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Convergence and stability of Jungcktype iterative procedures in convex bmetric spaces
 Abdolrahman Razani^{1}Email author and
 Mozhgan Bagherboum^{1}
https://doi.org/10.1186/168718122013331
© Razani and Bagherboum; licensee Springer. 2013
 Received: 19 February 2013
 Accepted: 13 November 2013
 Published: 3 December 2013
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 Jungcktype 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 Jungcktype 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 $(\phi ,\psi )$contractive mapping
 iterative procedure
 coincidence point
 stability
 convex bmetric space
1 Introduction
Czerwik [1] initiated the study of multivalued contractions in bmetric spaces.
 (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 bmetric 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, ℝtrees and $CAT(0)$ spaces are examples of bmetric spaces.
Throughout this paper, ${\mathbb{R}}^{+}$ is the set of nonnegative real numbers and Y is a nonempty arbitrary subset of a bmetric 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 bmetric spaces (for more details, see [1]).
The first result on stability of Tstable 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 Ishikawatype 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.
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 Tstability 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 bmetric 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 φquasinonexpansive 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 wellknown 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.
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 JungckZamfirescu 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:

(z_{1}) $d(Tx,Ty)\le \alpha d(Sx,Sy)$;

(z_{2}) $d(Tx,Ty)\le \beta [d(Sx,Tx)+d(Sy,Ty)]$;

(z_{3}) $d(Tx,Ty)\le \gamma [d(Sx,Ty)+d(Sy,Tx)]$.
It is worth mentioning that a JungckZamfirescu mapping is a (JR) mapping. In [[13], Proposition 3.3], a comparison of the above contractive conditions is established as follows.
 (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 Jungcktype contractive mappings. Then, using a JungckPicard 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 Jungcktype iterative procedures (such as JungckIshikawa 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.
 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:
holds for all $k\ge {k}_{0}$ and any $t\in {\mathbb{R}}^{+}$.
Berinde [16] expanded the concept of $(c)$comparison functions in bmetric spaces to scomparison 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 scomparison function if it satisfies (i_{ φ }) and
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 scomparison function is a comparison function.
In the sequel, some lemmas which are useful to obtain our main results are stated.
Lemma 2.2 ([17])
Lemma 2.3 ([18])
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.
where φ is a subadditive comparison function. If ${lim}_{n\to \mathrm{\infty}}{\epsilon}_{n}=0$, then ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$.
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 ${lim}_{n\to \mathrm{\infty}}{\sum}_{i=0}^{n}{\phi}^{ni}({\epsilon}_{i})=0$ from Lemma 2.2. Thus, inequality (2.2) implies that ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$. □
where φ is a convex subadditive comparison function. If ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, then ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$.
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 φquasinonexpansive mappings is studied. Furthermore, it is showed that this class includes the various classes of contractive mappings which is introduced in Section 1.
for all $x\in Y$.
The above definition was used in [19] when S is the identity mapping on $Y=X$.
It is obvious that any weak Jungck $(\phi ,\psi )$contraction is also Jungck φquasinonexpansive, but the reverse is not true. The next example illustrates this matter.
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 φquasinonexpansive 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 bmetric space with parameter s, let Y be an arbitrary subset of X, and let $S,T:Y\to X$. If T is a JungckZamfirescu 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 $t\in {\mathbb{R}}^{+}$.
for all $t\in {\mathbb{R}}^{+}$. It is clear that φ is an scomparison 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 $\phi (t)=\frac{q{s}^{2}}{1qs}t$ and $\psi (t)=\frac{q{s}^{2}}{1qs}t$ for all $t\in {\mathbb{R}}^{+}$.
for all $t\in {\mathbb{R}}^{+}$. It shows that φ is an scomparison 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 $\phi (t)=\psi (t)=\frac{q{s}^{2}}{1qs}t$ for all $t\in {\mathbb{R}}^{+}$. Also, every (JQC) mapping is a weak Jungck $(\phi ,\psi )$contractive mapping with $\phi (t)=\psi (t)=\frac{q{s}^{2}}{1qs}t$ 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 bmetric spaces is stated. Then, using some Jungcktype iterative procedures, we prove the existence of coincidence points as well as the strong convergence theorems for the weak Jungck $(\phi ,\psi )$contractive mappings.
A bmetric space X equipped with the convex structure W is called a convex bmetric space, which is denoted by $(X,d,W)$.
for all $x,y\in {l}^{p}$, is a bmetric space with $s={2}^{p1}>1$. 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 bmetric space with $W(x,y,\lambda )=\lambda x+(1\lambda )y$. (In a similar way, the space ${L}^{p}$ ($p>1$) is a convex bmetric space.)
Now, the iterative procedures in a convex bmetric space are ready to be illustrated. From now on, it is assumed that $(X,d)$ is a bmetric space (resp. $(X,d,W)$ is a convex bmetric 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.
where ${x}_{0}\in Y$ is the initial approximation and f is a function.
 1.
The Jungck iteration (or JungckPicard 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 JungckKrasnoselskij 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 JungckMann 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 JungckIshikawa 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 bmetric 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 JungckPicard iterative process converges strongly to the coincidence value.
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.
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 bmetric, 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 JungckIshikawa iterative process converges strongly to the coincidence value of S and T.
Since φ is a convex subadditive comparison function, we have the desired result from Lemma 2.5. □
 (1)
Based on Theorem 4.3, it is clear that the JungckMann iterative process as well as the JungckKrasnoselskij iterative process converge;
 (2)
 (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.
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 bmetric 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 bmetric spaces. In addition, Singh et al. [13] introduced the stability and almost stability of Jungcktype iterative procedures in metric spaces. Below, we state these concepts in convex bmetric spaces.
 (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 $Y=X$.
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 bmetric 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 bmetric 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 JungckPicard iteration is $(S,T)$stable.
Since φ is a subadditive scomparison 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.
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).
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 JungckMann iterative and JungckIshikawa 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 bmetric 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 JungckIshikawa iteration is almost $(S,T)$stable.
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.
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
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