or stable if and only if $\lim_{n\rightarrow \infty}\varepsilon _{n}=0$ implies $\lim_{n\rightarrow \infty}Ty_{n}=Tx^{*}$, where $x^{*}$ is the fixed point of S.

Lemma 1.2

([18])

Let $\lbrace p_{n} \rbrace _{n=0}^{\infty}$, $\lbrace q_{n} \rbrace _{n=0}^{\infty}$, $\lbrace r_{n} \rbrace _{n=0}^{\infty}$, and $\lbrace s_{n} \rbrace _{n=0}^{\infty}$ be sequences of nonnegative numbers satisfying the following conditions:

$$ p_{n+1}\leq (1-q_{n})p_{n}+q_{n}r_{n}+s_{n}, \quad n\geq 1. $$

If $\sum_{n=1}^{\infty }q_{n}=\infty$, $\lim_{n\rightarrow \infty}r_{n}=0$ and $\sum_{n=1}^{\infty }s_{n}<\infty$ hold, then $\lim_{n\rightarrow \infty}p_{n}=0$.

2 Main results

Now, we prove the following lemma, which allows us to perform the respective convergence and stability proofs.

Lemma 2.1

Let X be a $CAT(0)$ and $S, T : X \to X$ be two mappings and S is a T-Ciric quasicontraction mapping. Then,

$$ d\bigl(Tx^{*},TSu\bigr)\leq \delta d\bigl(Tx^{*},Tu\bigr), $$

(10)

where $\delta =\max \lbrace h,\frac{h}{2-h} \rbrace \in [0,1)$ and $x^{*}$ is the fixed point of S.

Proof

From Theorem 1.1, we obtain that S has a unique fixed point in X. Putting $x=x^{*}$ and $y=u$

$$ d\bigl(Tx^{*},TSu\bigr)\leq h \max \biggl\lbrace d \bigl(Tx^{*},Tu\bigr), \frac{d(Tu,TSu)}{2},\frac{d(Tx^{*},TSu)+d(Tu,Tx^{*})}{2} \biggr\rbrace . $$

(11)

Therefore, we have three cases.

Case 1.

$$ d\bigl(Tx^{*},TSu\bigr)\leq hd\bigl(Tx^{*},Tu \bigr).$$

Case 2.

$$ \begin{aligned} d\bigl(Tx^{*},TSu\bigr)&\leq \frac{h}{2}d(Tu,TSu) \\ &\leq \frac{h}{2} \bigl\lbrace d\bigl(Tx^{*},Tu\bigr)+d \bigl(Tx^{*},TSu\bigr) \bigr\rbrace \\ &\leq \frac{h}{2-h}d\bigl(Tx^{*},Tu\bigr). \end{aligned} $$

Case 3.

$$ \begin{aligned} d\bigl(Tx^{*},TSu\bigr)&\leq \frac{h}{2} \bigl\lbrace d\bigl(Tx^{*},TSu\bigr)+d \bigl(Tx^{*},Tu\bigr) \bigr\rbrace \\ &\leq \frac{h}{2-h}d\bigl(Tx^{*},Tu\bigr). \end{aligned} $$

If we take $\delta =\max \lbrace h,\frac{h}{2-h} \rbrace \in [0,1)$ we show that the mapping type T-ciric quasicontraction reduces to a TZ-opertor. □

Theorem 2.1

Let $(X,d)$ be a complete $\operatorname{CAT}(0)$ space. Let K be a nonempty, closed, and convex bounded subset of X. Let $S, T : K \to K$ be two mappings and S is a T-Ciric quasicontraction mapping. Let $\{Tx_{n}\}^{\infty}_{n=0}$ be defined by (8) satisfying the conditions:

• i. $\sum_{n=1}^{\infty} (\alpha _{n}\beta _{n}+\alpha _{n}\beta _{n} \gamma _{n} )=\infty$.

• ii. $\sum_{n=1}^{\infty }\alpha _{n}\beta _{n}\gamma _{n}^{\prime}< \infty$, $\sum_{n=1}^{\infty}\alpha _{n}\beta _{n}^{\prime}<\infty$, and $\sum_{n=1}^{\infty}\alpha _{n}^{\prime}<\infty$.

Then, $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $Tx^{*}$, where $x^{*}$ is the fixed point of S.

Proof

Let $x^{*}$ be the fixed point of S and

$$ \begin{aligned} &\varepsilon _{1}=\sup \bigl\lbrace d \bigl(\epsilon _{n},Tx^{*}\bigr):n\geq 1 \bigr\rbrace , \qquad \varepsilon _{3}=\sup \bigl\lbrace d\bigl(\epsilon _{n}^{\prime \prime},Tx^{*} \bigr):n \geq 1 \bigr\rbrace , \\ &\varepsilon _{2}=\sup \bigl\lbrace d\bigl(\epsilon _{n}^{\prime},Tx^{*}\bigr):n \geq 1 \bigr\rbrace , \qquad \varepsilon =\max \lbrace \varepsilon _{i}:i=1,2,3 \rbrace . \end{aligned} $$

Using (8), Lemma 2.1, and Lemma 1.1, we have

$$ \begin{aligned} d\bigl(Tz_{n},Tx^{*}\bigr)& = d \bigl(\bigl(1-\gamma _{n}-\gamma _{n}^{\prime} \bigr)Tx_{n} \oplus \gamma _{n}TSx_{n}\oplus \gamma _{n}^{\prime}\epsilon _{n}^{ \prime \prime},Tx^{*} \bigr) \\ & \leq \bigl(1-\gamma _{n}-\gamma _{n}^{\prime}\bigr)d \bigl(Tx_{n},Tx^{*}\bigr)+\gamma _{n} d \bigl(TSx_{n},Tx^{*}\bigr)+\gamma _{n}^{\prime} \varepsilon \\ & \leq \bigl(1-\gamma _{n}-\gamma _{n}^{\prime}\bigr)d \bigl(Tx_{n},Tx^{*}\bigr)+\delta \gamma _{n} d \bigl(Tx_{n},Tx^{*}\bigr)+\gamma _{n}^{\prime} \varepsilon \\ & \leq \bigl[1-\gamma _{n}(1-\delta )-\gamma _{n}^{\prime} \bigr] d\bigl(Tx_{n},Tx^{*}\bigr)+ \gamma _{n}^{\prime}\varepsilon , \end{aligned} $$

i.e.,

$$ \begin{aligned} d\bigl(Ty_{n},Tx^{*} \bigr)& = d \bigl(\bigl(1-\beta _{n}-\beta _{n}^{\prime} \bigr)Tx_{n} \oplus \beta _{n}TSz_{n}\oplus \beta _{n}^{\prime}\epsilon _{n}^{ \prime}, Tx^{*} \bigr) \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Tx_{n},Tx^{*}\bigr)+\beta _{n}d \bigl(TSz_{n},Tx^{*}\bigr)+ \beta _{n}^{\prime} \varepsilon \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Tx_{n},Tx^{*}\bigr)+\delta \beta _{n}d \bigl(Tz_{n},Tx^{*}\bigr)+\beta _{n}^{\prime} \varepsilon \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Tx_{n},Tx^{*}\bigr)+\delta \beta _{n} \bigl[ \bigl(1-\gamma _{n}(1-\delta )-\gamma _{n}^{\prime} \bigr)d\bigl(Tx_{n},Tx^{*}\bigr) \\ &\quad{}+\gamma _{n}^{\prime}\varepsilon \bigr]+\beta _{n}^{\prime} \varepsilon \\ & \leq \bigl[1-\beta _{n}(1-\delta )-\delta \beta _{n}\gamma _{n}(1- \delta )-\beta _{n}^{\prime}-\delta \beta _{n}\gamma _{n}^{\prime} \bigr] d\bigl(Tx_{n},Tx^{*} \bigr) \\ &\quad{}+\delta \beta _{n}\gamma _{n}^{\prime}\varepsilon + \beta _{n}^{ \prime}\varepsilon . \end{aligned} $$

(12)

Again using (8), (12), Lemma 2.1, and Lemma 1.1,

$$ \begin{aligned} d\bigl(Tx_{n+1},Tx^{*}\bigr)& = d \bigl(\bigl(1-\alpha _{n}-\alpha _{n}^{\prime} \bigr)TSx_{n} \oplus \alpha _{n}TSy_{n}\oplus \alpha _{n}^{\prime}\epsilon _{n},Tx^{*} \bigr) \\ & \leq \bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)d \bigl(TSx_{n},Tx^{*}\bigr)+\alpha _{n} d \bigl(TSy_{n},Tx^{*}\bigr)+\alpha _{n}^{\prime} \varepsilon \\ & \leq \bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr) \delta d\bigl(Tx_{n},Tx^{*}\bigr)+ \delta \alpha _{n} d\bigl(Ty_{n},Tx^{*}\bigr)+\alpha _{n}^{\prime}\varepsilon \\ & \leq \bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr) \delta d\bigl(Tx_{n},Tx^{*}\bigr)+ \delta \alpha _{n} \bigl[ \bigl(1-\beta _{n}(1-\delta )-\delta \beta _{n} \gamma _{n}(1-\delta ) \\ &\quad{}-\beta _{n}^{\prime}-\delta \beta _{n} \gamma _{n}^{ \prime} \bigr) d\bigl(Tx_{n},Tx^{*} \bigr)+\delta \beta _{n}\gamma _{n}^{\prime} \varepsilon +\beta _{n}^{\prime}\varepsilon \bigr]+\alpha _{n}^{ \prime} \varepsilon \\ & \leq \bigl[1- \bigl(\alpha _{n}\beta _{n}(1-\delta )+ \delta \alpha _{n} \beta _{n}\gamma _{n}(1-\delta ) \bigr) \bigr]d\bigl(Tx_{n},Tx^{*}\bigr) \\ &\quad{}+\varepsilon \bigl[\delta ^{2}\alpha _{n}\beta _{n}\gamma _{n}^{ \prime}+\delta \alpha _{n} \beta _{n}^{\prime}+\alpha _{n}^{\prime} \bigr]. \end{aligned} $$

We set $A_{n}=\alpha _{n}\beta _{n}(1-\delta )+\delta \alpha _{n}\beta _{n} \gamma _{n}(1-\delta )$ and $B_{n}=\delta ^{2}\alpha _{n}\beta _{n}\gamma _{n}^{\prime}+\delta \alpha _{n}\beta _{n}^{\prime}+\alpha _{n}^{\prime}$. Then, the above inequality reduces to

$$ \begin{aligned} d\bigl(Tx_{n+1},Tx^{*}\bigr)& \leq (1-A_{n} )d\bigl(Tx_{n},Tx^{*}\bigr)+ \varepsilon B_{n}, \end{aligned} $$

since $0\leq \delta <1$, by assumption of the theorem $\sum_{n=1}^{\infty} (\alpha _{n}\beta _{n}+\alpha _{n}\beta _{n} \gamma _{n} )=\infty$, $\sum_{n=1}^{\infty }\alpha _{n}\beta _{n}\gamma _{n}^{\prime}< \infty$, $\sum_{n=1}^{\infty}\alpha _{n}\beta _{n}^{\prime}<\infty$, and $\sum_{n=1}^{\infty}\alpha _{n}^{\prime}<\infty$, it follows that $\sum_{n=1}^{\infty }A_{n}=\infty$ and $\sum_{n=1}^{\infty }B_{n}<\infty$ therefore by Lemma 1.2 we obtain that $\lim_{n\rightarrow \infty}d(Tx_{n},Tx^{*})=0$, so that $\lbrace Tx_{n} \rbrace$ converges strongly to $Tx^{*}$, where $x^{*}$ is the fixed point of S. This completes the proof. □

Theorem 2.2

Let X, K, T, S, and $\lbrace x_{n} \rbrace ^{\infty}_{n=0}$ satisfy the hypothesis of Theorem 2.1. Then, the sequence by (8) is $(T,S)$-stable.

Proof

Since $\lbrace \epsilon _{n} \rbrace$, $\lbrace \epsilon _{n}^{\prime} \rbrace$, and $\lbrace \epsilon _{n}^{\prime \prime} \rbrace$ are bounded sequences in K, we can put,

$$ \begin{aligned} &\varepsilon _{1}=\sup \bigl\lbrace d \bigl(\epsilon _{n},Tx^{*}\bigr):n\geq 1 \bigr\rbrace , \qquad \varepsilon _{3}=\sup \bigl\lbrace d\bigl(\epsilon _{n}^{\prime \prime},Tx^{*} \bigr):n \geq 1 \bigr\rbrace , \\ &\varepsilon _{2}=\sup \bigl\lbrace d\bigl(\epsilon _{n}^{\prime},Tx^{*}\bigr):n \geq 1 \bigr\rbrace , \qquad \varepsilon =\max \lbrace \varepsilon _{i}=1,2,3 \rbrace . \end{aligned} $$

Suppose that $\lbrace Ty_{n} \rbrace \subset K$ is an arbitrary sequence,

$$ \xi _{n}=d \bigl(Ty_{n+1},\bigl(1-\alpha _{n}- \alpha _{n}^{\prime}\bigr)TSy_{n} \oplus \alpha _{n}TSb_{n}\oplus \alpha _{n}^{\prime}\epsilon _{n} \bigr), $$

where $Tb_{n}=(1-\beta _{n}-\beta _{n}^{\prime})Ty_{n}\oplus \beta _{n}TSc_{n} \oplus \beta _{n}^{\prime}\epsilon _{n}^{\prime}$, $Tc_{n}=(1-\gamma _{n}-\gamma _{n}^{\prime})Ty_{n}\oplus \gamma _{n}TSy_{n} \oplus \gamma _{n}^{\prime}\epsilon _{n}^{\prime \prime}$ and let $\lim_{n\rightarrow \infty}\xi _{n}=0$.

Then, for (8) we have

$$ \begin{aligned} d\bigl(Ty_{n+1},Tx^{*} \bigr)& \leq d \bigl(Ty_{n+1},\bigl(1-\alpha _{n}-\alpha _{n}^{ \prime}\bigr)TSy_{n}\oplus \alpha _{n}TSb_{n}\oplus \alpha _{n}^{\prime} \epsilon _{n} \bigr) \\ &\quad{}+d \bigl(\bigl(1-\alpha _{n}-\alpha _{n}^{\prime} \bigr)TSy_{n}\oplus \alpha _{n}TSb_{n} \oplus \alpha _{n}^{\prime}\epsilon _{n},Tx^{*} \bigr) \\ & \leq \xi _{n}+d \bigl(\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)TSy_{n} \oplus \alpha _{n}TSb_{n}\oplus \alpha _{n}^{\prime}\epsilon _{n},Tx^{*} \bigr) \\ & \leq \xi _{n}+\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)d\bigl(TSy_{n},Tx^{*} \bigr)+ \alpha _{n} d\bigl(TSb_{n},Tx^{*}\bigr)+ \alpha _{n}^{\prime }d\bigl(\epsilon _{n},Tx^{*} \bigr) \\ & \leq \xi _{n}+\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)\delta d\bigl(Ty_{n},Tx^{*} \bigr)+ \delta \alpha _{n} d\bigl(Tb_{n},Tx^{*} \bigr)+\alpha _{n}^{\prime }d\bigl(\epsilon _{n},Tx^{*} \bigr) \\ & \leq \xi _{n}+\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)\delta d\bigl(Ty_{n},Tx^{*} \bigr)+ \delta \alpha _{n} d\bigl(Tb_{n},Tx^{*} \bigr)+\alpha _{n}^{\prime }\varepsilon . \end{aligned} $$

(13)

Now, we have the following estimates:

$$ \begin{aligned} d\bigl(Tb_{n},Tx^{*}\bigr)& = d \bigl(\bigl(1-\beta _{n}-\beta _{n}^{\prime} \bigr)Ty_{n}\oplus \beta _{n}TSc_{n} \oplus \beta _{n}^{\prime}\epsilon _{n}^{\prime},Tx^{*} \bigr) \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Ty_{n},Tx^{*}\bigr)+\beta _{n}d \bigl(TSc_{n},Tx^{*}\bigr)+ \beta _{n}^{\prime }d \bigl(\epsilon _{n}^{\prime},Tx^{*}\bigr) \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Ty_{n},Tx^{*}\bigr)+\beta _{n} \delta d \bigl(Tc_{n},Tx^{*}\bigr)+\beta _{n}^{\prime }d \bigl(\epsilon _{n}^{\prime},Tx^{*}\bigr) \\ & \leq \bigl(1-\beta _{n}-\beta _{n}^{\prime}\bigr)d \bigl(Ty_{n},Tx^{*}\bigr)+\beta _{n} d \bigl(Tc_{n},Tx^{*}\bigr)+ \beta _{n}^{\prime} \varepsilon , \end{aligned} $$

i.e.,

$$ \begin{aligned} d\bigl(Tc_{n},Tx^{*} \bigr)& \leq d\bigl(\bigl(1-\gamma _{n}-\gamma _{n}^{\prime} \bigr)Ty_{n} \oplus \gamma _{n}TSy_{n}\oplus \gamma _{n}^{\prime}\epsilon _{n}^{ \prime \prime},Tx^{*} \bigr) \\ & \leq \bigl(1-\gamma _{n}-\gamma _{n}^{\prime}\bigr)d \bigl(Ty_{n},Tx^{*}\bigr)+\gamma _{n}d \bigl(TSy_{n},Tx^{*}\bigr)+ \gamma _{n}^{\prime }d \bigl(\epsilon _{n}^{\prime \prime},Tx^{*}\bigr) \\ & \leq \bigl(1-\gamma _{n}-\gamma _{n}^{\prime}\bigr) \delta d\bigl(Ty_{n},Tx^{*}\bigr)+ \gamma _{n}\delta d\bigl(Ty_{n},Tx^{*}\bigr)+\gamma _{n}^{\prime }d \bigl(\epsilon _{n}^{ \prime \prime},Tx^{*}\bigr) \\ & \leq \bigl(1-\gamma _{n}(1-\delta )-\gamma _{n}^{\prime} \bigr) d\bigl(Ty_{n},Tx^{*}\bigr)+ \gamma _{n}^{\prime } \varepsilon . \end{aligned} $$

(14)

It follows from (14) and (13) that

$$ \begin{aligned} d\bigl(Ty_{n+1},Tx^{*}\bigr)& \leq \bigl[1- \bigl(\alpha _{n}\beta _{n}(1- \delta )+\delta \alpha _{n}\beta _{n}\gamma _{n}(1-\delta ) \bigr) \bigr]d\bigl(Tx_{n},Tx^{*}\bigr) \\ &\quad{}+\varepsilon \bigl[\delta ^{2}\alpha _{n}\beta _{n}\gamma _{n}^{ \prime}+\delta \alpha _{n} \beta _{n}^{\prime}+\alpha _{n}^{\prime} \bigr]+\xi _{n} \\ & \leq (1-q_{n} )d\bigl(Ty_{n},Tx^{*} \bigr)+s_{n}+\xi _{n}, \end{aligned} $$

(15)

where $q_{n}=\alpha _{n}\beta _{n}(1-\delta )+\delta \alpha _{n}\beta _{n} \gamma _{n}(1-\delta )$ and $s_{n}=\delta ^{2}\alpha _{n}\beta _{n}\gamma _{n}^{\prime}+\delta \alpha _{n}\beta _{n}^{\prime}+\alpha _{n}^{\prime}$, since by assumption of the theorem $\lim_{n\rightarrow \infty}\xi _{n}=0$, it follows Lemma 1.2, then $\lim_{n\rightarrow \infty}Ty_{n}=Tx^{*}$.

Conversely, $\lim_{n\rightarrow \infty}d(Ty_{n+1},Tx^{*})=0$. Then, using Lemma 2.1 and the triangle inequality, we have:

$$ \begin{aligned} \xi _{n}& = d \bigl(Ty_{n+1},\bigl(1-\alpha _{n}-\alpha _{n}^{\prime} \bigr)TSy_{n} \oplus \alpha _{n}TSb_{n}\oplus \alpha _{n}^{\prime}\epsilon _{n} \bigr) \\ & \leq d\bigl(Ty_{n+1},Tx^{*}\bigr)+d\bigl(\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)TSy_{n} \oplus \alpha _{n}TSb_{n}\oplus \alpha _{n}^{\prime} \epsilon _{n},Tx^{*}\bigr) \\ & \leq d\bigl(Ty_{n+1},Tx^{*}\bigr)+\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)d\bigl(TSy_{n},Tx^{*} \bigr)+ \alpha _{n} d\bigl(TSb_{n},Tx^{*}\bigr)+ \alpha _{n}^{\prime }d\bigl(\epsilon _{n},Tx^{*} \bigr) \\ & \leq d\bigl(Ty_{n+1},Tx^{*}\bigr)+\bigl(1-\alpha _{n}-\alpha _{n}^{\prime}\bigr)\delta d \bigl(Ty_{n},Tx^{*}\bigr)+ \delta \alpha _{n} d \bigl(Tb_{n},Tx^{*}\bigr)+\alpha _{n}^{\prime} \varepsilon . \end{aligned} $$

(16)

Using estimates (14) and (16) yields

$$ \begin{aligned} \xi _{n}& \leq d\bigl(Ty_{n+1},Tx^{*} \bigr)+ \bigl(\alpha _{n}\beta _{n}(1- \delta )+\delta \alpha _{n}\beta _{n}\gamma _{n}(1-\delta ) \bigr)d \bigl(Ty_{n},Tx^{*}\bigr) \\ &\quad{}+\varepsilon \bigl[\delta ^{2}\alpha _{n}\beta _{n}\gamma _{n}^{ \prime}+\delta \alpha _{n} \beta _{n}^{\prime}+\alpha _{n}^{\prime} \bigr] \\ & \leq d\bigl(Ty_{n+1},Tx^{*}\bigr)+ (1-q_{n} )d \bigl(Ty_{n},Tx^{*}\bigr)+s_{n}. \end{aligned} $$

Again, from Lemma 1.2, hence $\lim_{n\rightarrow \infty}\xi _{n}=0$. Therefore, $\lbrace Tx_{n} \rbrace$ defined by (8) is $(TS)$-stable. □

3 Application to a contraction of integral type

Theorem 3.1

Let K be a nonempty, closed, and convex subset of a complete CAT(0) space X and let $S, T : K \to K$ be an operator satisfying the following condition:

$$ \int _{0}^{d(TSx,TSy)}\mu (t) \,dt\leq h \int _{0}^{\max \lbrace d(Tx,Ty), \frac{d(Tx,TSx)+d(Ty,TSy)}{2},\frac{d(Tx,TSy)+d(Ty,TSx)}{2} \rbrace}\mu (t) \,dt $$

(17)

for all $x, y\in X$ and $0 < h < 1$, where $\mu : [ 0,\infty )\to [ 0, \infty )$ is a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$. Let $\{x_{n}\}$ be defined by (8), where $\lbrace \epsilon _{n} \rbrace$, $\lbrace \epsilon _{n}^{ \prime} \rbrace$, $\lbrace \epsilon _{n}^{\prime \prime} \rbrace$ are bounded sequences in K and $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \gamma _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \alpha _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, and $\lbrace \gamma _{n}^{\prime } \rbrace ^{\infty}_{n=1}$ are appropriate sequences in $[0,1]$, furthermore satisfying the conditions:

• i. $\sum_{n=1}^{\infty} (\alpha _{n}\beta _{n}+\alpha _{n}\beta _{n} \gamma _{n} )=\infty$.

• ii. $\sum_{n=1}^{\infty }\alpha _{n}\beta _{n}\gamma _{n}^{\prime}< \infty$, $\sum_{n=1}^{\infty}\alpha _{n}\beta _{n}^{\prime}<\infty$, and $\sum_{n=1}^{\infty}\alpha _{n}^{\prime}<\infty$.

Then, $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $Tx^{*}$, where $x^{*}$ is the fixed point of S.

Proof

The proof of Theorem 3.1 fulfills Theorem 2.1 by putting $\mu (t) = 1$ over $[0,\infty )$, since the contraction condition of the integral type becomes the general contraction condition (2) that does not involve integrals. This completes the proof. □

Corollary 3.1

Let K be a nonempty, closed, and convex subset of a complete CAT(0) space X and let $S, T : K \to K$ be an operator satisfying the following condition:

$$ \int _{0}^{d(TSx,TSy)}\mu (t) \,dt\leq h \int _{0}^{ \frac{d(Tx,TSx)+d(Ty,TSy)}{2}}\mu (t) \,dt $$

(18)

for all $x, y\in X$ and $0 < h < 1$, where $\mu : [ 0,\infty )\to [ 0, \infty )$ is a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$. Let $\{x_{n}\}$ be defined by (8), where $\lbrace \epsilon _{n} \rbrace$, $\lbrace \epsilon _{n}^{ \prime} \rbrace$, $\lbrace \epsilon _{n}^{\prime \prime} \rbrace$ are bounded sequences in K and $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \gamma _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \alpha _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, and $\lbrace \gamma _{n}^{\prime } \rbrace ^{\infty}_{n=1}$ are appropriate sequences in $[0,1]$, furthermore satisfying the conditions:

• i. $\sum_{n=1}^{\infty} (\alpha _{n}\beta _{n}+\alpha _{n}\beta _{n} \gamma _{n} )=\infty$.

• ii. $\sum_{n=1}^{\infty }\alpha _{n}\beta _{n}\gamma _{n}^{\prime}< \infty$, $\sum_{n=1}^{\infty}\alpha _{n}\beta _{n}^{\prime}<\infty$, and $\sum_{n=1}^{\infty}\alpha _{n}^{\prime}<\infty$.

Then, $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $Tx^{*}$, where $x^{*}$ is the fixed point of S.

Proof

The proof of Corollary 3.1 fulfills Theorem 2.1 by putting $\mu (t) = 1$ over $[0,\infty )$ and

$$ \begin{aligned} &\max \biggl\lbrace d(Tx,Ty),\frac{d(Tx,TSx)+d(Ty,TSy)}{2}, \frac{d(Tx,TSy)+d(Ty,TSx)}{2} \biggr\rbrace \\ &\quad =\frac{d(Tx,TSx)+d(Ty,TSy)}{2}. \end{aligned} $$

This completes the proof. □

Corollary 3.2

Let K be a nonempty, closed, and convex subset of a complete CAT(0) space X and let $S, T : K \to K$ be an operator satisfying the following condition:

$$ \int _{0}^{d(TSx,TSy)}\mu (t) \,dt\leq h \int _{0}^{d(Tx,Ty)}\mu (t) \,dt $$

(19)

for all $x, y\in X$ and $0 < h < 1$, where $\mu : [ 0,\infty )\to [ 0, \infty )$ is a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$. Let $\{x_{n}\}$ be defined by (8), where $\lbrace \epsilon _{n} \rbrace$, $\lbrace \epsilon _{n}^{ \prime} \rbrace$, $\lbrace \epsilon _{n}^{\prime \prime} \rbrace$ are bounded sequences in K and $\lbrace \alpha _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \gamma _{n} \rbrace ^{\infty}_{n=1}$, $\lbrace \alpha _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, $\lbrace \beta _{n}^{\prime } \rbrace ^{\infty}_{n=1}$, and $\lbrace \gamma _{n}^{\prime } \rbrace ^{\infty}_{n=1}$ are appropriate sequences in $[0,1]$, furthermore satisfying the conditions:

• i. $\sum_{n=1}^{\infty} (\alpha _{n}\beta _{n}+\alpha _{n}\beta _{n} \gamma _{n} )=\infty$.

• ii. $\sum_{n=1}^{\infty }\alpha _{n}\beta _{n}\gamma _{n}^{\prime}< \infty$, $\sum_{n=1}^{\infty}\alpha _{n}\beta _{n}^{\prime}<\infty$, and $\sum_{n=1}^{\infty}\alpha _{n}^{\prime}<\infty$.

Then, $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $Tx^{*}$, where $x^{*}$ is the fixed point of S.

Proof

The proof of Corollary 3.2 fulfills Theorem 2.1 by putting $\mu (t) = 1$ over $[0,\infty )$ and

$$ \begin{aligned} &\max \biggl\lbrace d(Tx,Ty),\frac{d(Tx,TSx)+d(Ty,TSy)}{2}, \frac{d(Tx,TSy)+d(Ty,TSx)}{2} \biggr\rbrace \\ &\quad =d(Tx,Ty). \end{aligned} $$

This completes the proof. □

Example 3.1

Let $X=\{0,1,2,3,4,5\}$ and d be the usual metric of reals. Let $S, T : X \to X$ be given by

$$ Sx= \textstyle\begin{cases} 5, &\text{if } x=0, \\ 3,&\text{otherwise} \end{cases}\displaystyle \quad \text{and}\quad Tx=x. $$

Let $\mu : [ 0,\infty )\to [ 0, \infty )$ be a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$.

Let $x = 0$, $y = 1$ and using (17), we have

$$\begin{aligned} 2&= \int _{0}^{d(TSx,TSy)}\mu (t) \,dt \\ &\leq h \int _{0}^{\max \lbrace d(Tx,Ty), \frac{d(Tx,TSx)+d(Ty,TSy)}{2},\frac{d(Tx,TSy)+d(Ty,TSx)}{2} \rbrace}\mu (t) \,dt \\ &= h\max \biggl\lbrace 1,\frac {7}{2}, \frac {7}{2} \biggr\rbrace , \end{aligned}$$

which implies $h\geq \frac {4}{7}$. Now, if we take $0 < h < 1$, then condition (17) is satisfied and $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $T(3)$, where 3 is the fixed point of S.

Example 3.2

Let X be the real line with the usual metric d and suppose $C =[0, 1]$. Let $S, T : X \to X$ be given by

$$ Sx=\frac {x}{2} \quad \text{and}\quad Tx=x^{2}. $$

Let $\mu : [ 0,\infty )\to [ 0, \infty )$ be a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$. Let $x = 0$, $y = 1$ and using (18), we have

$$ \begin{aligned} \frac {1}{4}&= \int _{0}^{d(TSx,TSy)}\mu (t) \,dt \\ &\leq h \int _{0}^{\frac{d(Tx,TSx)+d(Ty,TSy)}{2}}\mu (t) \,dt \\ &=h \biggl( \frac {3}{8} \biggr), \end{aligned} $$

which implies $h\geq \frac {2}{3}$. Now, if we take $0 < h < 1$, then condition (18) is satisfied and $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $T(0)$, where 0 is the fixed point of S.

Example 3.3

Let X be the real line with the usual metric d and suppose $C =[0, 1]$. Let $S, T : X \to X$ be given by

$$ Sx=\frac {x+1}{2} \quad \text{and}\quad Tx=x^{2}. $$

Let $\mu : [ 0,\infty )\to [ 0, \infty )$ be a Lebesgue integrable mapping that is summable (i.e., with finite integral) on each compact subset of $[ 0, \infty )$, nonnegative, and such that for each $\epsilon >0$, $\int _{0}^{\epsilon}\mu (t) \,dt>0$. Let $x = 0$, $y = 1$ and using (19), we have

$$ \begin{aligned} \frac {3}{4}&= \int _{0}^{d(TSx,TSy)}\mu (t) \,dt \\ &\leq h \int _{0}^{d(Tx,Ty)}\mu (t) \,dt \\ &=h (1 ), \end{aligned} $$

which implies $h\geq \frac {3}{4}$. Now, if we take $0 < h < 1$, then condition (19) is satisfied and $\lbrace Tx_{n} \rbrace ^{\infty}_{n=0}$ converges strongly to $T(1)$, where 1 is the fixed point of S.