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

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 $(\phi ,\psi )$-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.

1 Introduction

Czerwik [1] 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 $d:X\times X\to {\mathbb{R}}^{+}$ is said to be a b-metric if and only if for all $x,y,z\in X$ the following conditions are satisfied:

• (1) $d(x,y)=0$ if and only if $x=y$;

• (2) $d(x,y)=d(y,x)$;

• (3) $d(x,z)\le s[d(x,y)+d(y,z)]$.

Then the pair $(X,d)$ 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, ℝ-trees and $CAT(0)$ 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 $(X,d)$. Moreover, $F(T)=\{x\in Y:Tx=x\}$ 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 [1]).

The first result on stability of T-stable mappings was introduced by Ostrowski [2] for the Banach contraction principle. Harder and Hicks [3] proved that the sequence $\{x_n\}$ 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 [4] extended the stability results of [3] to more general classes of contractive mappings. Ding [5] 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 $F(T)\ne \mathrm{\varnothing }$ and there exists a function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that

$$d(Tx,p)\le \phi (d(x,p))$$

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

Osilike [6] considered a mapping T from a metric space X into itself satisfying the condition $d(Tx,Ty)\le \delta d(x,y)+Ld(x,Tx)$ for some $\delta \in [0,1)$ and $L\ge 0$ for all $x,y\in X$. Furthermore, he extended some of the stability results in [4]. Indeed, he proved T-stability for such a mapping with respect to Picard, Kirk, Mann, and Ishikawa iterations. Thereafter, Olatinwo [7] improved this concept to the context of multivalued weak contraction for the Jungck iteration in a complete b-metric space. In [8] this contractive condition was generalized by replacing this condition with $d(Tx,Ty)\le \delta d(x,y)+\phi (d(x,Tx))$, where $0\le \delta <1$ and $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is monotone increasing with $\phi (0)=0$, and some stability results were proved. Recently, Olatinwo [9] extended this condition to $d(Tx,Ty)\le \phi (d(x,y))+\psi (d(x,Tx))$, where $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a subadditive comparison function and $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is monotone increasing with $\psi (0)=0$. He studied this contractive condition as a particular case of the class of φ-quasinonexpansive mappings (see [10]). 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 [11] 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 $T(X)\subset S(X)$. 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

$$\text{(JC)}d(Tx,Ty)\le \alpha d(Sx,Sy)$$

for all $x,y\in Y$. In addition, Jungck [12], 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) $d(Tx,Ty)\le \alpha d(Sx,Sy)$;

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

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

A mapping T is said to be a contractive mapping satisfying (JS), (JR) or (JQC) if there exists a constant $q\in [0,1)$ such that for any $x,y\in Y$,

$$\begin{array}{r}\text{(JS)}d(Tx,Ty)\le qmax\{d(Sx,Sy),\frac{1}{2}[d(Sx,Ty)+d(Sy,Tx)],d(Sx,Tx),d(Sy,Ty)\},\\ \text{(JR)}d(Tx,Ty)\le qmax\{d(Sx,Sy),\frac{1}{2}[d(Sx,Tx)+d(Sy,Ty)],d(Sx,Ty),d(Sy,Tx)\},\\ \text{(JQC)}d(Tx,Ty)\le qmax\{d(Sx,Sy),d(Sx,Tx),d(Sy,Ty),d(Sx,Ty),d(Sy,Tx)\}.\end{array}$$

A mapping T is said to be a weak Jungck contraction if there exist two constants $a\in [0,1)$ and $L\ge 0$ such that for all $x,y\in Y$,

$$\text{(WJC)}d(Tx,Ty)\le ad(Sx,Sy)+Ld(Sx,Tx).$$

It is worth mentioning that a Jungck-Zamfirescu mapping is a (JR) mapping. In [[13], Proposition 3.3], a comparison of the above contractive conditions is established as follows.

Proposition 1.2

• (i) (JC) ⇒ (JS) ⇒ (JQC);

• (ii) (JC) ⇒ (JR) ⇒ (JQC);

• (iii) (JS) and (JR) are independent;

• (iv) (JR) ⇒ (WJC);

• (v) (JS) and (WJC) are independent;

• (vi) (JQC) and (WJC) are independent;

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

In this paper, a special class of mappings called a weak Jungck $(\phi ,\psi )$-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 $(\phi ,\psi )$-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 [14], Berinde introduced the concepts of comparison function and $(c)$-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., $t_1<t_2\Rightarrow \phi (t_1)\le \phi (t_2)$;

• (ii _φ ) The sequence $\{{\phi }^n(t)\}\to 0$ 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 ) ${\sum }_{n=0}^{\mathrm{\infty }}{\phi }^n(t)$ converges for all $t\in {\mathbb{R}}^{+}$,

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

Several results regarding comparison functions can be found in [14] and [15]. Referring to [14] and [15], we have:

• 1. Any $(c)$-comparison function is a comparison function;

• 2. Any comparison function satisfies $\phi (0)=0$ and $\phi (t)<t$ for all $t>0$;

• 3. Any subadditive comparison function is continuous;

• 4. Condition (iii _φ ) is equivalent to the following one:

There exist $k_0\in \mathbb{N}$, $\alpha \in (0,1)$ and a convergent series of nonnegative terms $\sum v_n$ such that

$${\phi }^{k+1}(t)\le \alpha {\phi }^k(t)+v_k$$

holds for all $k\ge k_0$ and any $t\in {\mathbb{R}}^{+}$.

Berinde [16] expanded the concept of $(c)$-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 $k_0\in \mathbb{N}$, $\alpha \in (0,1)$, and a convergent series of nonnegative terms $\sum v_n$ such that

$$s^{k+1}{\phi }^{k+1}(t)\le \alpha s^k{\phi }^k(t)+v_k$$

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 ([17])

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

$$\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^n{\phi }^{n-k}({\epsilon }_k)=0.$$

Lemma 2.3 ([18])

Let $\{u_n\}$, $\{{\alpha }_n\}$, and $\{{\epsilon }_n\}$ be sequences of nonnegative real numbers satisfying the inequality

$$u_{n+1}\le {\alpha }_n{u}_n+{\epsilon }_n,n\in \mathbb{N}.$$

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

Lemma 2.4 Suppose that $\{u_n\}$ and $\{{\epsilon }_n\}$ are two sequences of nonnegative numbers such that

$$u_{n+1}\le \phi (u_n)+{\epsilon }_n,n=0,1,2,\dots ,$$

(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

$$\begin{array}{rcl}{u}_{n+1}& \le & \phi (u_n)+{\epsilon }_n\\ \le & \phi (\phi (u_{n-1})+{\epsilon }_{n-1})+{\epsilon }_n\\ \le & {\phi }^2(u_{n-1})+\phi ({\epsilon }_{n-1})+{\epsilon }_n\\ ⋮\\ \le & {\phi }^{n+1}(u_0)+\sum _{i=0}^n{\phi }^{n-i}({\epsilon }_i),\end{array}$$

(2.2)

where ${\phi }^0=I$ (identity mapping). Moreover, since any comparison function satisfies (ii _φ ), hence ${lim}_{n\to \mathrm{\infty }}{\phi }^{n+1}(u_0)=0$. Also, we have ${lim}_{n\to \mathrm{\infty }}{\sum }_{i=0}^n{\phi }^{n-i}({\epsilon }_i)=0$ from Lemma 2.2. Thus, inequality (2.2) implies that ${lim}_{n\to \mathrm{\infty }}{u}_n=0$. □

Lemma 2.5 Let $\{{\alpha }_n\}$ be a real sequence in $[0,1]$, let $\{{\epsilon }_n\}$ be a sequence of positive numbers such that ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_n$ converges, and let $\{u_n\}$ be a sequence of nonnegative numbers such that

$$u_{n+1}\le (1-{\alpha }_n)u_n+{\alpha }_n\phi (u_n)+{\epsilon }_n,n=0,1,2,\dots ,$$

(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 $\phi (t)\le t$ for all $t\ge 0$, using a straightforward induction and (2.3), one can obtain

$$\begin{array}{rcl}{u}_{n+p+1}& \le & (1-{\alpha }_{n+p})u_{n+p}+{\alpha }_{n+p}\phi (u_{n+p})+{\epsilon }_{n+p}\\ \le & (1-{\alpha }_{n+p})[(1-{\alpha }_{n+p-1})u_{n+p-1}+{\alpha }_{n+p-1}\phi (u_{n+p-1})+{\epsilon }_{n+p-1}]\\ +{\alpha }_{n+p}[(1-{\alpha }_{n+p-1})\phi (u_{n+p-1})+{\alpha }_{n+p-1}{\phi }^2(u_{n+p-1})+\phi ({\epsilon }_{n+p-1})]+{\epsilon }_{n+p}\\ \le & (1-{\alpha }_{n+p})(1-{\alpha }_{n+p-1})u_{n+p-1}+[1-(1-{\alpha }_{n+p})(1-{\alpha }_{n+p-1})]\phi (u_{n+p-1})\\ +{\epsilon }_{n+p-1}+{\epsilon }_{n+p}\\ ⋮\\ \le & (\prod _{i=n}^{n+p}(1-{\alpha }_i))u_n+(1-\prod _{i=n}^{n+p}(1-{\alpha }_i))\phi (u_n)+\sum _{i=n}^{n+p}{\epsilon }_i\\ \le & (\prod _{i=n}^{n+p}(1-{\alpha }_i))u_n+\phi (u_n)+\sum _{i=n}^{n+p}{\epsilon }_i\\ \le & exp(-\sum _{i=n}^{n+p}{\alpha }_i)u_n+\phi (u_n)+\sum _{i=n}^{n+p}{\epsilon }_i\end{array}$$

for all $n,p\in \mathbb{N}$. Now, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$ yields that ${lim}_{p\to \mathrm{\infty }}exp(-{\sum }_{i=n}^{n+p}{\alpha }_i)=0$. Then

$$\underset{p\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{u}_p=\underset{p\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{u}_{n+p+1}\le \phi (u_n)+\sum _{i=n}^{\mathrm{\infty }}{\epsilon }_i,n=0,1,2,\dots ,$$

(2.4)

which implies that

$$\underset{p\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{u}_p\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\phi (u_n)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{u}_n.$$

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 $u\le \phi (u)<u$, which is a contradiction. Hence $u=0$ and the desired conclusion follows. □

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

In this section, the class of weak Jungck $(\phi ,\psi )$-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 $(X,d)$, 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

$$d(Tx,p)\le \phi (d(Sx,p))$$

for all $x\in Y$.

The above definition was used in [19] when S is the identity mapping on $Y=X$.

Definition 3.2 Let Y be an arbitrary subset of a b-metric space $(X,d)$ and $S,T:Y\to X$. A mapping T is said to be a weak Jungck $(\phi ,\psi )$-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 $\psi (0)=0$ such that for all $x,y\in Y$,

$$d(Tx,Ty)\le \phi (d(Sx,Sy))+\psi (min\{d(Sx,Tx),d(Sx,Ty)\}).$$

(3.1)

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

Example 3.1 Let $S,T:[0,1]\to [0,1]$ be given by $Sx=x$ and

$$Tx=\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}<x\le 1,\end{array}$$

where $[0,1]$ is endowed with the usual metric. It is easy to see that T satisfies the following property:

$$d(Tx,p)\le \phi (d(x,p))$$

for all $x\in [0,1]$, $p\in F(T)=\{0\}$, and $\phi (x)=x$. But T is not a weak Jungck $(\phi ,\psi )$-contractive mapping. Indeed, if there exist a 1-comparison function φ and a monotone increasing function ψ with upper semicontinuity from the right at $\psi (0)=0$ such that for all $x,y\in [0,1]$,

$$d(Tx,Ty)\le \phi (d(x,y))+\psi (min\{d(x,Tx),d(x,Ty)\}),$$

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

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

Proposition 3.3 Let $(X,d)$ 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 $(\phi ,\psi )$-contractive mapping if $\alpha <\frac{1}{s}$ and $\beta ,\gamma <\frac{1}{s(1+s^2)}$. Moreover, it is a weak Jungck $(\phi ,\psi )$-contraction with $\phi (t)=max\{\alpha ,\frac{\beta s^2}{1-\beta s},\frac{\gamma s^2}{1-\gamma s}\}t$ and $\psi (t)=max\{\frac{\beta (1+s^2)}{1-\beta s},\frac{\gamma (1+s^2)}{1-\gamma s}\}t$ for all $t\in {\mathbb{R}}^{+}$.

Proof If $min\{d(Sx,Tx),d(Sx,Ty)\}=d(Sx,Tx)$, then for all $x,y\in Y$,

$$\begin{array}{rcl}d(Tx,Ty)& \le & \beta [d(Sx,Tx)+d(Sy,Ty)]\\ \le & \beta d(Sx,Tx)+\beta s[d(Sy,Tx)+d(Tx,Ty)]\\ \le & \beta d(Sx,Tx)+\beta s^2[d(Sy,Sx)+d(Sx,Tx)]+\beta sd(Tx,Ty),\end{array}$$

which implies that

$$d(Tx,Ty)\le \frac{\beta s^2}{1-\beta s}d(Sx,Sy)+\frac{\beta (1+s^2)}{1-\beta s}d(Sx,Tx).$$

Also

$$\begin{array}{rcl}d(Tx,Ty)& \le & \gamma [d(Sx,Ty)+d(Sy,Tx)]\\ \le & \gamma s[d(Sx,Tx)+d(Tx,Ty)]+\gamma s[d(Sy,Sx)+d(Sx,Tx)]\end{array}$$

yields that

$$d(Tx,Ty)\le \frac{\gamma s}{1-\gamma s}d(Sx,Sy)+\frac{2\gamma s}{1-\gamma s}d(Sx,Tx).$$

Similarly, if $min\{d(Sx,Tx),d(Sx,Ty)\}=d(Sx,Ty)$, then for all $x,y\in Y$,

$$\begin{array}{rcl}d(Tx,Ty)& \le & \beta [d(Sx,Tx)+d(Sy,Ty)]\\ \le & \beta s[d(Sx,Ty)+d(Ty,Tx)]+\beta s[d(Sy,Sx)+d(Sx,Ty)],\end{array}$$

thus

$$d(Tx,Ty)\le \frac{\beta s}{1-\beta s}d(Sx,Sy)+\frac{2\beta s}{1-\beta s}d(Sx,Ty).$$

In addition,

$$\begin{array}{rcl}d(Tx,Ty)& \le & \gamma [d(Sx,Ty)+d(Sy,Tx)]\\ \le & \gamma d(Sx,Ty)+\gamma s[d(Sy,Ty)+d(Ty,Tx)]\\ \le & \gamma d(Sx,Ty)+\gamma s^2[d(Sy,Sx)+d(Sx,Ty)]+\gamma sd(Tx,Ty)\end{array}$$

implies that

$$d(Tx,Ty)\le \frac{\gamma s^2}{1-\gamma s}d(Sx,Sy)+\frac{\gamma (1+s^2)}{1-\gamma s}d(Sx,Ty).$$

Now, let

$$\phi (t):=max\{\alpha ,\frac{\beta s}{1-\beta s},\frac{\beta s^2}{1-\beta s},\frac{\gamma s}{1-\gamma s},\frac{\gamma s^2}{1-\gamma s}\}t=max\{\alpha ,\frac{\beta s^2}{1-\beta s},\frac{\gamma s^2}{1-\gamma s}\}t$$

and

$$\begin{array}{rl}\psi (t)& :=max\{0,\frac{2\beta s}{1-\beta s},\frac{\beta (1+s^2)}{1-\beta s},\frac{2\gamma s}{1-\gamma s},\frac{\gamma (1+s^2)}{1-\gamma s}\}t\\ =max\{\frac{\beta (1+s^2)}{1-\beta s},\frac{\gamma (1+s^2)}{1-\gamma s}\}t\end{array}$$

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

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 $(\phi ,\psi )$-contractive mapping, provided that $q<\frac{1}{s(1+s^2)}$. Furthermore, it is a weak Jungck $(\phi ,\psi )$-contraction with $\phi (t)=\frac{q{s}^2}{1-qs}t$ and $\psi (t)=\frac{q{s}^2}{1-qs}t$ for all $t\in {\mathbb{R}}^{+}$.

Proof If $min\{d(Sx,Tx),d(Sx,Ty)\}=d(Sx,Tx)$, then according to the inequality

$$\begin{array}{rcl}d(Tx,Ty)& \le & qd(Sy,Ty)\le qs[d(Sy,Tx)+d(Tx,Ty)]\\ =& q{s}^2[d(Sy,Sx)+d(Sx,Tx)]+qsd(Tx,Ty),\end{array}$$

we have

$$d(Tx,Ty)\le \frac{q{s}^2}{1-qs}d(Sx,Sy)+\frac{q{s}^2}{1-qs}d(Sx,Tx)$$

for all $x,y\in Y$. Moreover,

$$\begin{array}{rl}d(Tx,Ty)& \le \frac{q}{2}[d(Sx,Ty)+d(Tx,Sy)]\\ \le \frac{qs}{2}[d(Sx,Tx)+d(Tx,Ty)]+\frac{qs}{2}[d(Tx,Sx)+d(Sx,Sy)]\end{array}$$

implies that

$$d(Tx,Ty)\le \frac{qs}{2-qs}d(Sx,Sy)+\frac{2qs}{2-qs}d(Sx,Tx).$$

On the other hand, if $min\{d(Sx,Tx),d(Sx,Ty)\}=d(Sx,Ty)$, then

$$d(Tx,Ty)\le qd(Sx,Tx)\le qs[d(Sx,Ty)+d(Ty,Tx)]$$

yields that

$$d(Tx,Ty)\le \frac{qs}{1-qs}d(Sx,Ty)$$

for all $x,y\in Y$. Also

$$d(Tx,Ty)\le qd(Sy,Ty)\le qs[d(Sy,Sx)+d(Sx,Ty)].$$

Moreover,

$$\begin{array}{rcl}d(Tx,Ty)& \le & \frac{q}{2}[d(Sx,Ty)+d(Tx,Sy)]\\ \le & \frac{q}{2}d(Sx,Ty)+\frac{qs}{2}[d(Tx,Ty)+d(Ty,Sy)]\\ \le & \frac{q}{2}d(Sx,Ty)+\frac{qs}{2}d(Tx,Ty)+\frac{q{s}^2}{2}[d(Ty,Sx)+d(Sx,Sy)]\end{array}$$

yields that

$$d(Tx,Ty)\le \frac{q{s}^2}{2-qs}d(Sx,Sy)+\frac{q(1+s^2)}{2-qs}d(Sx,Ty).$$

Now, we take

$$\phi (t):=max\{0,q,qs,\frac{qs}{2-qs},\frac{q{s}^2}{1-qs},\frac{q{s}^2}{2-qs}\}t=\frac{q{s}^2}{1-qs}t$$

and

$$\psi (t):=max\{0,q,qs,\frac{qs}{1-qs},\frac{2qs}{2-qs},\frac{q{s}^2}{1-qs},\frac{q(1+s^2)}{2-qs}\}t=\frac{q{s}^2}{1-qs}t$$

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

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

4 Convergence results

In 1970, Takahashi [20] 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 $(\phi ,\psi )$-contractive mappings.

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

$$d(z,W(x,y,\lambda ))\le \lambda d(z,x)+(1-\lambda )d(z,y).$$

(4.1)

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

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

$$d(x,y)=\sum _{n=1}^{\mathrm{\infty }}{|x_n-y_n|}^p,$$

for all $x,y\in l^p$, is a b-metric space with $s=2^{p-1}>1$. Also, regarding the convexity of $f(t)=t^p$, we obtain that $d(z,\lambda x+(1-\lambda )y)\le \lambda d(z,x)+(1-\lambda )d(z,y)$ for all $z\in l^p$, that is, $l^p$ ($p>1$) is a convex b-metric space with $W(x,y,\lambda )=\lambda x+(1-\lambda )y$. (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 $(X,d)$ is a b-metric space (resp. $(X,d,W)$ 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 $T(Y)\subset S(Y)$, where $S(Y)$ is a complete subspace of X.

Let $\{x_n\}$ be the sequence generated by an iterative procedure involving the mapping T and S, that is,

$$S{x}_{n+1}=f(T,x_n),n=0,1,2,\dots ,$$

(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. The Jungck iteration (or Jungck-Picard iteration) is given from (4.2) for $f(T,x_n)=T{x}_n$. This process was essentially introduced by Jungck [12] and it reduces to the Picard iterative process, when S is the identity mapping on $Y=X$;

• 2. The Jungck-Krasnoselskij iteration is defined by (4.2) with

  $$f(T,x_n)=W(S{x}_n,T{x}_n,\lambda ),$$

  (4.3)

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

• 3. The Jungck-Mann iteration is stated by (4.2) with

  $$f(T,x_n)=W(S{x}_n,T{x}_n,{\alpha }_n),$$

  (4.4)

  where $\{{\alpha }_n\}$ is a sequence of real numbers such that $0\le {\alpha }_n\le 1$;

• 4. The Jungck-Ishikawa iteration is introduced by (4.2) with

  $$\begin{array}{r}f(T,x_n)=W(S{x}_n,T{y}_n,{\alpha }_n),\\ S{y}_n=W(S{x}_n,T{x}_n,{\beta }_n),\end{array}$$

  (4.5)

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

It is worth noting that Olatinwo and Postolache [21] used the above iterative procedures in the setting of convex metric spaces.

Theorem 4.2 Suppose that $(X,d)$ is a b-metric space, and let $S,T:Y\to X$ be such that T is a weak Jungck $(\phi ,\psi )$-contractive mapping. Then S and T have a coincidence point. Moreover, for any $x_0\in Y$, the sequence $\{S{x}_n\}$ 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 $\{x_n\}$ 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

$$d(T{x}_n,T{x}_{n-1})\le \phi (d(S{x}_n,S{x}_{n-1}))+\psi (min\{d(S{x}_n,T{x}_n),d(S{x}_n,T{x}_{n-1})\}),$$

which implies that

$$d(S{x}_{n+1},S{x}_n)\le \phi (d(S{x}_n,S{x}_{n-1})),$$

and, inductively,

$$d(S{x}_{n+1},S{x}_n)\le {\phi }^n(d(S{x}_1,S{x}_0)).$$

Therefore

$$\begin{array}{rcl}d(S{x}_{n+p},S{x}_n)& \le & s^{p-1}d(S{x}_{n+p},S{x}_{n+p-1})+s^{p-1}d(S{x}_{n+p-1},S{x}_{n+p-2})\\ +\cdots +s^{2}d(S{x}_{n+2},S{x}_{n+1})+sd(S{x}_{n+1},S{x}_n)\\ \le & s^p{\phi }^{n+p-1}(d(S{x}_1,S{x}_0))+s^{p-1}{\phi }^{n+p-2}(d(S{x}_1,S{x}_0))\\ +\cdots +s^2{\phi }^{n+1}(d(S{x}_1,S{x}_0))+s{\phi }^n(d(S{x}_1,S{x}_0))\\ =& \sum _{i=1}^p{s}^i{\phi }^{n+i-1}(d(S{x}_1,S{x}_0))\\ =& \frac{1}{s^{n-1}}\sum _{i=n}^{n+p-1}{s}^i{\phi }^i(d(S{x}_1,S{x}_0)),n,p\in \mathbb{N},p\ne 0.\end{array}$$

Since ${\sum }_{i=1}^{\mathrm{\infty }}{s}^i{\phi }^i(t)<\mathrm{\infty }$ for all $t\in {\mathbb{R}}^{+}$, $\{S{x}_n\}$ is a Cauchy sequence. Also, $S(Y)$ is complete, so $\{S{x}_n\}$ has a limit in $S(Y)$, that is, there exists $z\in S^{-1}p$ such that $p={lim}_{n\to \mathrm{\infty }}S{x}_n$. Hence, $Sz=p$ and

$$\begin{array}{rcl}d(Sz,Tz)& \le & sd(Sz,S{x}_{n+1})+sd(S{x}_{n+1},Tz)=sd(S{x}_{n+1},Sz)+sd(Tz,T{x}_n)\\ \le & sd(S{x}_{n+1},Sz)+s\phi (d(Sz,S{x}_n))+s\psi (min\{d(Sz,Tz),d(Sz,T{x}_n)\})\\ \le & sd(S{x}_{n+1},p)+sd(S{x}_n,p)+s\psi (d(S{x}_{n+1},p)).\end{array}$$

Taking the upper limit in the above inequality, we obtain $d(Sz,Tz)=0$. 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

$$d(p,q)=d(T{z}_1,T{z}_2)\le \phi (d(S{z}_1,S{z}_2))+\psi (min\{d(S{z}_1,T{z}_1),d(S{z}_1,T{z}_2)\})=\phi (d(p,q)).$$

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

Using Proposition 3.3, one can conclude that the above theorem is a significant extension of [[22], Theorem 3.1] and [[23], Theorem 3.1].

Theorem 4.3 Let $(X,d,W)$ be a convex b-metric, and let $S,T:Y\to X$ be such that T is a weak Jungck $(\phi ,\psi )$-contractive mapping such that φ is a convex subadditive function. Let $\{{\alpha }_n\}$ be a real sequence in $[0,1]$ such that ${\sum }_{n=0}^{\mathrm{\infty }}(1-{\alpha }_n)=\mathrm{\infty }$. Then, for any $x_0\in Y$, the sequence $\{S{x}_n\}$ 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 $S{x}_{n+1}=W(S{x}_n,T{y}_n,{\alpha }_n)$, where $S{y}_n=W(S{x}_n,T{x}_n,{\beta }_n)$ 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

$$\begin{array}{rcl}d(S{x}_{n+1},p)& \le & {\alpha }_{n}d(S{x}_n,p)+(1-{\alpha }_n)d(T{y}_n,p)\\ \le & {\alpha }_{n}d(S{x}_n,p)+(1-{\alpha }_n)\\ \times [\phi (d(Sz,S{y}_n))+\psi (min\{d(Sz,Tz),d(Sz,T{y}_n)\})]\\ =& {\alpha }_{n}d(S{x}_n,p)+(1-{\alpha }_n)\phi (d(S{y}_n,p)),\end{array}$$

(4.6)

and

$$\begin{array}{rcl}d(S{y}_n,p)& \le & {\beta }_{n}d(S{x}_n,p)+(1-{\beta }_n)d(T{x}_n,p)\\ \le & {\beta }_{n}d(S{x}_n,p)+(1-{\beta }_n)\\ \times [\phi (d(Sz,S{x}_n))+\psi (min\{d(Sz,Tz),d(Sz,T{x}_n)\})]\\ \le & {\beta }_{n}d(S{x}_n,p)+(1-{\beta }_n)\phi (d(S{x}_n,p))\\ \le & {\beta }_{n}d(S{x}_n,p)+(1-{\beta }_n)d(S{x}_n,p)\\ =& d(S{x}_n,p).\end{array}$$

(4.7)

Substituting (4.7) in (4.6), it follows that

$$d(S{x}_{n+1},p)\le {\alpha }_{n}d(S{x}_n,p)+(1-{\alpha }_n)\phi (d(S{x}_n,p)),n=0,1,2,\dots .$$

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

Remark 4.1

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

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

• (3) In Hilbert spaces, assuming that $q<\frac{1}{s(1+s^2)}$ in (JQC), Theorem 4.3 is an extension of the results in [25].

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

Example 4.2 Let $S,T:[0,1]\to [0,1]$ be given by $Sx=x$ and

$$Tx=\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}<x\le 1,\end{array}$$

where $[0,1]$ is endowed with the usual metric. Let $x_0\in (\frac{1}{2},1]$ and $x_{n+1}=\lambda x_n+(1-\lambda )T{x}_n$ 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 [2] 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. [13] 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 $(X,d,W)$ be a convex b-metric space, let Y be a subset of X, and let $S,T:Y\to Y$ be such that $T(Y)\subset S(Y)$. For any $x_0\in Y$, let the sequence $\{S{x}_n\}$, generated by iterative procedure (4.2), converges to p. Also, let $\{S{y}_n\}\subset X$ be an arbitrary sequence and let ${\epsilon }_n=d(S{y}_{n+1},f(T,y_n))$, $n=0,1,2,\dots$ . Then

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

• (ii) Iterative procedure (4.2) will be called almost $(S,T)$-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 [3] when S is the identity mapping on $Y=X$.

Example 5.1 Let $S,T:[0,1]\to [0,\frac{3}{2}]$ be given by $Sx=x^2+\frac{x}{2}$ and

$$Tx=\{\begin{array}{ll}0,& 0\le x\le \frac{1}{2},\\ \frac{1}{2},& \frac{1}{2}<x\le 1,\end{array}$$

where $[0,\frac{3}{2}]$ is endowed with the usual metric. Let $x_0\in [0,1]$ 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 ${lim}_{n\to \mathrm{\infty }}S{x}_n=0=S(0)=T(0)$; i.e., the Picard iteration converges strongly to the coincidence value. But the Picard iteration is not $(S,T)$-stable. Indeed, take the sequence $\{y_n\}$ given by $y_n=\frac{n+2}{2n}$, $n\in \mathbb{N}$. One can see easily that the sequence $\{S{y}_n\}$ does not converge to the coincidence value, while ${\epsilon }_n=d(S{y}_{n+1},T{y}_n)=\frac{1}{{(n+1)}^2}+\frac{3}{2(n+1)}\to 0$ 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 [[23], Theorem 4.2]. Further, this theorem reduces the condition $s^{2}q<1$ to the condition $sq<1$.

Theorem 5.2 Let $(X,d)$ be a b-metric space and T be a weak Jungck $(\phi ,\psi )$-contractive mapping such that φ is subadditive. For $x_0\in Y$, let $\{S{x}_n\}$ be the Picard iterative process defined by $S{x}_{n+1}=T{x}_n$. Then the Jungck-Picard iteration is $(S,T)$-stable.

Proof Note that, by Theorem 4.2, there exists a coincidence point $z\in Y$ such that $\{S{x}_n\}$ converges to $p=Sz=Tz$. Suppose that $\{S{y}_n\}\subset X$ and define ${\epsilon }_n=d(S{y}_{n+1},f(T,y_n))$, where $f(T,y_n)=T{y}_n$. Assume that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_n=0$. Then we have

$$\begin{array}{rcl}d(S{y}_{n+1},p)& \le & s[d(S{y}_{n+1},T{y}_n)+d(T{y}_n,p)]\\ \le & s{\epsilon }_n+s[\phi (d(Sz,S{y}_n))+\psi (min\{d(Sz,Tz),d(Sz,T{y}_n)\})]\\ =& s{\epsilon }_n+s\phi (d(S{y}_n,p)).\end{array}$$

Since φ is a subadditive s-comparison function, we get that sφ is a subadditive comparison function. Therefore, Lemma 2.4 yields that ${lim}_{n\to \mathrm{\infty }}d(S{y}_n,p)=0$, 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 [22], Theorem 3.4 of Singh et al. [13], Theorems 4.1 and 4.2 of Singh and Prasad [23], Theorem 1 of Osilike [6], Theorem 2 of Berinde [28], Theorem 2.1 of Bosede and Rhoades [29] as well as Corollary 2 of Qing and Rhoades [30].

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

Example 5.2 Let $S,T:[0,1]\to \mathbb{R}$ be given by $Sx=x$ and $Tx=\frac{-x}{2}$, where ℝ is again endowed with the usual metric. Then T is a weak Jungck $(\frac{I}{2},0)$-contraction. Let $\{x_n\}$ be a sequence generated by the Ishikawa iterative process with ${\alpha }_n={\beta }_n=1-\frac{1}{n+1}$ and $x_0\in [0,1]$. Then

$$\{\begin{array}{l}{z}_n=S{z}_n={\beta }_{n}S{x}_n+(1-{\beta }_n)T{x}_n=(1-\frac{1}{n+1})x_n+\frac{1}{n+1}T{x}_n=(1-\frac{3}{2(n+1)})x_n,\\ x_{n+1}=S{x}_{n+1}={\alpha }_{n}S{x}_n+(1-{\alpha }_n)T{z}_n=(1-\frac{1}{n+1})x_n+\frac{1}{n+1}T{z}_n=(1-\frac{3}{2(n+1)}+\frac{3}{4{(n+1)}^2})x_n.\end{array}$$

Suppose that $t_n=\frac{3}{2(n+1)}-\frac{3}{4{(n+1)}^2}$. As $t_n\in (0,1)$ and ${\sum }_{n=0}^{\mathrm{\infty }}{t}_n=\mathrm{\infty }$, Lemma 2 of [31] implies that ${lim}_{n\to \mathrm{\infty }}{x}_n=0=S(0)=T(0)$ (the unique coincidence value of S and T).

To prove the fact that the Ishikawa iteration is not $(S,T)$-stable, we use the sequence $\{y_n\}$ given by $y_n=\frac{n+1}{n+2}$. Then

$$\begin{array}{rcl}{\epsilon }_n& =& |y_{n+1}-f(T,y_n)|\\ =& |y_{n+1}-(1-\frac{3}{2(n+1)}+\frac{3}{4{(n+1)}^2})y_n|\\ =& |\frac{n+2}{n+3}-(1-\frac{3}{2(n+1)}+\frac{3}{4{(n+1)}^2})\frac{n+1}{n+2}|\\ =& \frac{6n^2+25n+13}{4(n+1)(n+2)(n+3)}.\end{array}$$

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 $(S,T)$-stable, but it is almost $(S,T)$-stable. (The almost $(S,T)$-stability is shown in the following.)

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

Theorem 5.3 Let $(X,d,W)$ be a convex b-metric space and let T be a weak Jungck $(\phi ,\psi )$-contractive mapping such that φ is a convex subadditive function. Let $\{{\alpha }_n\}$ be a real sequence in $[0,1]$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n<\mathrm{\infty }$. For $x_0\in Y$, let $\{S{x}_n\}$ be the Ishikawa iterative process given by (4.5). Then the Jungck-Ishikawa iteration is almost $(S,T)$-stable.

Proof In view of Theorem 4.3, there exists a coincidence point $z\in Y$ such that $\{S{x}_n\}$ converges to $p=Sz=Tz$. Suppose that $\{S{y}_n\}\subset X$, ${\epsilon }_n=d(S{y}_{n+1},W(S{y}_n,T{u}_n,{\alpha }_n))$, $n=0,1,2,\dots$ , where $S{u}_n=W(S{y}_n,T{y}_n,{\beta }_n)$. Assume that ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_n<\mathrm{\infty }$. Then

$$\begin{array}{rcl}d(S{y}_{n+1},p)& \le & s[d(S{y}_{n+1},W(S{y}_n,T{u}_n,{\alpha }_n))+d(W(S{y}_n,T{u}_n,{\alpha }_n),p)]\\ \le & s{\epsilon }_n+s[{\alpha }_{n}d(S{y}_n,p)+(1-{\alpha }_n)d(T{u}_n,p)]\\ \le & s{\epsilon }_n+s{\alpha }_{n}d(S{y}_n,p)+s(1-{\alpha }_n)\\ \times [\phi (d(Sz,S{u}_n))+\psi (min\{d(Sz,Tz),d(Sz,T{u}_n)\})]\\ \le & s{\epsilon }_n+s{\alpha }_{n}d(S{y}_n,p)+s(1-{\alpha }_n)\phi (d(S{u}_n,p)),\end{array}$$

(5.1)

and

$$\begin{array}{rcl}d(S{u}_n,p)& \le & {\beta }_{n}d(S{y}_n,p)+(1-{\beta }_n)d(T{y}_n,p)\\ \le & {\beta }_{n}d(S{y}_n,p)+(1-{\beta }_n)\\ \times [\phi (d(Sz,S{y}_n))+\psi (min\{d(Sz,Tz),d(Sz,T{y}_n)\})]\\ \le & {\beta }_{n}d(S{y}_n,p)+(1-{\beta }_n)\phi (d(S{y}_n,p))\\ \le & {\beta }_{n}d(S{y}_n,p)+(1-{\beta }_n)d(S{y}_n,p)\\ =& d(S{y}_n,p).\end{array}$$

(5.2)

From (5.1) and (5.2), we conclude that

$$d(S{y}_{n+1},p)\le s{\epsilon }_n+s{\alpha }_{n}d(S{y}_n,p)+s(1-{\alpha }_n)\phi (d(S{y}_n,p)).$$

(5.3)

Since φ is an s-comparison function, sφ is a comparison function. Thus, inequality (5.3) implies that

$$d(S{y}_{n+1},p)\le s{\epsilon }_n+s{\alpha }_{n}d(S{y}_n,p)+(1-{\alpha }_n)d(S{y}_n,p)=(1+(s-1){\alpha }_n)d(S{y}_n,p)+s{\epsilon }_n.$$

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

In a similar way, using Lemma 1 of [32] 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 $(S,T)$-stable.