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# Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces

Fixed Point Theory and Applications20132013:331

https://doi.org/10.1186/1687-1812-2013-331

• Received: 19 February 2013
• Accepted: 13 November 2013
• Published:

## Abstract

The purpose of this paper is to investigate some strong convergence as well as stability results of some iterative procedures for a special class of mappings. First, this class of mappings called weak Jungck $\left(\phi ,\psi \right)$-contractive mappings, which is a generalization of some known classes of Jungck-type contractive mappings, is introduced. Then, using an iterative procedure, we prove the existence of coincidence points for such mappings. Finally, we investigate the strong convergence of some iterative Jungck-type procedures and study stability and almost stability of these procedures. Our results improve and extend many known results in other spaces.

MSC:47H06, 47H10, 54H25, 65D15.

## Keywords

• weak Jungck $\left(\phi ,\psi \right)$-contractive mapping
• iterative procedure
• coincidence point
• stability
• convex b-metric space

## 1 Introduction

Czerwik  initiated the study of multivalued contractions in b-metric spaces.

Definition 1.1 Let X be a set and let $s\ge 1$ be a given real number. A function $d:X×X\to {\mathbb{R}}^{+}$ is said to be a b-metric if and only if for all $x,y,z\in X$ the following conditions are satisfied:
1. (1)

$d\left(x,y\right)=0$ if and only if $x=y$;

2. (2)

$d\left(x,y\right)=d\left(y,x\right)$;

3. (3)

$d\left(x,z\right)\le s\left[d\left(x,y\right)+d\left(y,z\right)\right]$.

Then the pair $\left(X,d\right)$ is called a b-metric space.

It is clear that normed linear spaces, ${l}^{p}$ (or ${L}^{p}$) spaces ($p>0$), ${l}^{\mathrm{\infty }}$ (or ${L}^{\mathrm{\infty }}$) spaces, Hilbert spaces, Banach spaces, hyperbolic spaces, -trees and $CAT\left(0\right)$ spaces are examples of b-metric spaces.

Throughout this paper, ${\mathbb{R}}^{+}$ is the set of nonnegative real numbers and Y is a nonempty arbitrary subset of a b-metric space $\left(X,d\right)$. Moreover, $F\left(T\right)=\left\{x\in Y:Tx=x\right\}$ will be denoted as the set of fixed points of $T:Y\to X$. Approximately, all the concepts and results in metric spaces are extended to the setting of b-metric spaces (for more details, see ).

The first result on stability of T-stable mappings was introduced by Ostrowski  for the Banach contraction principle. Harder and Hicks  proved that the sequence $\left\{{x}_{n}\right\}$ generated by the Picard iterative process in a complete metric space converges strongly to the fixed point of T and is stable with respect to T, provided that T is a Zamfirescu mapping. Rhoades  extended the stability results of  to more general classes of contractive mappings. Ding  constructed the Ishikawa-type iterative process in a convex metric space. He showed that this process converges to the fixed point of T, provided that T belongs in the class which is defined by Rhoades.

A mapping T is said to be a φ-quasinonexpansive if $F\left(T\right)\ne \mathrm{\varnothing }$ and there exists a function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that
$d\left(Tx,p\right)\le \phi \left(d\left(x,p\right)\right)$

for all $x\in X$ and $p\in F\left(T\right)$.

Osilike  considered a mapping T from a metric space X into itself satisfying the condition $d\left(Tx,Ty\right)\le \delta d\left(x,y\right)+Ld\left(x,Tx\right)$ for some $\delta \in \left[0,1\right)$ and $L\ge 0$ for all $x,y\in X$. Furthermore, he extended some of the stability results in . Indeed, he proved T-stability for such a mapping with respect to Picard, Kirk, Mann, and Ishikawa iterations. Thereafter, Olatinwo  improved this concept to the context of multivalued weak contraction for the Jungck iteration in a complete b-metric space. In  this contractive condition was generalized by replacing this condition with $d\left(Tx,Ty\right)\le \delta d\left(x,y\right)+\phi \left(d\left(x,Tx\right)\right)$, where $0\le \delta <1$ and $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is monotone increasing with $\phi \left(0\right)=0$, and some stability results were proved. Recently, Olatinwo  extended this condition to $d\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)+\psi \left(d\left(x,Tx\right)\right)$, where $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a subadditive comparison function and $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is monotone increasing with $\psi \left(0\right)=0$. He studied this contractive condition as a particular case of the class of φ-quasinonexpansive mappings (see ). Also, he proved some stability results as well as strong convergence results for the pair of nonself mappings in a complete metric space.

In 1968, Goebel  generalized the well-known Banach contraction principle by taking a continuous mapping S in place of the identity mapping, where S commuted with T and $T\left(X\right)\subset S\left(X\right)$. In fact, he used two mappings $S,T:Y\to X$ for introducing the contractive condition as follows.

A mapping T is called a Jungck contraction if there exists a real number $0\le \alpha <1$ such that
$\text{(JC)}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le \alpha d\left(Sx,Sy\right)$

for all $x,y\in Y$. In addition, Jungck , using a constructive method, proved the existence of a unique common fixed point of S and T, where $Y=X$.

A mapping T is said to be a Jungck-Zamfirescu contraction (JZ) if there exist real numbers α, β, and γ satisfying $0\le \alpha <1$, $0\le \beta ,\gamma <\frac{1}{2}$ such that for each $x,y\in Y$, one has at least one of the following:

• (z1) $d\left(Tx,Ty\right)\le \alpha d\left(Sx,Sy\right)$;

• (z2) $d\left(Tx,Ty\right)\le \beta \left[d\left(Sx,Tx\right)+d\left(Sy,Ty\right)\right]$;

• (z3) $d\left(Tx,Ty\right)\le \gamma \left[d\left(Sx,Ty\right)+d\left(Sy,Tx\right)\right]$.

A mapping T is said to be a contractive mapping satisfying (JS), (JR) or (JQC) if there exists a constant $q\in \left[0,1\right)$ such that for any $x,y\in Y$,
$\begin{array}{r}\text{(JS)}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le qmax\left\{d\left(Sx,Sy\right),\frac{1}{2}\left[d\left(Sx,Ty\right)+d\left(Sy,Tx\right)\right],d\left(Sx,Tx\right),d\left(Sy,Ty\right)\right\},\\ \text{(JR)}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le qmax\left\{d\left(Sx,Sy\right),\frac{1}{2}\left[d\left(Sx,Tx\right)+d\left(Sy,Ty\right)\right],d\left(Sx,Ty\right),d\left(Sy,Tx\right)\right\},\\ \text{(JQC)}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le qmax\left\{d\left(Sx,Sy\right),d\left(Sx,Tx\right),d\left(Sy,Ty\right),d\left(Sx,Ty\right),d\left(Sy,Tx\right)\right\}.\end{array}$
A mapping T is said to be a weak Jungck contraction if there exist two constants $a\in \left[0,1\right)$ and $L\ge 0$ such that for all $x,y\in Y$,
$\text{(WJC)}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le ad\left(Sx,Sy\right)+Ld\left(Sx,Tx\right).$

It is worth mentioning that a Jungck-Zamfirescu mapping is a (JR) mapping. In [, Proposition 3.3], a comparison of the above contractive conditions is established as follows.

Proposition 1.2
1. (i)

(JC) (JS) (JQC);

2. (ii)

(JC) (JR) (JQC);

3. (iii)

(JS) and (JR) are independent;

4. (iv)

(JR) (WJC);

5. (v)

(JS) and (WJC) are independent;

6. (vi)

(JQC) and (WJC) are independent;

7. (vii)

reverse implications of (i), (ii), and (iv) are not true.

In this paper, a special class of mappings called a weak Jungck $\left(\phi ,\psi \right)$-contraction is introduced, and it is shown that it contains other known classes of Jungck-type contractive mappings. Then, using a Jungck-Picard iterative procedure, we investigate the existence of coincidence points and the uniqueness of the coincidence value of weak Jungck $\left(\phi ,\psi \right)$-contractive mappings. Also, some strong convergence as well as stability results of some Jungck-type iterative procedures (such as Jungck-Ishikawa etc.) are studied. These results play a crucial role in numerical computations for approximation of coincidence values of two nonlinear mappings.

## 2 Preliminary

In , Berinde introduced the concepts of comparison function and $\left(c\right)$-comparison function with respect to the function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. A function φ is called a comparison function if it satisfies the following:

• (i φ ) φ is monotone increasing, i.e., ${t}_{1}<{t}_{2}⇒\phi \left({t}_{1}\right)\le \phi \left({t}_{2}\right)$;

• (ii φ ) The sequence $\left\{{\phi }^{n}\left(t\right)\right\}\to 0$ for all $t\in {\mathbb{R}}^{+}$, where ${\phi }^{n}$ stands for the n th iterate of φ.

If φ satisfies (i φ ) and

(iii φ ) ${\sum }_{n=0}^{\mathrm{\infty }}{\phi }^{n}\left(t\right)$ converges for all $t\in {\mathbb{R}}^{+}$,

then φ is said to be a $\left(c\right)$-comparison function.

Several results regarding comparison functions can be found in  and . Referring to  and , we have:
1. 1.

Any $\left(c\right)$-comparison function is a comparison function;

2. 2.

Any comparison function satisfies $\phi \left(0\right)=0$ and $\phi \left(t\right) for all $t>0$;

3. 3.

Any subadditive comparison function is continuous;

4. 4.

Condition (iii φ ) is equivalent to the following one:

There exist ${k}_{0}\in \mathbb{N}$, $\alpha \in \left(0,1\right)$ and a convergent series of nonnegative terms $\sum {v}_{n}$ such that
${\phi }^{k+1}\left(t\right)\le \alpha {\phi }^{k}\left(t\right)+{v}_{k}$

holds for all $k\ge {k}_{0}$ and any $t\in {\mathbb{R}}^{+}$.

Berinde  expanded the concept of $\left(c\right)$-comparison functions in b-metric spaces to s-comparison functions as follows.

Definition 2.1 Let $s\ge 1$ be a real number. A mapping $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is called an s-comparison function if it satisfies (i φ ) and

(iv φ ) There exist ${k}_{0}\in \mathbb{N}$, $\alpha \in \left(0,1\right)$, and a convergent series of nonnegative terms $\sum {v}_{n}$ such that
${s}^{k+1}{\phi }^{k+1}\left(t\right)\le \alpha {s}^{k}{\phi }^{k}\left(t\right)+{v}_{k}$

holds for all $k\ge {k}_{0}$ and any $t\in {\mathbb{R}}^{+}$.

Applying results 4 and 1 regarding comparison functions, it is easy to conclude that every s-comparison function is a comparison function.

In the sequel, some lemmas which are useful to obtain our main results are stated.

Lemma 2.2 ()

Let $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a comparison function, and let ${\epsilon }_{n}$ be a sequence of positive numbers such that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$. Then
$\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^{n}{\phi }^{n-k}\left({\epsilon }_{k}\right)=0.$

Lemma 2.3 ()

Let $\left\{{u}_{n}\right\}$, $\left\{{\alpha }_{n}\right\}$, and $\left\{{\epsilon }_{n}\right\}$ be sequences of nonnegative real numbers satisfying the inequality
${u}_{n+1}\le {\alpha }_{n}{u}_{n}+{\epsilon }_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$

If ${\alpha }_{n}\ge 1$, ${\sum }_{n=1}^{\mathrm{\infty }}\left({\alpha }_{n}-1\right)<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{u}_{n}$ exists.

Lemma 2.4 Suppose that $\left\{{u}_{n}\right\}$ and $\left\{{\epsilon }_{n}\right\}$ are two sequences of nonnegative numbers such that
${u}_{n+1}\le \phi \left({u}_{n}\right)+{\epsilon }_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(2.1)

where φ is a subadditive comparison function. If ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$, then ${lim}_{n\to \mathrm{\infty }}{u}_{n}=0$.

Proof The monotone increasing and the subadditivity of φ together with inequality (2.1) imply that
$\begin{array}{rcl}{u}_{n+1}& \le & \phi \left({u}_{n}\right)+{\epsilon }_{n}\\ \le & \phi \left(\phi \left({u}_{n-1}\right)+{\epsilon }_{n-1}\right)+{\epsilon }_{n}\\ \le & {\phi }^{2}\left({u}_{n-1}\right)+\phi \left({\epsilon }_{n-1}\right)+{\epsilon }_{n}\\ ⋮\\ \le & {\phi }^{n+1}\left({u}_{0}\right)+\sum _{i=0}^{n}{\phi }^{n-i}\left({\epsilon }_{i}\right),\end{array}$
(2.2)

where ${\phi }^{0}=I$ (identity mapping). Moreover, since any comparison function satisfies (ii φ ), hence ${lim}_{n\to \mathrm{\infty }}{\phi }^{n+1}\left({u}_{0}\right)=0$. Also, we have ${lim}_{n\to \mathrm{\infty }}{\sum }_{i=0}^{n}{\phi }^{n-i}\left({\epsilon }_{i}\right)=0$ from Lemma 2.2. Thus, inequality (2.2) implies that ${lim}_{n\to \mathrm{\infty }}{u}_{n}=0$. □

Lemma 2.5 Let $\left\{{\alpha }_{n}\right\}$ be a real sequence in $\left[0,1\right]$, let $\left\{{\epsilon }_{n}\right\}$ be a sequence of positive numbers such that ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}$ converges, and let $\left\{{u}_{n}\right\}$ be a sequence of nonnegative numbers such that
${u}_{n+1}\le \left(1-{\alpha }_{n}\right){u}_{n}+{\alpha }_{n}\phi \left({u}_{n}\right)+{\epsilon }_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(2.3)

where φ is a convex subadditive comparison function. If ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{u}_{n}=0$.

Proof Since $\phi \left(t\right)\le t$ for all $t\ge 0$, using a straightforward induction and (2.3), one can obtain
$\begin{array}{rcl}{u}_{n+p+1}& \le & \left(1-{\alpha }_{n+p}\right){u}_{n+p}+{\alpha }_{n+p}\phi \left({u}_{n+p}\right)+{\epsilon }_{n+p}\\ \le & \left(1-{\alpha }_{n+p}\right)\left[\left(1-{\alpha }_{n+p-1}\right){u}_{n+p-1}+{\alpha }_{n+p-1}\phi \left({u}_{n+p-1}\right)+{\epsilon }_{n+p-1}\right]\\ +{\alpha }_{n+p}\left[\left(1-{\alpha }_{n+p-1}\right)\phi \left({u}_{n+p-1}\right)+{\alpha }_{n+p-1}{\phi }^{2}\left({u}_{n+p-1}\right)+\phi \left({\epsilon }_{n+p-1}\right)\right]+{\epsilon }_{n+p}\\ \le & \left(1-{\alpha }_{n+p}\right)\left(1-{\alpha }_{n+p-1}\right){u}_{n+p-1}+\left[1-\left(1-{\alpha }_{n+p}\right)\left(1-{\alpha }_{n+p-1}\right)\right]\phi \left({u}_{n+p-1}\right)\\ +{\epsilon }_{n+p-1}+{\epsilon }_{n+p}\\ ⋮\\ \le & \left(\prod _{i=n}^{n+p}\left(1-{\alpha }_{i}\right)\right){u}_{n}+\left(1-\prod _{i=n}^{n+p}\left(1-{\alpha }_{i}\right)\right)\phi \left({u}_{n}\right)+\sum _{i=n}^{n+p}{\epsilon }_{i}\\ \le & \left(\prod _{i=n}^{n+p}\left(1-{\alpha }_{i}\right)\right){u}_{n}+\phi \left({u}_{n}\right)+\sum _{i=n}^{n+p}{\epsilon }_{i}\\ \le & exp\left(-\sum _{i=n}^{n+p}{\alpha }_{i}\right){u}_{n}+\phi \left({u}_{n}\right)+\sum _{i=n}^{n+p}{\epsilon }_{i}\end{array}$
for all $n,p\in \mathbb{N}$. Now, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ yields that ${lim}_{p\to \mathrm{\infty }}exp\left(-{\sum }_{i=n}^{n+p}{\alpha }_{i}\right)=0$. Then
$\underset{p\to \mathrm{\infty }}{lim sup}{u}_{p}=\underset{p\to \mathrm{\infty }}{lim sup}{u}_{n+p+1}\le \phi \left({u}_{n}\right)+\sum _{i=n}^{\mathrm{\infty }}{\epsilon }_{i},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(2.4)
which implies that
$\underset{p\to \mathrm{\infty }}{lim sup}{u}_{p}\le \underset{n\to \mathrm{\infty }}{lim inf}\phi \left({u}_{n}\right)\le \underset{n\to \mathrm{\infty }}{lim inf}{u}_{n}.$

Therefore, there exists $u\in {\mathbb{R}}^{+}$ such that ${lim}_{n\to \mathrm{\infty }}{u}_{n}=u$. Assume that $u>0$. Since φ is continuous and ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}$ converges, letting $n\to \mathrm{\infty }$ in (2.4), we get that $u\le \phi \left(u\right), which is a contradiction. Hence $u=0$ and the desired conclusion follows. □

## 3 Weak Jungck $\left(\phi ,\psi \right)$-contractive mappings

In this section, the class of weak Jungck $\left(\phi ,\psi \right)$-contractive mappings which contains the class of Jungck φ-quasinonexpansive mappings is studied. Furthermore, it is showed that this class includes the various classes of contractive mappings which is introduced in Section 1.

Definition 3.1 Let Y be an arbitrary subset of a b-metric space $\left(X,d\right)$, and let $S,T:Y\to X$ be such that z is a coincidence point of S and T, i.e., $Sz=Tz=p$. We say that T is a Jungck φ-quasinonexpansive mapping with respect to S if there exists a function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that
$d\left(Tx,p\right)\le \phi \left(d\left(Sx,p\right)\right)$

for all $x\in Y$.

The above definition was used in  when S is the identity mapping on $Y=X$.

Definition 3.2 Let Y be an arbitrary subset of a b-metric space $\left(X,d\right)$ and $S,T:Y\to X$. A mapping T is said to be a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping with respect to S if there exist an s-comparison function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and a monotone increasing function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with upper semicontinuity from the right at $\psi \left(0\right)=0$ such that for all $x,y\in Y$,
$d\left(Tx,Ty\right)\le \phi \left(d\left(Sx,Sy\right)\right)+\psi \left(min\left\{d\left(Sx,Tx\right),d\left(Sx,Ty\right)\right\}\right).$
(3.1)

It is obvious that any weak Jungck $\left(\phi ,\psi \right)$-contraction is also Jungck φ-quasinonexpansive, but the reverse is not true. The next example illustrates this matter.

Example 3.1 Let $S,T:\left[0,1\right]\to \left[0,1\right]$ be given by $Sx=x$ and
$Tx=\left\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}
where $\left[0,1\right]$ is endowed with the usual metric. It is easy to see that T satisfies the following property:
$d\left(Tx,p\right)\le \phi \left(d\left(x,p\right)\right)$
for all $x\in \left[0,1\right]$, $p\in F\left(T\right)=\left\{0\right\}$, and $\phi \left(x\right)=x$. But T is not a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping. Indeed, if there exist a 1-comparison function φ and a monotone increasing function ψ with upper semicontinuity from the right at $\psi \left(0\right)=0$ such that for all $x,y\in \left[0,1\right]$,
$d\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)+\psi \left(min\left\{d\left(x,Tx\right),d\left(x,Ty\right)\right\}\right),$

then, taking $x=\frac{1}{2}$, $y=1$, we have $\frac{1}{2}\le \phi \left(\frac{1}{2}\right)+\psi \left(0\right)$. This shows that the class of φ-quasinonexpansive mappings properly includes the class of weak Jungck $\left(\phi ,\psi \right)$-contractive mappings.

In what follows, we prove that all the mappings introduced in Section 1 are in the class of weak Jungck $\left(\phi ,\psi \right)$-contractive mappings. It is clear that every Jungck contractive mapping is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping with $\phi \left(t\right)=\alpha t$ and $\psi \left(t\right)=0$, where $0\le \alpha <\frac{1}{s}$.

Proposition 3.3 Let $\left(X,d\right)$ be a b-metric space with parameter s, let Y be an arbitrary subset of X, and let $S,T:Y\to X$. If T is a Jungck-Zamfirescu contraction (JZ), then T is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping if $\alpha <\frac{1}{s}$ and $\beta ,\gamma <\frac{1}{s\left(1+{s}^{2}\right)}$. Moreover, it is a weak Jungck $\left(\phi ,\psi \right)$-contraction with $\phi \left(t\right)=max\left\{\alpha ,\frac{\beta {s}^{2}}{1-\beta s},\frac{\gamma {s}^{2}}{1-\gamma s}\right\}t$ and $\psi \left(t\right)=max\left\{\frac{\beta \left(1+{s}^{2}\right)}{1-\beta s},\frac{\gamma \left(1+{s}^{2}\right)}{1-\gamma s}\right\}t$ for all $t\in {\mathbb{R}}^{+}$.

Proof If $min\left\{d\left(Sx,Tx\right),d\left(Sx,Ty\right)\right\}=d\left(Sx,Tx\right)$, then for all $x,y\in Y$,
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & \beta \left[d\left(Sx,Tx\right)+d\left(Sy,Ty\right)\right]\\ \le & \beta d\left(Sx,Tx\right)+\beta s\left[d\left(Sy,Tx\right)+d\left(Tx,Ty\right)\right]\\ \le & \beta d\left(Sx,Tx\right)+\beta {s}^{2}\left[d\left(Sy,Sx\right)+d\left(Sx,Tx\right)\right]+\beta sd\left(Tx,Ty\right),\end{array}$
which implies that
$d\left(Tx,Ty\right)\le \frac{\beta {s}^{2}}{1-\beta s}d\left(Sx,Sy\right)+\frac{\beta \left(1+{s}^{2}\right)}{1-\beta s}d\left(Sx,Tx\right).$
Also
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & \gamma \left[d\left(Sx,Ty\right)+d\left(Sy,Tx\right)\right]\\ \le & \gamma s\left[d\left(Sx,Tx\right)+d\left(Tx,Ty\right)\right]+\gamma s\left[d\left(Sy,Sx\right)+d\left(Sx,Tx\right)\right]\end{array}$
yields that
$d\left(Tx,Ty\right)\le \frac{\gamma s}{1-\gamma s}d\left(Sx,Sy\right)+\frac{2\gamma s}{1-\gamma s}d\left(Sx,Tx\right).$
Similarly, if $min\left\{d\left(Sx,Tx\right),d\left(Sx,Ty\right)\right\}=d\left(Sx,Ty\right)$, then for all $x,y\in Y$,
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & \beta \left[d\left(Sx,Tx\right)+d\left(Sy,Ty\right)\right]\\ \le & \beta s\left[d\left(Sx,Ty\right)+d\left(Ty,Tx\right)\right]+\beta s\left[d\left(Sy,Sx\right)+d\left(Sx,Ty\right)\right],\end{array}$
thus
$d\left(Tx,Ty\right)\le \frac{\beta s}{1-\beta s}d\left(Sx,Sy\right)+\frac{2\beta s}{1-\beta s}d\left(Sx,Ty\right).$
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & \gamma \left[d\left(Sx,Ty\right)+d\left(Sy,Tx\right)\right]\\ \le & \gamma d\left(Sx,Ty\right)+\gamma s\left[d\left(Sy,Ty\right)+d\left(Ty,Tx\right)\right]\\ \le & \gamma d\left(Sx,Ty\right)+\gamma {s}^{2}\left[d\left(Sy,Sx\right)+d\left(Sx,Ty\right)\right]+\gamma sd\left(Tx,Ty\right)\end{array}$
implies that
$d\left(Tx,Ty\right)\le \frac{\gamma {s}^{2}}{1-\gamma s}d\left(Sx,Sy\right)+\frac{\gamma \left(1+{s}^{2}\right)}{1-\gamma s}d\left(Sx,Ty\right).$
Now, let
$\phi \left(t\right):=max\left\{\alpha ,\frac{\beta s}{1-\beta s},\frac{\beta {s}^{2}}{1-\beta s},\frac{\gamma s}{1-\gamma s},\frac{\gamma {s}^{2}}{1-\gamma s}\right\}t=max\left\{\alpha ,\frac{\beta {s}^{2}}{1-\beta s},\frac{\gamma {s}^{2}}{1-\gamma s}\right\}t$
and
$\begin{array}{rl}\psi \left(t\right)& :=max\left\{0,\frac{2\beta s}{1-\beta s},\frac{\beta \left(1+{s}^{2}\right)}{1-\beta s},\frac{2\gamma s}{1-\gamma s},\frac{\gamma \left(1+{s}^{2}\right)}{1-\gamma s}\right\}t\\ =max\left\{\frac{\beta \left(1+{s}^{2}\right)}{1-\beta s},\frac{\gamma \left(1+{s}^{2}\right)}{1-\gamma s}\right\}t\end{array}$

for all $t\in {\mathbb{R}}^{+}$. It is clear that φ is an s-comparison function, where $\alpha <\frac{1}{s}$ and $\beta ,\gamma <\frac{1}{s\left(1+{s}^{2}\right)}$ and ψ is a monotone increasing function which is continuous from the right at $\psi \left(0\right)=0$. □

The following result shows that this fact is still true for a more general class of mappings.

Proposition 3.4 Let X, Y and $S,T:Y\to X$ be as in the above proposition. If T satisfies (JS), then T is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping, provided that $q<\frac{1}{s\left(1+{s}^{2}\right)}$. Furthermore, it is a weak Jungck $\left(\phi ,\psi \right)$-contraction with $\phi \left(t\right)=\frac{q{s}^{2}}{1-qs}t$ and $\psi \left(t\right)=\frac{q{s}^{2}}{1-qs}t$ for all $t\in {\mathbb{R}}^{+}$.

Proof If $min\left\{d\left(Sx,Tx\right),d\left(Sx,Ty\right)\right\}=d\left(Sx,Tx\right)$, then according to the inequality
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & qd\left(Sy,Ty\right)\le qs\left[d\left(Sy,Tx\right)+d\left(Tx,Ty\right)\right]\\ =& q{s}^{2}\left[d\left(Sy,Sx\right)+d\left(Sx,Tx\right)\right]+qsd\left(Tx,Ty\right),\end{array}$
we have
$d\left(Tx,Ty\right)\le \frac{q{s}^{2}}{1-qs}d\left(Sx,Sy\right)+\frac{q{s}^{2}}{1-qs}d\left(Sx,Tx\right)$
for all $x,y\in Y$. Moreover,
$\begin{array}{rl}d\left(Tx,Ty\right)& \le \frac{q}{2}\left[d\left(Sx,Ty\right)+d\left(Tx,Sy\right)\right]\\ \le \frac{qs}{2}\left[d\left(Sx,Tx\right)+d\left(Tx,Ty\right)\right]+\frac{qs}{2}\left[d\left(Tx,Sx\right)+d\left(Sx,Sy\right)\right]\end{array}$
implies that
$d\left(Tx,Ty\right)\le \frac{qs}{2-qs}d\left(Sx,Sy\right)+\frac{2qs}{2-qs}d\left(Sx,Tx\right).$
On the other hand, if $min\left\{d\left(Sx,Tx\right),d\left(Sx,Ty\right)\right\}=d\left(Sx,Ty\right)$, then
$d\left(Tx,Ty\right)\le qd\left(Sx,Tx\right)\le qs\left[d\left(Sx,Ty\right)+d\left(Ty,Tx\right)\right]$
yields that
$d\left(Tx,Ty\right)\le \frac{qs}{1-qs}d\left(Sx,Ty\right)$
for all $x,y\in Y$. Also
$d\left(Tx,Ty\right)\le qd\left(Sy,Ty\right)\le qs\left[d\left(Sy,Sx\right)+d\left(Sx,Ty\right)\right].$
Moreover,
$\begin{array}{rcl}d\left(Tx,Ty\right)& \le & \frac{q}{2}\left[d\left(Sx,Ty\right)+d\left(Tx,Sy\right)\right]\\ \le & \frac{q}{2}d\left(Sx,Ty\right)+\frac{qs}{2}\left[d\left(Tx,Ty\right)+d\left(Ty,Sy\right)\right]\\ \le & \frac{q}{2}d\left(Sx,Ty\right)+\frac{qs}{2}d\left(Tx,Ty\right)+\frac{q{s}^{2}}{2}\left[d\left(Ty,Sx\right)+d\left(Sx,Sy\right)\right]\end{array}$
yields that
$d\left(Tx,Ty\right)\le \frac{q{s}^{2}}{2-qs}d\left(Sx,Sy\right)+\frac{q\left(1+{s}^{2}\right)}{2-qs}d\left(Sx,Ty\right).$
Now, we take
$\phi \left(t\right):=max\left\{0,q,qs,\frac{qs}{2-qs},\frac{q{s}^{2}}{1-qs},\frac{q{s}^{2}}{2-qs}\right\}t=\frac{q{s}^{2}}{1-qs}t$
and
$\psi \left(t\right):=max\left\{0,q,qs,\frac{qs}{1-qs},\frac{2qs}{2-qs},\frac{q{s}^{2}}{1-qs},\frac{q\left(1+{s}^{2}\right)}{2-qs}\right\}t=\frac{q{s}^{2}}{1-qs}t$

for all $t\in {\mathbb{R}}^{+}$. It shows that φ is an s-comparison function provided that $q<\frac{1}{s\left(1+{s}^{2}\right)}$ and ψ is a monotone increasing function which is continuous at $\psi \left(0\right)=0$. □

Similar arguments illustrate that every (JR) mapping is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping, provided that $q<\frac{1}{s\left(1+{s}^{2}\right)}$. In fact, it is a weak Jungck $\left(\phi ,\psi \right)$-contraction with $\phi \left(t\right)=\psi \left(t\right)=\frac{q{s}^{2}}{1-qs}t$ for all $t\in {\mathbb{R}}^{+}$. Also, every (JQC) mapping is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping with $\phi \left(t\right)=\psi \left(t\right)=\frac{q{s}^{2}}{1-qs}t$ for all $t\in {\mathbb{R}}^{+}$, provided that $q<\frac{1}{s\left(1+{s}^{2}\right)}$.

## 4 Convergence results

In 1970, Takahashi  defined a convex structure on metric spaces. In this section a version of the convexity notion in b-metric spaces is stated. Then, using some Jungck-type iterative procedures, we prove the existence of coincidence points as well as the strong convergence theorems for the weak Jungck $\left(\phi ,\psi \right)$-contractive mappings.

Definition 4.1 Let $\left(X,d\right)$ be a b-metric space. A mapping $W:X×X×\left[0,1\right]\to X$ is said to be a convex structure on X if for each $\left(x,y,\lambda \right)\in X×X×\left[0,1\right]$ and $z\in X$,
$d\left(z,W\left(x,y,\lambda \right)\right)\le \lambda d\left(z,x\right)+\left(1-\lambda \right)d\left(z,y\right).$
(4.1)

A b-metric space X equipped with the convex structure W is called a convex b-metric space, which is denoted by $\left(X,d,W\right)$.

Example 4.1 The space ${l}^{p}$ ($p>1$) consisting of all the sequences $\left\{{x}_{n}\right\}$ of real numbers for which ${\sum }_{n=1}^{\mathrm{\infty }}{|{x}_{n}|}^{p}$ converges, with the function $d:{l}^{p}×{l}^{p}\to \mathbb{R}$ given by
$d\left(x,y\right)=\sum _{n=1}^{\mathrm{\infty }}{|{x}_{n}-{y}_{n}|}^{p},$

for all $x,y\in {l}^{p}$, is a b-metric space with $s={2}^{p-1}>1$. Also, regarding the convexity of $f\left(t\right)={t}^{p}$, we obtain that $d\left(z,\lambda x+\left(1-\lambda \right)y\right)\le \lambda d\left(z,x\right)+\left(1-\lambda \right)d\left(z,y\right)$ for all $z\in {l}^{p}$, that is, ${l}^{p}$ ($p>1$) is a convex b-metric space with $W\left(x,y,\lambda \right)=\lambda x+\left(1-\lambda \right)y$. (In a similar way, the space ${L}^{p}$ ($p>1$) is a convex b-metric space.)

Now, the iterative procedures in a convex b-metric space are ready to be illustrated. From now on, it is assumed that $\left(X,d\right)$ is a b-metric space (resp. $\left(X,d,W\right)$ is a convex b-metric space) with parameter s and that $S,T:Y\to X$ are two nonself mappings on a subset Y of X such that $T\left(Y\right)\subset S\left(Y\right)$, where $S\left(Y\right)$ is a complete subspace of X.

Let $\left\{{x}_{n}\right\}$ be the sequence generated by an iterative procedure involving the mapping T and S, that is,
$S{x}_{n+1}=f\left(T,{x}_{n}\right),\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$
(4.2)

where ${x}_{0}\in Y$ is the initial approximation and f is a function.

In the sequel, we discuss several special cases of (4.2):
1. 1.

The Jungck iteration (or Jungck-Picard iteration) is given from (4.2) for $f\left(T,{x}_{n}\right)=T{x}_{n}$. This process was essentially introduced by Jungck  and it reduces to the Picard iterative process, when S is the identity mapping on $Y=X$;

2. 2.
The Jungck-Krasnoselskij iteration is defined by (4.2) with
$f\left(T,{x}_{n}\right)=W\left(S{x}_{n},T{x}_{n},\lambda \right),$
(4.3)

where $0\le \lambda \le 1$;

3. 3.
The Jungck-Mann iteration is stated by (4.2) with
$f\left(T,{x}_{n}\right)=W\left(S{x}_{n},T{x}_{n},{\alpha }_{n}\right),$
(4.4)

where $\left\{{\alpha }_{n}\right\}$ is a sequence of real numbers such that $0\le {\alpha }_{n}\le 1$;

4. 4.
The Jungck-Ishikawa iteration is introduced by (4.2) with
$\begin{array}{r}f\left(T,{x}_{n}\right)=W\left(S{x}_{n},T{y}_{n},{\alpha }_{n}\right),\\ S{y}_{n}=W\left(S{x}_{n},T{x}_{n},{\beta }_{n}\right),\end{array}$
(4.5)

where $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are two sequences of real numbers such that $0\le {\alpha }_{n},{\beta }_{n}\le 1$.

It is worth noting that Olatinwo and Postolache  used the above iterative procedures in the setting of convex metric spaces.

Theorem 4.2 Suppose that $\left(X,d\right)$ is a b-metric space, and let $S,T:Y\to X$ be such that T is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping. Then S and T have a coincidence point. Moreover, for any ${x}_{0}\in Y$, the sequence $\left\{S{x}_{n}\right\}$ generated by the Jungck-Picard iterative process converges strongly to the coincidence value.

Proof First, we prove that S and T have at least one coincidence point in Y. To do this, let $\left\{{x}_{n}\right\}$ be the Jungck-Picard iterative process defined by $S{x}_{n+1}=T{x}_{n}$ and ${x}_{0}\in Y$. Taking $x={x}_{n}$ and $y={x}_{n-1}$ in (3.1), we obtain
$d\left(T{x}_{n},T{x}_{n-1}\right)\le \phi \left(d\left(S{x}_{n},S{x}_{n-1}\right)\right)+\psi \left(min\left\{d\left(S{x}_{n},T{x}_{n}\right),d\left(S{x}_{n},T{x}_{n-1}\right)\right\}\right),$
which implies that
$d\left(S{x}_{n+1},S{x}_{n}\right)\le \phi \left(d\left(S{x}_{n},S{x}_{n-1}\right)\right),$
and, inductively,
$d\left(S{x}_{n+1},S{x}_{n}\right)\le {\phi }^{n}\left(d\left(S{x}_{1},S{x}_{0}\right)\right).$
Therefore
$\begin{array}{rcl}d\left(S{x}_{n+p},S{x}_{n}\right)& \le & {s}^{p-1}d\left(S{x}_{n+p},S{x}_{n+p-1}\right)+{s}^{p-1}d\left(S{x}_{n+p-1},S{x}_{n+p-2}\right)\\ +\cdots +{s}^{2}d\left(S{x}_{n+2},S{x}_{n+1}\right)+sd\left(S{x}_{n+1},S{x}_{n}\right)\\ \le & {s}^{p}{\phi }^{n+p-1}\left(d\left(S{x}_{1},S{x}_{0}\right)\right)+{s}^{p-1}{\phi }^{n+p-2}\left(d\left(S{x}_{1},S{x}_{0}\right)\right)\\ +\cdots +{s}^{2}{\phi }^{n+1}\left(d\left(S{x}_{1},S{x}_{0}\right)\right)+s{\phi }^{n}\left(d\left(S{x}_{1},S{x}_{0}\right)\right)\\ =& \sum _{i=1}^{p}{s}^{i}{\phi }^{n+i-1}\left(d\left(S{x}_{1},S{x}_{0}\right)\right)\\ =& \frac{1}{{s}^{n-1}}\sum _{i=n}^{n+p-1}{s}^{i}{\phi }^{i}\left(d\left(S{x}_{1},S{x}_{0}\right)\right),\phantom{\rule{1em}{0ex}}n,p\in \mathbb{N},p\ne 0.\end{array}$
Since ${\sum }_{i=1}^{\mathrm{\infty }}{s}^{i}{\phi }^{i}\left(t\right)<\mathrm{\infty }$ for all $t\in {\mathbb{R}}^{+}$, $\left\{S{x}_{n}\right\}$ is a Cauchy sequence. Also, $S\left(Y\right)$ is complete, so $\left\{S{x}_{n}\right\}$ has a limit in $S\left(Y\right)$, that is, there exists $z\in {S}^{-1}p$ such that $p={lim}_{n\to \mathrm{\infty }}S{x}_{n}$. Hence, $Sz=p$ and
$\begin{array}{rcl}d\left(Sz,Tz\right)& \le & sd\left(Sz,S{x}_{n+1}\right)+sd\left(S{x}_{n+1},Tz\right)=sd\left(S{x}_{n+1},Sz\right)+sd\left(Tz,T{x}_{n}\right)\\ \le & sd\left(S{x}_{n+1},Sz\right)+s\phi \left(d\left(Sz,S{x}_{n}\right)\right)+s\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{x}_{n}\right)\right\}\right)\\ \le & sd\left(S{x}_{n+1},p\right)+sd\left(S{x}_{n},p\right)+s\psi \left(d\left(S{x}_{n+1},p\right)\right).\end{array}$

Taking the upper limit in the above inequality, we obtain $d\left(Sz,Tz\right)=0$. Hence, $Tz=Sz=p$, i.e., z is a coincidence point.

Now, we show that S and T have a unique coincidence value. Assume that S and T have two coincidence values $p,q\in X$ such that $p\ne q$. Then there exist ${z}_{1},{z}_{2}\in Y$ such that $S{z}_{1}=T{z}_{1}=p$ and $S{z}_{2}=T{z}_{2}=q$. Thus, we conclude that
$d\left(p,q\right)=d\left(T{z}_{1},T{z}_{2}\right)\le \phi \left(d\left(S{z}_{1},S{z}_{2}\right)\right)+\psi \left(min\left\{d\left(S{z}_{1},T{z}_{1}\right),d\left(S{z}_{1},T{z}_{2}\right)\right\}\right)=\phi \left(d\left(p,q\right)\right).$

From our assumptions on φ, it is impossible unless $d\left(p,q\right)=0$, that is, $p=q$, which is a contradiction. □

Using Proposition 3.3, one can conclude that the above theorem is a significant extension of [, Theorem 3.1] and [, Theorem 3.1].

Theorem 4.3 Let $\left(X,d,W\right)$ be a convex b-metric, and let $S,T:Y\to X$ be such that T is a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping such that φ is a convex subadditive function. Let $\left\{{\alpha }_{n}\right\}$ be a real sequence in $\left[0,1\right]$ such that ${\sum }_{n=0}^{\mathrm{\infty }}\left(1-{\alpha }_{n}\right)=\mathrm{\infty }$. Then, for any ${x}_{0}\in Y$, the sequence $\left\{S{x}_{n}\right\}$ defined by the Jungck-Ishikawa iterative process converges strongly to the coincidence value of S and T.

Proof Theorem 4.3 states the existence of coincidence points in Y and one can obtain the uniqueness of coincidence value in a similar way. We now show that the Jungck-Ishikawa iteration given by $S{x}_{n+1}=W\left(S{x}_{n},T{y}_{n},{\alpha }_{n}\right)$, where $S{y}_{n}=W\left(S{x}_{n},T{x}_{n},{\beta }_{n}\right)$ for each ${x}_{0}\in Y$, converges to $p=Sz=Tz$, where z is a coincidence point of S and T. Using (3.1), we have
$\begin{array}{rcl}d\left(S{x}_{n+1},p\right)& \le & {\alpha }_{n}d\left(S{x}_{n},p\right)+\left(1-{\alpha }_{n}\right)d\left(T{y}_{n},p\right)\\ \le & {\alpha }_{n}d\left(S{x}_{n},p\right)+\left(1-{\alpha }_{n}\right)\\ ×\left[\phi \left(d\left(Sz,S{y}_{n}\right)\right)+\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{y}_{n}\right)\right\}\right)\right]\\ =& {\alpha }_{n}d\left(S{x}_{n},p\right)+\left(1-{\alpha }_{n}\right)\phi \left(d\left(S{y}_{n},p\right)\right),\end{array}$
(4.6)
and
$\begin{array}{rcl}d\left(S{y}_{n},p\right)& \le & {\beta }_{n}d\left(S{x}_{n},p\right)+\left(1-{\beta }_{n}\right)d\left(T{x}_{n},p\right)\\ \le & {\beta }_{n}d\left(S{x}_{n},p\right)+\left(1-{\beta }_{n}\right)\\ ×\left[\phi \left(d\left(Sz,S{x}_{n}\right)\right)+\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{x}_{n}\right)\right\}\right)\right]\\ \le & {\beta }_{n}d\left(S{x}_{n},p\right)+\left(1-{\beta }_{n}\right)\phi \left(d\left(S{x}_{n},p\right)\right)\\ \le & {\beta }_{n}d\left(S{x}_{n},p\right)+\left(1-{\beta }_{n}\right)d\left(S{x}_{n},p\right)\\ =& d\left(S{x}_{n},p\right).\end{array}$
(4.7)
Substituting (4.7) in (4.6), it follows that
$d\left(S{x}_{n+1},p\right)\le {\alpha }_{n}d\left(S{x}_{n},p\right)+\left(1-{\alpha }_{n}\right)\phi \left(d\left(S{x}_{n},p\right)\right),\phantom{\rule{1em}{0ex}}n=0,1,2,\dots .$

Since φ is a convex subadditive comparison function, we have the desired result from Lemma 2.5. □

Remark 4.1
1. (1)

Based on Theorem 4.3, it is clear that the Jungck-Mann iterative process as well as the Jungck-Krasnoselskij iterative process converge;

2. (2)

In normed linear spaces, the generalization of this theorem is stated by Olatinwo [9, 24];

3. (3)

In Hilbert spaces, assuming that $q<\frac{1}{s\left(1+{s}^{2}\right)}$ in (JQC), Theorem 4.3 is an extension of the results in .

The following example shows that condition (3.1) in Theorem 4.3 is necessary.

Example 4.2 Let $S,T:\left[0,1\right]\to \left[0,1\right]$ be given by $Sx=x$ and
$Tx=\left\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}

where $\left[0,1\right]$ is endowed with the usual metric. Let ${x}_{0}\in \left(\frac{1}{2},1\right]$ and ${x}_{n+1}=\lambda {x}_{n}+\left(1-\lambda \right)T{x}_{n}$ for $n=0,1,2,\dots$ . Then ${x}_{n+1}={\lambda }^{n+1}{x}_{0}+\frac{1-{\lambda }^{n+1}}{2}$, which implies that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\frac{1}{2}$ if $0\le \lambda <1$ and ${lim}_{n\to \mathrm{\infty }}{x}_{n}={x}_{0}\ne 0$ if $\lambda =1$. Therefore, the Krasnoselskij iteration associated to T does not converge strongly to the coincidence value.

## 5 Stability results

This section is devoted entirely to the stability of some various iterative procedures in b-metric spaces. This concept was first proposed by Ostrowski  in metric spaces. Then, Czerwik et al. [26, 27] extended Ostrowski’s classical theorem in the setting of b-metric spaces. In addition, Singh et al.  introduced the stability and almost stability of Jungck-type iterative procedures in metric spaces. Below, we state these concepts in convex b-metric spaces.

Definition 5.1 Let $\left(X,d,W\right)$ be a convex b-metric space, let Y be a subset of X, and let $S,T:Y\to Y$ be such that $T\left(Y\right)\subset S\left(Y\right)$. For any ${x}_{0}\in Y$, let the sequence $\left\{S{x}_{n}\right\}$, generated by iterative procedure (4.2), converges to p. Also, let $\left\{S{y}_{n}\right\}\subset X$ be an arbitrary sequence and let ${\epsilon }_{n}=d\left(S{y}_{n+1},f\left(T,{y}_{n}\right)\right)$, $n=0,1,2,\dots$ . Then
1. (i)

Iterative procedure (4.2) will be called $\left(S,T\right)$-stable if ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$ implies that ${lim}_{n\to \mathrm{\infty }}S{y}_{n}=p$.

2. (ii)

Iterative procedure (4.2) will be called almost $\left(S,T\right)$-stable if ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$ implies that ${lim}_{n\to \mathrm{\infty }}S{y}_{n}=p$.

The above definition reduces to the concept of the stability of iterative procedure due to Harder and Hicks  when S is the identity mapping on $Y=X$.

Example 5.1 Let $S,T:\left[0,1\right]\to \left[0,\frac{3}{2}\right]$ be given by $Sx={x}^{2}+\frac{x}{2}$ and
$Tx=\left\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}

where $\left[0,\frac{3}{2}\right]$ is endowed with the usual metric. Let ${x}_{0}\in \left[0,1\right]$ and $S{x}_{n+1}=T{x}_{n}$ for $n=0,1,2,\dots$ . If $0\le {x}_{0}\le \frac{1}{2}$, then $S{x}_{n+1}=T{x}_{n}=0$, and if $\frac{1}{2}<{x}_{0}\le 1$, we have $S{x}_{1}=T{x}_{0}=\frac{1}{2}$ and $S{x}_{n+1}=T{x}_{n}=0$ for all $n\in \mathbb{N}$. Thus ${lim}_{n\to \mathrm{\infty }}S{x}_{n}=0=S\left(0\right)=T\left(0\right)$; i.e., the Picard iteration converges strongly to the coincidence value. But the Picard iteration is not $\left(S,T\right)$-stable. Indeed, take the sequence $\left\{{y}_{n}\right\}$ given by ${y}_{n}=\frac{n+2}{2n}$, $n\in \mathbb{N}$. One can see easily that the sequence $\left\{S{y}_{n}\right\}$ does not converge to the coincidence value, while ${\epsilon }_{n}=d\left(S{y}_{n+1},T{y}_{n}\right)=\frac{1}{{\left(n+1\right)}^{2}}+\frac{3}{2\left(n+1\right)}\to 0$ as $n\to \mathrm{\infty }$.

Our next theorem is presented for a pair of mappings on a nonempty subset with values in b-metric spaces under a condition more general than the condition stated by Singh and Prasad [, Theorem 4.2]. Further, this theorem reduces the condition ${s}^{2}q<1$ to the condition $sq<1$.

Theorem 5.2 Let $\left(X,d\right)$ be a b-metric space and T be a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping such that φ is subadditive. For ${x}_{0}\in Y$, let $\left\{S{x}_{n}\right\}$ be the Picard iterative process defined by $S{x}_{n+1}=T{x}_{n}$. Then the Jungck-Picard iteration is $\left(S,T\right)$-stable.

Proof Note that, by Theorem 4.2, there exists a coincidence point $z\in Y$ such that $\left\{S{x}_{n}\right\}$ converges to $p=Sz=Tz$. Suppose that $\left\{S{y}_{n}\right\}\subset X$ and define ${\epsilon }_{n}=d\left(S{y}_{n+1},f\left(T,{y}_{n}\right)\right)$, where $f\left(T,{y}_{n}\right)=T{y}_{n}$. Assume that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$. Then we have
$\begin{array}{rcl}d\left(S{y}_{n+1},p\right)& \le & s\left[d\left(S{y}_{n+1},T{y}_{n}\right)+d\left(T{y}_{n},p\right)\right]\\ \le & s{\epsilon }_{n}+s\left[\phi \left(d\left(Sz,S{y}_{n}\right)\right)+\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{y}_{n}\right)\right\}\right)\right]\\ =& s{\epsilon }_{n}+s\phi \left(d\left(S{y}_{n},p\right)\right).\end{array}$

Since φ is a subadditive s-comparison function, we get that is a subadditive comparison function. Therefore, Lemma 2.4 yields that ${lim}_{n\to \mathrm{\infty }}d\left(S{y}_{n},p\right)=0$, that is, ${lim}_{n\to \mathrm{\infty }}S{y}_{n}=p$. □

Remark 5.1 Theorem 5.2 is a generalization of Theorem 3.2 of Singh and Alam , Theorem 3.4 of Singh et al. , Theorems 4.1 and 4.2 of Singh and Prasad , Theorem 1 of Osilike , Theorem 2 of Berinde , Theorem 2.1 of Bosede and Rhoades  as well as Corollary 2 of Qing and Rhoades .

The following example shows that the Ishikawa iterative process is not $\left(S,T\right)$-stable.

Example 5.2 Let $S,T:\left[0,1\right]\to \mathbb{R}$ be given by $Sx=x$ and $Tx=\frac{-x}{2}$, where is again endowed with the usual metric. Then T is a weak Jungck $\left(\frac{I}{2},0\right)$-contraction. Let $\left\{{x}_{n}\right\}$ be a sequence generated by the Ishikawa iterative process with ${\alpha }_{n}={\beta }_{n}=1-\frac{1}{n+1}$ and ${x}_{0}\in \left[0,1\right]$. Then
$\left\{\begin{array}{l}{z}_{n}=S{z}_{n}={\beta }_{n}S{x}_{n}+\left(1-{\beta }_{n}\right)T{x}_{n}=\left(1-\frac{1}{n+1}\right){x}_{n}+\frac{1}{n+1}T{x}_{n}=\left(1-\frac{3}{2\left(n+1\right)}\right){x}_{n},\\ {x}_{n+1}=S{x}_{n+1}={\alpha }_{n}S{x}_{n}+\left(1-{\alpha }_{n}\right)T{z}_{n}=\left(1-\frac{1}{n+1}\right){x}_{n}+\frac{1}{n+1}T{z}_{n}=\left(1-\frac{3}{2\left(n+1\right)}+\frac{3}{4{\left(n+1\right)}^{2}}\right){x}_{n}.\end{array}$

Suppose that ${t}_{n}=\frac{3}{2\left(n+1\right)}-\frac{3}{4{\left(n+1\right)}^{2}}$. As ${t}_{n}\in \left(0,1\right)$ and ${\sum }_{n=0}^{\mathrm{\infty }}{t}_{n}=\mathrm{\infty }$, Lemma 2 of  implies that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=0=S\left(0\right)=T\left(0\right)$ (the unique coincidence value of S and T).

To prove the fact that the Ishikawa iteration is not $\left(S,T\right)$-stable, we use the sequence $\left\{{y}_{n}\right\}$ given by ${y}_{n}=\frac{n+1}{n+2}$. Then
$\begin{array}{rcl}{\epsilon }_{n}& =& |{y}_{n+1}-f\left(T,{y}_{n}\right)|\\ =& |{y}_{n+1}-\left(1-\frac{3}{2\left(n+1\right)}+\frac{3}{4{\left(n+1\right)}^{2}}\right){y}_{n}|\\ =& |\frac{n+2}{n+3}-\left(1-\frac{3}{2\left(n+1\right)}+\frac{3}{4{\left(n+1\right)}^{2}}\right)\frac{n+1}{n+2}|\\ =& \frac{6{n}^{2}+25n+13}{4\left(n+1\right)\left(n+2\right)\left(n+3\right)}.\end{array}$

It is clear that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}=\mathrm{\infty }$, while ${lim}_{n\to \mathrm{\infty }}{y}_{n}=1$. Therefore, the Ishikawa iterative procedure is not $\left(S,T\right)$-stable, but it is almost $\left(S,T\right)$-stable. (The almost $\left(S,T\right)$-stability is shown in the following.)

The following theorem states that Jungck-Mann iterative and Jungck-Ishikawa iterative process are almost $\left(S,T\right)$-stable provided that ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}<\mathrm{\infty }$.

Theorem 5.3 Let $\left(X,d,W\right)$ be a convex b-metric space and let T be a weak Jungck $\left(\phi ,\psi \right)$-contractive mapping such that φ is a convex subadditive function. Let $\left\{{\alpha }_{n}\right\}$ be a real sequence in $\left[0,1\right]$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}<\mathrm{\infty }$. For ${x}_{0}\in Y$, let $\left\{S{x}_{n}\right\}$ be the Ishikawa iterative process given by (4.5). Then the Jungck-Ishikawa iteration is almost $\left(S,T\right)$-stable.

Proof In view of Theorem 4.3, there exists a coincidence point $z\in Y$ such that $\left\{S{x}_{n}\right\}$ converges to $p=Sz=Tz$. Suppose that $\left\{S{y}_{n}\right\}\subset X$, ${\epsilon }_{n}=d\left(S{y}_{n+1},W\left(S{y}_{n},T{u}_{n},{\alpha }_{n}\right)\right)$, $n=0,1,2,\dots$ , where $S{u}_{n}=W\left(S{y}_{n},T{y}_{n},{\beta }_{n}\right)$. Assume that ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$. Then
$\begin{array}{rcl}d\left(S{y}_{n+1},p\right)& \le & s\left[d\left(S{y}_{n+1},W\left(S{y}_{n},T{u}_{n},{\alpha }_{n}\right)\right)+d\left(W\left(S{y}_{n},T{u}_{n},{\alpha }_{n}\right),p\right)\right]\\ \le & s{\epsilon }_{n}+s\left[{\alpha }_{n}d\left(S{y}_{n},p\right)+\left(1-{\alpha }_{n}\right)d\left(T{u}_{n},p\right)\right]\\ \le & s{\epsilon }_{n}+s{\alpha }_{n}d\left(S{y}_{n},p\right)+s\left(1-{\alpha }_{n}\right)\\ ×\left[\phi \left(d\left(Sz,S{u}_{n}\right)\right)+\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{u}_{n}\right)\right\}\right)\right]\\ \le & s{\epsilon }_{n}+s{\alpha }_{n}d\left(S{y}_{n},p\right)+s\left(1-{\alpha }_{n}\right)\phi \left(d\left(S{u}_{n},p\right)\right),\end{array}$
(5.1)
and
$\begin{array}{rcl}d\left(S{u}_{n},p\right)& \le & {\beta }_{n}d\left(S{y}_{n},p\right)+\left(1-{\beta }_{n}\right)d\left(T{y}_{n},p\right)\\ \le & {\beta }_{n}d\left(S{y}_{n},p\right)+\left(1-{\beta }_{n}\right)\\ ×\left[\phi \left(d\left(Sz,S{y}_{n}\right)\right)+\psi \left(min\left\{d\left(Sz,Tz\right),d\left(Sz,T{y}_{n}\right)\right\}\right)\right]\\ \le & {\beta }_{n}d\left(S{y}_{n},p\right)+\left(1-{\beta }_{n}\right)\phi \left(d\left(S{y}_{n},p\right)\right)\\ \le & {\beta }_{n}d\left(S{y}_{n},p\right)+\left(1-{\beta }_{n}\right)d\left(S{y}_{n},p\right)\\ =& d\left(S{y}_{n},p\right).\end{array}$
(5.2)
From (5.1) and (5.2), we conclude that
$d\left(S{y}_{n+1},p\right)\le s{\epsilon }_{n}+s{\alpha }_{n}d\left(S{y}_{n},p\right)+s\left(1-{\alpha }_{n}\right)\phi \left(d\left(S{y}_{n},p\right)\right).$
(5.3)
Since φ is an s-comparison function, is a comparison function. Thus, inequality (5.3) implies that
$d\left(S{y}_{n+1},p\right)\le s{\epsilon }_{n}+s{\alpha }_{n}d\left(S{y}_{n},p\right)+\left(1-{\alpha }_{n}\right)d\left(S{y}_{n},p\right)=\left(1+\left(s-1\right){\alpha }_{n}\right)d\left(S{y}_{n},p\right)+s{\epsilon }_{n}.$

Now, according to Lemma 2.3, ${lim}_{n\to \mathrm{\infty }}d\left(S{y}_{n},p\right)$ exists. Therefore, there exists $u\in {\mathbb{R}}^{+}$ such that ${lim}_{n\to \mathrm{\infty }}d\left(S{y}_{n},p\right)=u$. Assume that $u>0$. Since is a subadditive comparison function, φ is continuous and $s\phi \left(t\right) for all $t>0$. Then, letting $n\to \mathrm{\infty }$ in (5.3), we get $u\le s\phi \left(u\right), which is a contradiction. Hence, $u=0$ and this completes the proof. □

In a similar way, using Lemma 1 of  in place of Lemma 2.3 in the previous proof, by omitting the condition $\sum {\alpha }_{n}<\mathrm{\infty }$, one can prove that Theorem 5.3 holds in convex metric spaces. This indicates that the Ishikawa iterative process given Example 5.2 is almost $\left(S,T\right)$-stable.

## Declarations

### Acknowledgements

The authors express their gratitude to the referees for reading this paper carefully, providing valuable suggestions and comments, which improved the contents of this paper.

## Authors’ Affiliations

(1)
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

## References 