Fixed Point Results in Convex Double-Controlled Metric-Type Spaces and Applications

Abstract

This paper investigates fixed point results in convex double-controlled metric-type spaces. By introducing a convex structure on double-controlled metric-type spaces, we study the convergence of the Mann iteration process for contractive mappings in this framework. Under suitable conditions on the control functions, we establish the existence and uniqueness of fixed points and prove that the Mann iterative sequence converges to the fixed point. In addition, we investigate the stability of the Mann iteration process and establish a data dependence result. Finally, an application to a Fredholm integral equation is presented to illustrate the applicability of the obtained results.

1. Introduction

Fixed point theory constitutes an important area of research with numerous applications in nonlinear analysis, optimization theory, as well as differential and integral equations. A central result in this field is the contraction principle introduced by Banach [1]. Owing to its simplicity and effectiveness, the Banach fixed point theorem has become one of the fundamental tools of fixed point theory and has motivated a wide range of subsequent investigations.

Over time, several researchers have focused on extending the Banach contraction principle in two main directions. One approach consists of modifying or weakening the contractive conditions, while the other involves enlarging the class of spaces in which fixed point results can be established. In this context, many generalizations of metric spaces have been proposed. Among these, b-metric spaces, introduced by Bakhtin [2] and later investigated by Czerwik [3], have received considerable attention. Numerous fixed point results for various classes of contractive mappings have been obtained in this setting, making b-metric spaces an active topic of research [4,5,6,7,8,9,10,11].

Further developments in this direction led to the introduction of more flexible structures. Kamran et al. [12] introduced extended b-metric spaces, where the triangle inequality is controlled by a function instead of a constant. Later, Mlaiki et al. [13] introduced controlled metric-type spaces, where control functions are used in the triangle inequality to obtain a more general framework for fixed point results. As a further generalization, Abdeljawad et al. [14] introduced double-controlled metric-type spaces in which two different control functions appear in the triangle inequality. Various fixed point results for different types of contractive mappings have been established in this setting [15,16,17,18,19,20,21,22,23].

In 1970, Takahashi [24] introduced the concept of convex structure in metric spaces. Later, Goebel and Kirk [25] studied the convergence properties of different iterative processes using such convex structures. Based on these ideas, Chen et al. [26] studied convex b-metric spaces and obtained convergence results using Mann’s iteration process in these spaces, together with applications to integral equations.

Motivated by these developments, it is natural to study convex structures in more general metric-type spaces. In particular, the investigation of iterative methods in double-controlled metric-type spaces equipped with a convex structure is of considerable interest. However, studies involving iterative processes in such spaces are still limited in the existing literature. This motivates further research on convex double-controlled metric-type spaces and the behavior of iterative methods in this setting.

The aim of this paper is to investigate fixed point results in convex double-controlled metric-type spaces. Using Mann’s iteration process, we establish convergence results for contractive mappings in this framework. Furthermore, we investigate certain stability properties of this iteration process and present illustrative examples demonstrating the applicability of the obtained results. These results extend some existing fixed point results in the literature and contribute to the study of iterative methods in generalized metric-type spaces.

In particular, the main novelty of this work lies in extending the results obtained in convex b-metric spaces to the more general framework of convex double-controlled metric-type spaces. Unlike the setting of Yildirim [11], where the control is governed by a single constant, the present framework involves two control functions, which leads to additional technical difficulties in the analysis of iterative processes.

Moreover, we also establish stability and data dependence results for the Mann iteration process in this generalized setting, which have not been addressed in the context of convex double-controlled metric-type spaces.

2. Preliminaries

Definition 1

([3]). Let G be a nonempty set and $s\ge 1$ be a given real number. A mapping $d_c:G\times G⟶[0,\infty )$ is called a bmetric on G if the following conditions hold for all $ð,\hslash ,\mathsf{\ell }\in G$:

1. 

$d_c(ð,\hslash )=0⟺ð=\hslash$;

2. 

$d_c(ð,\hslash )=d_c(\hslash ,ð)$;

3. 

$d_c(ð,\hslash )\le s\left[d_c(ð,\mathsf{\ell })+d_c(\mathsf{\ell },\hslash )\right]$.

Then, the pair $(G,d_c)$ is called a b-metric space.

The notion of extended b-metric spaces was introduced by Kamran et al. in 2017 [12].

Definition 2

([12]). Let G be a nonempty set and $\theta :G\times G⟶[1,\infty )$ be a function. A mapping $d_c:G\times G⟶[0,\infty )$ is called an extended b metric on G if the following conditions hold for all $ð,\hslash ,\mathsf{\ell }\in G$:

1. 

$d_c(ð,\hslash )=0⟺ð=\hslash$;

2. 

$d_c(ð,\hslash )=d_c(\hslash ,ð)$;

3. 

$d_c(ð,\hslash )\le \theta (ð,\hslash )\left[d_c(ð,\mathsf{\ell })+d_c(\mathsf{\ell },\hslash )\right]$.

The pair $(G,d_c)$ is then referred to as an extended b-metric space.

Later, Mlaiki et al. [13] proposed a broader framework extending b-metric spaces.

Definition 3

([13]). Let G be a nonempty set, and let $\alpha :G\times G⟶[1,\infty )$ be a function. A mapping $d_c:G\times G⟶[0,\infty )$ is said to define a controlled metric-type structure on G if for every $ð,\hslash ,\mathsf{\ell }\in G$, the following are true:

1. 

$d_c(ð,\hslash )=0$ if and only if $ð=\hslash$;

2. 

$d_c(ð,\hslash )=d_c(\hslash ,ð)$;

3. 

$d_c(ð,\hslash )\le \alpha (ð,\mathsf{\ell })d_c(ð,\mathsf{\ell })+\alpha (\mathsf{\ell },\hslash )d_c(\mathsf{\ell },\hslash )$.

In such a case, the pair $(G,d_c)$ is called a controlled metric-type space.

A further generalization was presented by Abdeljawad et al. [14] in 2018.

Definition 4

([14]). Let $\alpha ,\mu :G\times G⟶[1,\infty )$ be two control functions. A mapping $d_c:G\times G⟶[0,\infty )$ is called a double controlled metric-type if the following hold for all $ð,\hslash ,\mathsf{\ell }\in G$:

1. 

$d_c(ð,\hslash )=0⟺ð=\hslash$;

2. 

$d_c(ð,\hslash )=d_c(\hslash ,ð)$;

3. 

$d_c(ð,\hslash )\le \alpha (ð,\mathsf{\ell })d_c(ð,\mathsf{\ell })+\mu (\mathsf{\ell },\hslash )d_c(\mathsf{\ell },\hslash )$.

The space $(G,d_c)$ equipped with such a function will be called a double-controlled metric-type space (DCMTS).

Remark 1

([14]). Every controlled metric-type space may be viewed as a particular case of a double-controlled metric-type space when the two control functions coincide. The reverse implication does not hold in general.

The standard topological notions such as convergence, Cauchy sequences, and completeness in double-controlled metric-type spaces are defined in the usual manner.

Definition 5

([14]). Let $(G,d_c)$ be a DCMTS:

(1) 

A sequence $\left\{{ð}_n\right\}\subset G$ is said to converge to $ð\in G$ if for every $\epsilon >0$, there exists an integer $N_{\epsilon }$ such that $d_c({ð}_n,ð)<\epsilon$ whenever $n\ge N_{\epsilon }$. It is written as ${lim}_{n\to \infty }{ð}_n=ð$.

(2) 

A sequence $\left\{{ð}_n\right\}$ is called a Cauchy sequence if for every $\epsilon >0$, there exists an integer $N_{\epsilon }$ such that $d_c({ð}_n,{ð}_m)<\epsilon$ for all $m,n\ge N_{\epsilon }$.

(3) 

$(G,d_c)$ is said to be complete if every Cauchy sequence is convergent.

Definition 6

([24]). Let $G\ne ⌀$, and let $I=[0,1]$. Define the mapping $d_c:G\times G⟶[0,\infty ).$ Then, a continuous mapping $\omega :G\times G\times I⟶G$ is called a convex structure on G if for every $o,ð,\hslash \in G$ and $\lambda \in I$, we have

$$d_c(o,\omega (ð,\hslash ;\lambda ))\le \lambda d_c(o,ð)+(1-\lambda )d_c(o,\hslash ).$$

(1)

3. Main Results

3.1. Convergence Results

We start by defining the concept of a convex double-controlled metric-type space.

Definition 7.

Let $(G,d_c)$ be a double-controlled metric-type space with control functions $\alpha ,\mu :G\times G⟶[1,\infty ).$ Assume that $\omega :G\times G\times I⟶G$ is a convex structure on G. Then, the triple $(G,d_c,\omega )$ will be called a convex double-controlled metric-type space (convex DCMTS).

For a mapping $\Re :G⟶G$, the classical Mann iteration can be adapted to the convex DCMTS framework as follows:

$${ð}_{n+1}=\omega ({ð}_n,\Re {ð}_n;{\delta }_n),n\in \mathbb{N},$$

where $\left\{{\delta }_n\right\}\subset [0,1]$. The sequence $\left\{{ð}_n\right\}$ generated in this way will be referred to as the Mann iteration sequence associated with ℜ.

We now provide examples illustrating a convex DCMTS.

Example 1.

Let $G={\mathbb{R}}^2$, and for any $ð,\hslash \in G$, define the metric $d_c:G\times G⟶[0,\infty )$ by

$$d_c(ð,\hslash )={∥ð-\hslash ∥}_{\infty }^p,p>1,$$

where ${∥ð-\hslash ∥}_{\infty }=max\left\{\left|{ð}_1-{\hslash }_1\right|,\left|{ð}_2-{\hslash }_2\right|\right\}$ for $ð=({ð}_1,{ð}_2)$ and $\hslash =({\hslash }_1,{\hslash }_2)$ in ${\mathbb{R}}^2.$

Let the convex structure $\omega :G\times G\times [0,1]⟶G$ be given by

$$\omega (ð,\hslash ;\lambda )=\lambda ð+(1-\lambda )\hslash .$$

Then, $(G,d_c,\omega )$ is a convex DCMTS with control functions $\alpha (ð,\hslash )=\mu (ð,\hslash )=2^{p-1}$ for all $ð,\hslash \in G$, where $\alpha ,\mu :G\times G⟶[1,\infty ).$

Clearly, for all $ð,\hslash \in G$, $d_c(ð,\hslash )=0$ if and only if $ð=\hslash$ and $d_c$ is symmetric.

Also, for any $ð,\hslash ,\mathsf{\ell }\in G$, we have

$${∥ð-\hslash ∥}_{\infty }\le {∥ð-\mathsf{\ell }∥}_{\infty }+{∥\mathsf{\ell }-\hslash ∥}_{\infty }$$

(2)

and since $p>1,$ by taking the pth power and using ${\left(a+b\right)}^p\le 2^{p-1}\left(a^p+b^p\right),$ for $a,b\ge 0,$ we deduce that

$${∥ð-\hslash ∥}_{\infty }^p\le 2^{p-1}\left({∥ð-\mathsf{\ell }∥}_{\infty }^p+{∥\mathsf{\ell }-\hslash ∥}_{\infty }^p\right).$$

(3)

From the inequalities in Equations (2) and (3), we get

$$\begin{array}{ccc}\hfill d_c(ð,\hslash )& =& {∥ð-\hslash ∥}_{\infty }^p\hfill \\ & \le & 2^{p-1}\left({∥ð-\mathsf{\ell }∥}_{\infty }^p+{∥\mathsf{\ell }-\hslash ∥}_{\infty }^p\right)\hfill \\ & \le & 2^{p-1}\left[d_c(ð,\mathsf{\ell })+d_c(\mathsf{\ell },\hslash )\right].\hfill \end{array}$$

Thus, $d_c$ satisfies the double-controlled metric-type inequality with control functions $\alpha (ð,\hslash )=\mu (ð,\hslash )=2^{p-1}.$ Therefore, $(G,d_c,\omega )$ is a DCMTS with α and $\mu .$

It remains to verify that the function ω satisfies the inequality in Equation (1).

For any $o,ð,\hslash \in G$, we get

$$\begin{array}{ccc}\hfill d_c(o,\omega (ð,\hslash ;\lambda ))& =& {∥o-\omega ∥}_{\infty }^p\hfill \\ & \le & {\left[\lambda {∥o-ð∥}_{\infty }+(1-\lambda ){∥o-\hslash ∥}_{\infty }\right]}^p\hfill \\ & \le & \lambda {∥o-ð∥}_{\infty }^p+(1-\lambda ){∥o-\hslash ∥}_{\infty }^p\hfill \\ & \le & \lambda d_c(o,ð)+(1-\lambda )d_c(o,\hslash ).\hfill \end{array}$$

Hence, $(G,d_c,\omega )$ is a convex DCMTS with α and $\mu .$

Example 2.

Let $G=[0,\infty )$, and for any $ð,\hslash \in G$, define the metric $d_c:G\times G⟶[0,\infty )$ by

$$d_c(ð,\hslash )=e^{\left|ð-\hslash \right|}-1.$$

Also, define the convex structure $\omega :G\times G\times [0,1]⟶G$, given by

$$\omega (ð,\hslash ;\lambda )=\lambda ð+(1-\lambda )\hslash ,\mathrm{for}\lambda \in [0,1].$$

Let the control functions $\alpha ,\mu :G\times G⟶[1,\infty )$ be given by

$$\alpha (ð,\hslash )={\displaystyle \frac{e^{\left|ð-\hslash \right|}+1}{2}}and\mu (ð,\hslash )={\displaystyle \frac{e^{\left|ð-\hslash \right|}+1}{2}}.$$

Then, $(G,d_c,\omega )$ is a convex DCMTS with control functions $\alpha ,\mu .$

Indeed, it is clear that for all $ð,\hslash \in G$, $d_c(ð,\hslash )=0$ if and only if $ð=\hslash$ and $d_c$ is symmetric.

For any $ð,\hslash ,\mathsf{\ell }\in G$, using the triangle inequality $\left|ð-\hslash \right|\le \left|ð-\mathsf{\ell }\right|+\left|\mathsf{\ell }-\hslash \right|$ and the fact that the function $t\mapsto e^t-1$ is increasing and convex on $[0,\infty ),$ we obtain

$$\begin{array}{ccc}\hfill d_c(ð,\hslash )& =& e^{\left|ð-\hslash \right|}-1\hfill \\ & \le & e^{\left|ð-\mathsf{\ell }\right|+\left|\mathsf{\ell }-\hslash \right|}-1\hfill \\ & =& \left(e^{{\displaystyle \frac{2^{\left|ð-\mathsf{\ell }\right|+2\left|\mathsf{\ell }-\hslash \right|}}{2}}}-1\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\left(e^{2\left|ð-\mathsf{\ell }\right|}-1\right)+{\displaystyle \frac{1}{2}}\left(e^{2\left|\mathsf{\ell }-\hslash \right|}-1\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(e^{\left|ð-\mathsf{\ell }\right|}-1\right)\left(e^{\left|ð-\mathsf{\ell }\right|}+1\right)+{\displaystyle \frac{1}{2}}\left(e^{\left|\mathsf{\ell }-\hslash \right|}-1\right)\left(e^{\left|\mathsf{\ell }-\hslash \right|}+1\right)\hfill \\ & \le & {\displaystyle \frac{e^{\left|ð-\mathsf{\ell }\right|}+1}{2}}{d}_c(ð,\mathsf{\ell })+{\displaystyle \frac{e^{\left|\mathsf{\ell }-\hslash \right|}+1}{2}}{d}_c(\mathsf{\ell },\hslash )\hfill \\ & =& \alpha (ð,\mathsf{\ell })d_c(ð,\mathsf{\ell })+\mu (\mathsf{\ell },\hslash )d_c(\mathsf{\ell },\hslash ).\hfill \end{array}$$

Thus, the double-controlled metric-type inequality holds. Therefore, $(G,d_c,\omega )$ is a DCMTS with α and $\mu .$

Next, for any $o,ð,\hslash \in G$ and $\lambda \in [0,1],$ we have

$$\begin{array}{ccc}\hfill d_c(o,\omega (ð,\hslash ;\lambda ))& =& e^{\left|o-\omega (ð,\hslash ;\lambda )\right|}-1\hfill \\ & \le & e^{\lambda \left|o-ð\right|+(1-\lambda )\left|o-\hslash \right|}-1,\hfill \end{array}$$

and since the function $t\mapsto e^t-1$ is increasing and convex on $[0,\infty ),$ it follows that

$$\begin{array}{ccc}\hfill d_c(o,\omega (ð,\hslash ;\lambda ))& \le & \lambda \left(e^{\left|o-ð\right|}-1\right)+(1-\lambda )\left(e^{\left|o-\hslash \right|}-1\right)\hfill \\ & =& \lambda d_c(o,ð)+(1-\lambda )d_c(o,\hslash ).\hfill \end{array}$$

Hence, $(G,d_c,\omega )$ is a convex DCMTS with α and $\mu .$

On the other hand, $(G,d_c)$ is not a b-metric space. Indeed, assume that there exists $s\ge 1$ such that

$$d_c(ð,\hslash )\le s\left[d_c(ð,\mathsf{\ell })+d_c(\mathsf{\ell },\hslash )\right]$$

for all $ð,\hslash ,\mathsf{\ell }\in G$. By taking $ð=0,$ $\hslash =2n,$ $\mathsf{\ell }=n$, we obtain

$$e^{2n}-1\le 2s\left(e^n-1\right),$$

which implies

$${\displaystyle \frac{e^n+1}{2}}\le s.$$

However, for $n⟶\infty$, we have ${\displaystyle \frac{e^n+1}{2}}⟶\infty ,$ which is a contradiction. Therefore, $(G,d_c)$ is not a b-metric space.

We now present a Banach contraction result for complete convex double-controlled metric-type spaces via Mann’s iteration process.

Theorem 1.

Let $(G,d_c,\omega )$ be a complete convex DCMTS with control functions $\alpha ,\mu :G\times G⟶[1,\infty )$ and ℜ$:G⟶G$ be a contraction mapping; that is, there exists $\beta \in [0,1)$ such that

$$d_c(\Re ð,\Re \hslash )\le \beta d_c(ð,\hslash ),forallð,\hslash \in G.$$

Fix ${ð}_0\in G$, and define Mann’s iteration process by

$${ð}_{n+1}=\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right),n\in \mathbb{N},$$

(4)

where $\left\{{\delta }_n\right\}\subset [0,1).$ Let $ð\in G$ be an arbitrary point, and set

$$\Omega :=\left\{ð,{ð}_n,\Re {ð}_n:n\in \mathbb{N}\right\}.$$

Assume that the control functions are bounded on $\Omega \times \Omega ,$ i.e., there exist constants $\xi ,\varpi \ge 1$ such that

$$\alpha (ð,\hslash )\le \xi ,\mu (ð,\hslash )\le \varpi ,forallð,\hslash ,\mathsf{\ell }\in \Omega .$$

Since the control functions are bounded on $\Omega \times \Omega ,$ let us define

$$\kappa :=max\left\{\xi ,\varpi \right\}.$$

In addition, if $\beta {\kappa }^4<1$ and $0\le {\delta }_n<{\displaystyle \frac{\frac{1}{{\kappa }^4}-\beta }{1-\beta }}$ for each $n\in \mathbb{N},$ then ℜ has a unique fixed point ${ð}^{\ast }\in G$, and Mann’s iteration sequence $\left\{{ð}_n\right\}$ converges to ${ð}^{\ast }.$

Proof. 

Observe that for each $n\in \mathbb{N}$, we have

$$d_c({ð}_n,{ð}_{n+1})=d_c\left({ð}_n,\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right)\right)\le \left(1-{\delta }_n\right)d_c({ð}_n,\Re {ð}_n).$$

By using the double-controlled metric-type inequality together with the convex structure, we estimate the term $d_c({ð}_n,\Re {ð}_n)$ as follows:

$$\begin{array}{ccc}\hfill d_c({ð}_n,\Re {ð}_n)& \le & \alpha \left({ð}_n,\Re {ð}_{n-1}\right)d_c\left({ð}_n,\Re {ð}_{n-1}\right)+\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)d_c\left(\Re {ð}_{n-1},\Re {ð}_n\right)\hfill \\ & \le & \alpha \left({ð}_n,\Re {ð}_{n-1}\right)d_c\left(\omega \left({ð}_{n-1},\Re {ð}_{n-1};{\delta }_{n-1}\right),\Re {ð}_{n-1}\right)\hfill \\ & & +\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)\beta d_c\left({ð}_{n-1},{ð}_n\right)\hfill \\ & =& \alpha \left({ð}_n,\Re {ð}_{n-1}\right)d_c\left(\omega \left({ð}_{n-1},\Re {ð}_{n-1};{\delta }_{n-1}\right),\Re {ð}_{n-1}\right)\hfill \\ & & +\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)\beta d_c\left({ð}_{n-1},\omega \left({ð}_{n-1},\Re {ð}_{n-1};{\delta }_{n-1}\right)\right)\hfill \\ & \le & \alpha \left({ð}_n,\Re {ð}_{n-1}\right)\left[{\delta }_{n-1}{d}_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)+\left(1-{\delta }_{n-1}\right)d_c\left(\Re {ð}_{n-1},\Re {ð}_{n-1}\right)\right]\hfill \\ & & +\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)\beta \left[{\delta }_{n-1}{d}_c\left({ð}_{n-1},{ð}_{n-1}\right)+\left(1-{\delta }_{n-1}\right)d_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)\right]\hfill \\ & =& \alpha \left({ð}_n,\Re {ð}_{n-1}\right){\delta }_{n-1}{d}_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)\hfill \\ & & +\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)\beta \left(1-{\delta }_{n-1}\right)d_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)\hfill \\ & =& \left[\alpha \left({ð}_n,\Re {ð}_{n-1}\right){\delta }_{n-1}+\mu \left(\Re {ð}_{n-1},\Re {ð}_n\right)\beta \left(1-{\delta }_{n-1}\right)\right]d_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)\hfill \\ & \le & \left[\xi {\delta }_{n-1}+\varpi \beta \left(1-{\delta }_{n-1}\right)\right]d_c\left({ð}_{n-1},\Re {ð}_{n-1}\right).\hfill \end{array}$$

Let us take $\kappa :=max\left\{\xi ,\varpi \right\}$ and define ${\gamma }_{n-1}=\kappa \left[{\delta }_{n-1}+\beta \left(1-{\delta }_{n-1}\right)\right].$ Under the assumptions $\beta {\kappa }^4<1$ and $0\le {\delta }_n<{\displaystyle \frac{\frac{1}{{\kappa }^4}-\beta }{1-\beta }}$, we get

$$d_c({ð}_n,\Re {ð}_n)\le {\gamma }_{n-1}{d}_c\left({ð}_{n-1},\Re {ð}_{n-1}\right)<{\displaystyle \frac{1}{{\kappa }^3}}{d}_c\left({ð}_{n-1},\Re {ð}_{n-1}\right),$$

(5)

from which it follows that $\left\{d_c({ð}_n,\Re {ð}_n)\right\}$ is decreasing and bounded below by zero. Therefore, there exists $c\ge 0$ such that

$$\underset{n\to \infty }{lim}{d}_c({ð}_n,\Re {ð}_n)=c.$$

We now prove that $c=0$. Assume, on the contrary, that $c>0$. Passing to the limit as $n⟶\infty$ in the inequality in Equation (5) gives

$$c\le {\displaystyle \frac{1}{{\kappa }^3}}c<c,$$

which is a contradiction. Hence, we get that $c=0$. Moreover, we have

$$d_c({ð}_n,{ð}_{n+1})\le \left(1-{\delta }_n\right)d_c({ð}_n,\Re {ð}_n)<d_c({ð}_n,\Re {ð}_n),$$

which shows that ${lim}_{n\to \infty }{d}_c({ð}_n,{ð}_{n+1})=0.$

We now verify that $\left\{{ð}_n\right\}$ forms a Cauchy sequence. Suppose the contrary. Then, there exist ${\epsilon }_0>0$ and the subsequences $\left\{{ð}_{\theta \left(s\right)}\right\}$ and $\left\{{ð}_{\eta \left(s\right)}\right\}$ of $\left\{{ð}_n\right\}$ such that $\theta \left(s\right)>\eta \left(s\right)>s$, with

$$d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)}\right)\ge {\epsilon }_0$$

and

$$d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)}\right)<{\epsilon }_0.$$

Then, we conclude that

$$\begin{array}{ccc}\hfill {\epsilon }_0& \le & d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)}\right)\hfill \\ & \le & \alpha \left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & & +\mu \left({ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)}\right)d_c\left({ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)}\right)\hfill \\ & \le & \xi d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)+\varpi d_c\left({ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)}\right)\hfill \\ & \le & \kappa \left[d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)+d_c\left({ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)}\right)\right].\hfill \end{array}$$

By taking the limit superior on both sides of the above inequality and using the fact that ${lim}_{n\to \infty }{d}_c({ð}_n,{ð}_{n+1})=0$, we obtain

$${\displaystyle \frac{{\epsilon }_0}{\kappa }}\le \underset{n⟶\infty }{lim}sup{d}_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right).$$

In noticing that

$$\begin{array}{ccc}\hfill d_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)& =& d_c\left(\omega \left({ð}_{\theta \left(s\right)-1},\Re {ð}_{\theta \left(s\right)-1};{\delta }_{\theta \left(s\right)-1}\right),{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & \le & {\delta }_{\theta \left(s\right)-1}{d}_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)+\left(1-{\delta }_{\theta \left(s\right)-1}\right)d_c\left(\Re {ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & \le & {\delta }_{\theta \left(s\right)-1}{d}_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\left[\begin{array}{c}\alpha \left(\Re {ð}_{\theta \left(s\right)-1},\Re {ð}_{\eta \left(s\right)+1}\right)d_c\left(\Re {ð}_{\theta \left(s\right)-1},\Re {ð}_{\eta \left(s\right)+1}\right)\\ +\mu \left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)\end{array}\right]\hfill \\ & \le & {\delta }_{\theta \left(s\right)-1}{d}_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\left[\begin{array}{c}\alpha \left(\Re {ð}_{\theta \left(s\right)-1},\Re {ð}_{\eta \left(s\right)+1}\right)\beta d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)\\ +\mu \left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)\end{array}\right]\hfill \\ & \le & \left[{\delta }_{\theta \left(s\right)-1}+\left(1-{\delta }_{\theta \left(s\right)-1}\right)\alpha \left(\Re {ð}_{\theta \left(s\right)-1},\Re {ð}_{\eta \left(s\right)+1}\right)\beta \right]d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\mu \left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & \le & \left[{\delta }_{\theta \left(s\right)-1}+\left(1-{\delta }_{\theta \left(s\right)-1}\right)\alpha \left(\Re {ð}_{\theta \left(s\right)-1},\Re {ð}_{\eta \left(s\right)+1}\right)\beta \right]\hfill \\ & & \left[\begin{array}{c}\alpha \left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)}\right)d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)}\right)\\ +\mu \left({ð}_{\eta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)d_c\left({ð}_{\eta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)\end{array}\right]\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\mu \left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & \le & \left[{\delta }_{\theta \left(s\right)-1}+\left(1-{\delta }_{\theta \left(s\right)-1}\right)\xi \beta \right]\left[\xi d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)}\right)+\varpi d_c\left({ð}_{\eta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)\right]\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\varpi d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right)\hfill \\ & \le & {\kappa }^2\left[{\delta }_{\theta \left(s\right)-1}+\left(1-{\delta }_{\theta \left(s\right)-1}\right)\beta \right]\left[d_c\left({ð}_{\theta \left(s\right)-1},{ð}_{\eta \left(s\right)}\right)+d_c\left({ð}_{\eta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)\right]\hfill \\ & & +\left(1-{\delta }_{\theta \left(s\right)-1}\right)\kappa d_c\left(\Re {ð}_{\eta \left(s\right)+1},{ð}_{\eta \left(s\right)+1}\right),\hfill \end{array}$$

we deduce that

$${\displaystyle \frac{1}{\kappa }}{\epsilon }_0\le \underset{n⟶\infty }{lim}sup{d}_c\left({ð}_{\theta \left(s\right)},{ð}_{\eta \left(s\right)+1}\right)\le {\kappa }^2{\displaystyle \frac{1}{{\kappa }^4}}{\epsilon }_0<{\displaystyle \frac{1}{\kappa }}{\epsilon }_0,$$

which is a contradiction. Hence, the sequence $\left\{{ð}_n\right\}$ is a Cauchy sequence in G. Since G is complete, there exists ${ð}^{\ast }\in G$ such that ${lim}_{n\to \infty }{d}_c({ð}_n,{ð}^{\ast })=0.$

Finally, we verify that ${ð}^{\ast }$ is a fixed point of ℜ. Observe that

$$\begin{array}{ccc}\hfill d_c\left({ð}^{\ast },\Re {ð}^{\ast }\right)& \le & \alpha \left({ð}^{\ast },{ð}_n\right)d_c\left({ð}^{\ast },{ð}_n\right)+\mu \left({ð}_n,\Re {ð}^{\ast }\right)d_c\left({ð}_n,\Re {ð}^{\ast }\right)\hfill \\ & \le & \alpha \left({ð}^{\ast },{ð}_n\right)d_c\left({ð}^{\ast },{ð}_n\right)\hfill \\ & & +\mu \left({ð}_n,\Re {ð}^{\ast }\right)\left[\alpha \left({ð}_n,\Re {ð}_n\right)d_c\left({ð}_n,\Re {ð}_n\right)+\mu \left(\Re {ð}_n,\Re {ð}^{\ast }\right)d_c\left(\Re {ð}_n,\Re {ð}^{\ast }\right)\right]\hfill \\ & \le & \alpha \left({ð}^{\ast },{ð}_n\right)d_c\left({ð}^{\ast },{ð}_n\right)\hfill \\ & & +\mu \left({ð}_n,\Re {ð}^{\ast }\right)\left[\alpha \left({ð}_n,\Re {ð}_n\right)d_c\left({ð}_n,\Re {ð}_n\right)+\mu \left(\Re {ð}_n,\Re {ð}^{\ast }\right)\beta d_c\left({ð}_n,{ð}^{\ast }\right)\right]\hfill \\ & \le & \xi d_c\left({ð}^{\ast },{ð}_n\right)+\varpi \left[\xi d_c\left({ð}_n,\Re {ð}_n\right)+\varpi \beta d_c\left({ð}_n,{ð}^{\ast }\right)\right]\hfill \\ & \le & \kappa d_c\left({ð}^{\ast },{ð}_n\right)+{\kappa }^2{d}_c\left({ð}_n,\Re {ð}_n\right)+{\kappa }^2\beta d_c\left({ð}_n,{ð}^{\ast }\right).\hfill \end{array}$$

Letting $n⟶\infty$ yields $d_c\left({ð}^{\ast },\Re {ð}^{\ast }\right)=0.$ Therefore, $\Re {ð}^{\ast }={ð}^{\ast }$. Thus, ${ð}^{\ast }$ is a fixed point of ℜ.

To prove the uniqueness, suppose that ${\hslash }^{\ast }\in G$ is another fixed point of ℜ, i.e., $\Re {\hslash }^{\ast }={\hslash }^{\ast }$. Then, we have

$$d_c\left({ð}^{\ast },{\hslash }^{\ast }\right)=d_c\left(\Re {ð}^{\ast },\Re {\hslash }^{\ast }\right)\le \beta d_c\left({ð}^{\ast },{\hslash }^{\ast }\right).$$

Since 0 $\le \beta <1$, this implies $d_c\left({ð}^{\ast },{\hslash }^{\ast }\right)=0$ and hence ${ð}^{\ast }={\hslash }^{\ast }.$ Therefore, the fixed point is unique.

This completes the proof. □

An example is provided below to show how the above theorem can be applied.

Example 3.

Let $G=[0,\infty )$ and fix ${ð}_0\in G$ and choose $\lambda \in (0,1)$ such that $\sqrt{\lambda }{\left({\displaystyle \frac{e^{\sqrt{{ð}_0}}+1}{2}}\right)}^4<1.$ Define $\Re :G⟶G$ by

$$\Re ð={\displaystyle \frac{\lambda ð}{1+ð}}.$$

For any $ð,\hslash ,\mathsf{\ell }\in G,$ we define the function $d_c:G\times G⟶[0,\infty )$ by

$$d_c(ð,\hslash )=e^{\sqrt{\left|ð-\hslash \right|}}-1$$

while the mapping $\omega :G\times G\times [0,1]⟶G$ is defined by

$$\omega (ð,\hslash ;\delta )=\delta ð+(1-\delta )\hslash .$$

Let the control functions be given by

$$\alpha (ð,\hslash )={\displaystyle \frac{e^{\sqrt{\left|ð-\hslash \right|}}+1}{2}}and\mu (\mathsf{\ell },\hslash )={\displaystyle \frac{e^{\sqrt{\left|ð-\hslash \right|}}+1}{2}},$$

and Mann’s iteration be defined by

$${ð}_{n+1}=\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right),n\in \mathbb{N},$$

where $\left\{{\delta }_n\right\}\subset [0,1)$.

Then, $(G,d_c,\omega )$ is a complete convex DCMTS with control functions α and μ, and ℜ has a unique fixed point in G.

Indeed, for all $ð,\hslash \in G,$ it is clear that $d_c(ð,\hslash )=0$ if and only if $ð=\hslash$ and $d_c$ is symmetric.

For any $ð,\hslash ,\mathsf{\ell }\in G$, using the triangle inequality $\left|ð-\hslash \right|\le \left|ð-\mathsf{\ell }\right|+\left|\mathsf{\ell }-\hslash \right|,$ we obtain

$$\begin{array}{ccc}\hfill d_c\left(ð,\hslash \right)& =& e^{\sqrt{\left|ð-\hslash \right|}}-1\hfill \\ & \le & e^{\sqrt{\left|ð-\mathsf{\ell }\right|}+\sqrt{\left|\mathsf{\ell }-\hslash \right|}}-1.\hfill \end{array}$$

Set $a=e^{\sqrt{\left|ð-\mathsf{\ell }\right|}},$ $b=e^{\sqrt{\left|\mathsf{\ell }-\hslash \right|}}.$ Then, $a,b\ge 1$, and therefore

$$d_c\left(ð,\hslash \right)\le ab-1.$$

Now, using

$$ab-1\le {\displaystyle \frac{\left(a^2-1\right)+\left(b^2-1\right)}{2}},$$

we obtain

$$\begin{array}{ccc}\hfill d_c\left(ð,\hslash \right)& \le & {\displaystyle \frac{e^{2\sqrt{\left|ð-\mathsf{\ell }\right|}}-1}{2}}+{\displaystyle \frac{e^{2\sqrt{\left|\mathsf{\ell }-\hslash \right|}}-1}{2}}\hfill \\ & \le & \left({\displaystyle \frac{e^{\sqrt{\left|ð-\mathsf{\ell }\right|}}+1}{2}}\right)\left(e^{\sqrt{\left|ð-\mathsf{\ell }\right|}}-1\right)+\left({\displaystyle \frac{e^{\sqrt{\left|\mathsf{\ell }-\hslash \right|}}+1}{2}}\right)\left(e^{\sqrt{\left|\mathsf{\ell }-\hslash \right|}}-1\right)\hfill \\ & =& \alpha (ð,\mathsf{\ell })d_c(ð,\mathsf{\ell })+\mu (\mathsf{\ell },\hslash )d_c(\mathsf{\ell },\hslash ).\hfill \end{array}$$

Thus, $d_c$ satisfies the double-controlled metric-type inequality. Therefore, $(G,d_c,\omega )$ is a DCMTS with α and $\mu .$

Next, for any $o,ð,\hslash \in G$ and $\lambda \in [0,1],$ we have

$$\begin{array}{ccc}\hfill d_c(o,\omega (ð,\hslash ;\lambda ))& =& e^{\sqrt{\left|o-\omega (ð,\hslash ;\lambda )\right|}}-1\hfill \\ & \le & e^{\lambda \sqrt{\left|o-ð\right|}+(1-\lambda )\sqrt{\left|o-\hslash \right|}}-1,\hfill \end{array}$$

and since the exponential function is convex, it follows that

$$\begin{array}{ccc}\hfill d_c(o,\omega (ð,\hslash ;\lambda ))& \le & \lambda \left(e^{\sqrt{\left|o-ð\right|}}-1\right)+(1-\lambda )\left(e^{\sqrt{\left|o-\hslash \right|}}-1\right)\hfill \\ & =& \lambda d_c(o,ð)+(1-\lambda )d_c(o,\hslash ).\hfill \end{array}$$

Hence, $(G,d_c,\omega )$ is a convex DCMTS with α and $\mu .$

We now show that $\left(G,d_c\right)$ is complete. Let $\left\{{ð}_n\right\}$ be a Cauchy sequence in $G.$ Then, we have

$$\underset{n,m\to \infty }{lim}{d}_c\left({ð}_n,{ð}_m\right)=\underset{n,m\to \infty }{lim}{e}^{\sqrt{\left|{ð}_n-{ð}_m\right|}}-1=0.$$

Hence, it follows that

$$\underset{n,m\to \infty }{lim}\left|{ð}_n-{ð}_m\right|=0.$$

Thus, $\left\{{ð}_n\right\}$ is a Cauchy sequence in the usual metric on $[0,\infty ).$ Since $[0,\infty )$ is complete, there exists $ð\in [0,\infty )$ such that ${lim}_{n\to \infty }{ð}_n=ð$ in the usual metric. Consequently, we have

$$\underset{n\to \infty }{lim}{d}_c\left({ð}_n,ð\right)=\underset{n\to \infty }{lim}{e}^{\sqrt{\left|{ð}_n-ð\right|}}-1=0.$$

Therefore, $\left(G,d_c\right)$ is complete.

Now, for all $ð,\hslash \in G,$ we have

$$\begin{array}{ccc}\hfill \left|\Re ð-\Re \hslash \right|& =& \left|{\displaystyle \frac{\lambda ð}{1+ð}}-{\displaystyle \frac{\lambda \hslash }{1+\hslash }}\right|\hfill \\ & \le & \lambda {\displaystyle \frac{\left|ð-\hslash \right|}{\left(1+ð\right)\left(1+\hslash \right)}}\hfill \\ & \le & \lambda \left|ð-\hslash \right|\hfill \end{array}$$

and therefore

$$d_c\left(\Re ð,\Re \hslash \right)=e^{\sqrt{\left|\Re ð-\Re \hslash \right|}}-1\le e^{\sqrt{\lambda }\sqrt{\left|ð-\hslash \right|}}-1.$$

Now, since the exponential function is convex and $0<\sqrt{\lambda }<1$, we have

$$\begin{array}{ccc}\hfill e^{\sqrt{\lambda }\sqrt{\left|ð-\hslash \right|}}& =& e^{\left(1-\sqrt{\lambda }\right)0+\sqrt{\lambda }\sqrt{\left|ð-\hslash \right|}}\hfill \\ & \le & \left(1-\sqrt{\lambda }\right)e^0+\sqrt{\lambda }{e}^{\sqrt{\left|ð-\hslash \right|}}.\hfill \end{array}$$

Therefore, we have

$$e^{\sqrt{\lambda }\sqrt{\left|ð-\hslash \right|}}-1\le \sqrt{\lambda }\left(e^{\sqrt{\left|ð-\hslash \right|}}-1\right)$$

and hence, we obtain

$$d_c\left(\Re ð,\Re \hslash \right)\le \sqrt{\lambda }{d}_c\left(ð,\hslash \right).$$

Thus, ℜ is a contraction with a constant $\beta =\sqrt{\lambda }.$

Mann’s iteration is

$${ð}_{n+1}=\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right)={\delta }_n{ð}_n+\left(1-{\delta }_n\right){\displaystyle \frac{\lambda {ð}_n}{1+{ð}_n}}.$$

Since ${\displaystyle \frac{{ð}_n}{1+{ð}_n}}\le {ð}_n$ for ${ð}_n\ge 0$, we get

$${ð}_{n+1}\le {\delta }_n{ð}_n+\left(1-{\delta }_n\right)\lambda {ð}_n=\left[{\delta }_n+\left(1-{\delta }_n\right)\lambda \right]{ð}_n.$$

Let ${\rho }_n={\delta }_n+\left(1-{\delta }_n\right)\lambda .$ Since ${\rho }_n\in (0,1)$, by induction, we get

$$0\le {ð}_n\le {\rho }_n^n{ð}_0\le {ð}_{0,}$$

and hence the sequence $\left\{{ð}_n\right\}$ is bounded, and for $n⟶\infty ,$ we get that ${ð}_n⟶0.$

Set $\Omega :=\left\{0,{ð}_n,\Re {ð}_n:n\in \mathbb{N}\right\}.$ Using $0\le {ð}_n\le {ð}_0$, we find that

$$\alpha ({ð}_n,{ð}_{n+1})={\displaystyle \frac{e^{\sqrt{\left|{ð}_n-{ð}_{n+1}\right|}}+1}{2}}\le {\displaystyle \frac{e^{\sqrt{{ð}_0}}+1}{2}},$$

and

$$\mu ({ð}_n,{ð}_{n+1})={\displaystyle \frac{e^{\sqrt{\left|{ð}_n-{ð}_{n+1}\right|}}+1}{2}}\le {\displaystyle \frac{e^{\sqrt{{ð}_0}}+1}{2}}.$$

Thus, the control functions are bounded on $\Omega \times \Omega$, and one may take

$$\kappa ={\displaystyle \frac{e^{\sqrt{{ð}_0}}+1}{2}}<\infty .$$

Through the choice of λ, we have $\beta {\kappa }^4=\sqrt{\lambda }{\left({\displaystyle \frac{e^{\sqrt{{ð}_0}}+1}{2}}\right)}^4<1$.

Finally, solving

$$ð=\Re ð={\displaystyle \frac{\lambda ð}{1+ð}}$$

gives $ð=0.$ Hence, the only fixed point of ℜ is zero in $G.$

Therefore, all hypotheses of Theorem 1 are satisfied, and the Mann iteration converges strongly to a unique fixed point of ℜ.

3.2. Stability Results

In this section, we study the stability properties of the Mann iteration process in convex double-controlled metric-type spaces. We first establish the weak ℜ stability of the iteration process. Afterward, a data dependence result is obtained. To proceed, we recall the following auxiliary lemma.

Lemma 1

([27]). Let $\left\{k_n\right\}$ and $\left\{l_n\right\}$ be the sequences of non-negative real numbers satisfying $k_{n+1}\le h{k}_n+l_n$ for all $n\in \mathbb{N}$, where $h\in [0,1)$ and ${lim}_{n\to \infty }{l}_n=0.$ Then, ${lim}_{n\to \infty }{k}_n=0.$

The above lemma will play a key role in establishing the stability of the Mann iteration process.

We now investigate the weak ℜ stability of the iteration process defined in the previous section.

The notion of ℜ stability for iteration procedures in complete metric spaces was introduced by Qing and Rhoades [28]. We recall the definition below.

Definition 8

([28]). Let $(G,d_c)$ be a complete metric space, and let $\Re :G⟶G$ be a mapping with a fixed point ${ð}^{\ast }\in G.$ An iteration sequence $\left\{{ð}_n\right\}$ defined by ${ð}_{n+1}$ $=F(\Re ,{ð}_n)$, for all $n\in \mathbb{N},$ where F denotes an iteration function, is said to be ℜ-stable if $\left\{{ð}_n\right\}$ converges to a fixed point ${ð}^{\ast }$ of ℜ, and if every sequence $\left\{{\hslash }_n\right\}\subset G$ satisfies ${lim}_{n\to \infty }{d}_c\left({\hslash }_{n+1},F(\Re ,{\hslash }_n)\right)=0$, then we have ${lim}_{n\to \infty }{\hslash }_n={ð}^{\ast }.$

A related notion, called weak ℜ stability of iteration procedures, was introduced by Chen et al. [26] in 2020. The definition is recalled below.

Definition 9

([26]). Let $(G,d_c)$ be a complete metric space, and let $\Re :G⟶G$ be a mapping with a fixed point ${ð}^{\ast }\in G.$ An iteration sequence $\left\{{ð}_n\right\}$ defined by ${ð}_{n+1}$ $=F(\Re ,{ð}_n)$, for all $n\in \mathbb{N},$ where F denotes an iteration function, is said to be weakly ℜ-stable if $\left\{{ð}_n\right\}$ converges to a fixed point ${ð}^{\ast }$ of ℜ, and if every sequence $\left\{{\hslash }_n\right\}\subset G$ satisfies ${lim}_{n\to \infty }{d}_c\left({\hslash }_{n+1},F(\Re ,{\hslash }_n)\right)=0$, and the sequence $\left\{d_c\left({\hslash }_n,\Re {\hslash }_n\right)\right\}$ is bounded, then we have ${lim}_{n\to \infty }{\hslash }_n={ð}^{\ast }.$

Remark 2

([26]). Every ℜ-stable iterative procedure is automatically weakly ℜ-stable. In general, however, the reverse implication remains unclear.

Theorem 2.

Let $(G,d_c,\omega )$ be a complete convex DCMTS. Suppose that the hypotheses of Theorem 1 are satisfied, and assume in addition that

$$\underset{n\to \infty }{lim}{\delta }_n=0.$$

Then, the Mann iteration process is weakly ℜ-stable.

Proof. 

From Theorem 1, we know that the mapping ℜ possesses a unique fixed point ${ð}^{\ast }$ in G. Let $\left\{{\hslash }_n\right\}\subset G$ be a sequence satisfying ${lim}_{n\to \infty }{d}_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)=0$, and assume that the sequence $\left\{d_c\left({\hslash }_n,\Re {\hslash }_n\right)\right\}$ is bounded. We obtain

$$\begin{array}{ccc}\hfill d_c\left({\hslash }_{n+1},{ð}^{\ast }\right)& \le & \alpha \left({\hslash }_{n+1},{\hslash }_{n+1}\right)d_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)\hfill \\ & & +\mu \left({\hslash }_{n+1},{ð}^{\ast }\right)d_c\left(\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right),{ð}^{\ast }\right)\hfill \\ & \le & \alpha \left({\hslash }_{n+1},{\hslash }_{n+1}\right)d_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)\hfill \\ & & +\mu \left({\hslash }_{n+1},{ð}^{\ast }\right)\left[\begin{array}{c}\alpha \left({\hslash }_{n+1},\Re {\hslash }_n\right)d_c\left(\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right),\Re {\hslash }_n\right)\\ +\mu \left(\Re {\hslash }_n,{ð}^{\ast }\right)d_c\left(\Re {\hslash }_n,{ð}^{\ast }\right)\end{array}\right]\hfill \\ & \le & \alpha \left({\hslash }_{n+1},{\hslash }_{n+1}\right)d_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)\hfill \\ & & +\mu \left({\hslash }_{n+1},{ð}^{\ast }\right)\left[\alpha \left({\hslash }_{n+1},\Re {\hslash }_n\right){\delta }_n{d}_c\left({\hslash }_n,\Re {\hslash }_n\right)+\mu \left(\Re {\hslash }_n,{ð}^{\ast }\right)\beta d_c\left({\hslash }_n,{ð}^{\ast }\right)\right]\hfill \\ & =& \alpha \left({\hslash }_{n+1},{\hslash }_{n+1}\right)d_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)\hfill \\ & & +\mu \left({\hslash }_{n+1},{ð}^{\ast }\right)\alpha \left({\hslash }_{n+1},\Re {\hslash }_n\right){\delta }_n{d}_c\left({\hslash }_n,\Re {\hslash }_n\right)\hfill \\ & & +\mu \left({\hslash }_{n+1},{ð}^{\ast }\right)\mu \left(\Re {\hslash }_n,{ð}^{\ast }\right)\beta d_c\left({\hslash }_n,{ð}^{\ast }\right)\hfill \\ & \le & \kappa d_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)+{\kappa }^2{\delta }_n{d}_c\left({\hslash }_n,\Re {\hslash }_n\right)+{\kappa }^2\beta d_c\left({\hslash }_n,{ð}^{\ast }\right).\hfill \end{array}$$

Noticing that $\left\{d_c\left({\hslash }_n,\Re {\hslash }_n\right)\right\}$ is bounded, $\beta {\kappa }^2<{\displaystyle \frac{1}{{\kappa }^2}}<1$, ${lim}_{n\to \infty }{d}_c\left({\hslash }_{n+1},\omega \left({\hslash }_n,\Re {\hslash }_n;{\delta }_n\right)\right)=0$, and ${lim}_{n\to \infty }{\delta }_n=0,$ we can apply Lemma 1.

Consequently, we have

$$\underset{n\to \infty }{lim}{d}_c\left({\hslash }_n,{ð}^{\ast }\right)=0.$$

This finishes the proof. □

Next, we turn to the study of data dependence for the Mann iteration process in convex double-controlled metric-type spaces.

In order to establish the data dependence result, we first present the following auxiliary lemma and introduce the notion of an approximate operator.

Lemma 2

([29]). Let $\left\{r_n\right\}$ be a non-negative real sequence where there exists $n_0\in \mathbb{N}$ such that for all $n>n_0$ satisfying

$$r_{n+1}\le \left(1-{\psi }_n\right)r_n+{\psi }_n{\sigma }_n,$$

where ${\psi }_n\in \left(0,1\right)$ such that ${\sum }_{n=0}^{\infty }{\psi }_n=\infty$ and ${\sigma }_n\ge 0$ for all $n\in \mathbb{N},$ then

$$0\le \underset{n⟶\infty }{lim}sup{r}_n\le \underset{n⟶\infty }{lim}sup{\sigma }_n.$$

Definition 10

([29]). Let $\Re ,\wp :G⟶G$ be two operators. We say that ℘ is an approximate operator of ℜ for all $ð\in G$ and a fixed $\epsilon >0$ if $d_c(\Re ð,\wp ð)\le \epsilon .$

We are now ready to prove the following data dependence theorem.

Theorem 3.

Let G be a complete convex DCMTS with control functions $\alpha ,\mu :G\times G⟶[1,\infty )$, ℜ $:G⟶G$ be a contraction mapping with the fixed point ${ð}^{\ast }$, and ℘$:G⟶G$ be an approximate operator of ℜ with a fixed point ${\hslash }^{\ast }$. Let $\left\{{ð}_n\right\}$ be an iterative sequence generated by Equation (4) for ℜ, and define an iterative sequence $\left\{{\hslash }_n\right\}$ for ℘ as follows:

$${\hslash }_{n+1}=\omega \left({\hslash }_n,\wp {\hslash }_n;{\delta }_n\right),n\in \mathbb{N},$$

where $\left\{{\delta }_n\right\}$ is a real sequence in $[0,1)$ satisfying ${\sum }_{n=0}^{\infty }{\delta }_n=\infty .$ Assume that the control functions $\alpha ,\mu$ are bounded on $\Omega \times \Omega ,$ where Ω is defined in Theorem 1 and $\alpha ,\mu$ satisfy $\alpha (ð,\hslash )\le \xi ,$ $\mu (ð,\hslash )\le \varpi ,$ for all $ð,\hslash \in \Omega$, $\kappa :=max\left\{\xi ,\varpi \right\}$ and $\kappa \beta <1.$ Then, we have

$$d_c\left({ð}^{\ast },{\hslash }^{\ast }\right)\le {\displaystyle \frac{\kappa \epsilon }{1-\kappa \beta }}.$$

Proof. 

From the definition of the Mann iteration process for ℜ and $\wp ,$ we obtain

$$\begin{array}{ccc}\hfill d_c\left({ð}_{n+1},{\hslash }_{n+1}\right)& =& d_c\left(\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right),\omega \left({\hslash }_n,\wp {\hslash }_n;{\delta }_n\right)\right)\hfill \\ & \le & {\delta }_n{d}_c\left({ð}_n,{\hslash }_n\right)+\left(1-{\delta }_n\right)d_c\left(\Re {ð}_n,\wp {\hslash }_n\right)\hfill \\ & \le & {\delta }_n{d}_c\left({ð}_n,{\hslash }_n\right)+\left(1-{\delta }_n\right)\left[\begin{array}{c}\alpha \left(\Re {ð}_n,\Re {\hslash }_n\right)d_c\left(\Re {ð}_n,\Re {\hslash }_n\right)\\ +\mu \left(\Re {\hslash }_n,\wp {\hslash }_n\right)d_c\left(\Re {\hslash }_n,\wp {\hslash }_n\right)\end{array}\right]\hfill \\ & \le & {\delta }_n{d}_c\left({ð}_n,{\hslash }_n\right)+\left(1-{\delta }_n\right)\left[\varpi \beta d_c\left({ð}_n,{\hslash }_n\right)+\wp \epsilon \right]\hfill \\ & \le & {\delta }_n{d}_c\left({ð}_n,{\hslash }_n\right)+\left(1-{\delta }_n\right)\left[\kappa \beta d_c\left({ð}_n,{\hslash }_n\right)+\kappa \epsilon \right]\hfill \\ & \le & \left[{\delta }_n+\left(1-{\delta }_n\right)\kappa \beta \right]d_c\left({ð}_n,{\hslash }_n\right)+\left(1-{\delta }_n\right)\kappa \epsilon .\hfill \end{array}$$

By taking ${\rho }_n=1-{\delta }_n$, from last inequality, we have

$$d_c\left({ð}_{n+1},{\hslash }_{n+1}\right)\le \left[1-{\rho }_n\left(1-\kappa \beta \right)\right]d_c\left({ð}_n,{\hslash }_n\right)+{\rho }_n\left(1-\kappa \beta \right){\displaystyle \frac{\kappa \epsilon }{\left(1-\kappa \beta \right)}}.$$

For $\kappa \beta <1$, it can be easily seen that all conditions in Lemma 2 are satisfied. Therefore, we get

$$\underset{n⟶\infty }{lim}sup{d}_c\left({ð}_n,{\hslash }_n\right)\le \underset{n⟶\infty }{lim}sup{\displaystyle \frac{\kappa \epsilon }{\left(1-\kappa \beta \right)}}.$$

Since ${ð}_n⟶{ð}^{\ast }$ and ${ð}_n⟶{\hslash }^{\ast }$ as $n⟶\infty$, consequently, we deduce that

$$d_c\left({ð}^{\ast },{\hslash }^{\ast }\right)\le {\displaystyle \frac{\kappa \epsilon }{1-\kappa \beta }}.$$

□

The following example illustrates the above data dependence result.

Example 4.

Let $G=[0,1]$, and define the metric $d_c:G\times G⟶[0,\infty )$ by

$$d_c\left(ð,\hslash \right)=e^{\left|ð-\hslash \right|}-1,$$

for all $ð,\hslash \in G.$ The convex structure on G is given by

$$\omega (ð,\hslash ;\lambda )=\lambda ð+(1-\lambda )\hslash ,\forall ð,\hslash \in G,\lambda \in [0,1],$$

and the control functions $\alpha ,$ $\mu :G\times G⟶[1,\infty )$ are defined by

$$\alpha (ð,\hslash )={\displaystyle \frac{e^{\left|ð-\hslash \right|}+1}{2}}and\mu (ð,\hslash )={\displaystyle \frac{e^{\left|ð-\hslash \right|}+1}{2}}.$$

Under Example 2, $(G,d_c,\omega )$ is a convex DCMTS with the control functions $\alpha ,$ $\mu .$

Define the mapping $\Re :G⟶G$ and the approximate operator ℘$:G⟶G$ by

$$\Re ð={\displaystyle \frac{ð}{5}}and\wp ð={\displaystyle \frac{ð}{5}}+{\displaystyle \frac{1}{10}}.$$

For any $ð,\hslash \in G,$ we find that

$$\begin{array}{ccc}\hfill d_c\left(\Re ð,\Re \hslash \right)& =& e^{\left|\Re ð-\Re \hslash \right|}-1\hfill \\ & =& e^{{\displaystyle \frac{\left|ð-\hslash \right|}{5}}}-1\hfill \\ & \le & {\displaystyle \frac{1}{5}}{d}_c\left(ð,\hslash \right).\hfill \end{array}$$

Thus, ℜ is a contraction mapping with $\beta ={\displaystyle \frac{1}{5}}$, and ${ð}^{\ast }=0$ is the unique fixed point of ℜ. Similarly, ℘ is also a contraction with the same constant, and ${\hslash }^{\ast }={\displaystyle \frac{1}{8}}$ is the unique fixed point of ℘.

Moreover, for all $ð\in G$ and $\epsilon =e^{1/10}-1$, we get $d_c\left(\Re ð,\wp ð\right)\le \epsilon .$

Since $ð,\hslash \in G,$ we have $\left|ð-\hslash \right|\le 1.$ Hence, we find that

$$\alpha (ð,\hslash )\le {\displaystyle \frac{e+1}{2}}and\mu (ð,\hslash )\le {\displaystyle \frac{e+1}{2}}.$$

Thus, the control functions are bounded on $G\times G$ and therefore on $\Omega \times \Omega$. Hence, we may take $\kappa ={\displaystyle \frac{e+1}{2}}$, and we have

$$\kappa \beta ={\displaystyle \frac{e+1}{10}}<1.$$

Therefore, all assumptions of the Theorem 3 are satisfied.

Then, we can find the following estimate:

$$d_c\left({ð}^{\ast },{\hslash }^{\ast }\right)=e^{\left|0-{\displaystyle \frac{1}{8}}\right|}=e^{1/8}-1\le {\displaystyle \frac{\kappa \epsilon }{1-\kappa \beta }}.$$

Table 1. Convergence of the Mann iteration for ℜ and ℘.

$\mathit{n}\mathbf{\left(}\mathit{Iterative}\mathit{Step}\mathbf{\right)}$	${\mathit{ð}}_{\mathit{n}}$	${\mathit{\hslash }}_{\mathit{n}}$	${\mathit{d}}_{\mathit{c}}\left({\mathit{ð}}_{\mathit{n}},{\mathit{\hslash }}_{\mathit{n}}\right)$
1	0.7333333333	0.7666666667	0.0338951135
2	0.5866666667	0.6383333333	0.0530251894
5	0.3782826667	0.4559973333	0.0808141187
10	0.2469795540	0.3411078903	0.0987003712
50	0.0768159332	0.1922140585	0.1223204036
200	0.0259702147	0.1477239379	0.1294757932
1000	0.0072137245	0.1313125089	0.1321274041
5000	0.0019931755	0.1267440286	0.1328657342
10,000	0.0011446508	0.1260022118	0.1329857313
⋮	⋮	⋮	⋮
$n⟶\infty$	0.0000000000	0.1250000000	0.1331484531

shows that the sequences $\left\{{ð}_n\right\}$ and $\left\{{\hslash }_n\right\}$ generated by Equation (4) for ℜ and $\wp ,$ respectively, converge to the fixed points ${ð}^{\ast }=0$ and ${\hslash }^{\ast }={\displaystyle \frac{1}{8}},$ where ${\delta }_n={\displaystyle \frac{n+1}{n+2}},$ ${ð}_0={\hslash }_0=1.$

The above computations were obtained by using the MATLAB R2023b program.

4. Applications

As an application of Theorem 1, we studied the solvability of the following Fredholm integral equation:

$$ð\left(s\right)=f\left(s\right)+\varrho {\int }_i^{j}K(s,\tau )ð\left(\tau \right)d\tau ,s,\tau \in \left[i,j\right].$$

(6)

where $K:$ $\left[i,j\right]\times \left[i,j\right]⟶\mathbb{R}$ is a continuous function and $f\in C\left(\left[i,j\right]\right).$

Let $G=C\left(\left[i,j\right]\right)$ denote the space of all real-valued continuous functions on the interval $\left[i,j\right]$. Let the mapping $d_c:G\times G⟶[0,\infty )$ be defined by

$$d_c\left(ð,\hslash \right)=\underset{s\in \left[i,j\right]}{max}{\left|ð\left(s\right)-\hslash \left(s\right)\right|}^2,ð,\hslash \in G.$$

The convex structure is given by

$$\omega (ð,\hslash ;\lambda )=\lambda ð+(1-\lambda )\hslash ,\forall ð,\hslash \in G,\lambda \in [0,1].$$

It follows that $(G,d_c,\omega )$ forms a complete convex DCMTS with control functions $\alpha ,\mu :G\times G⟶[1,\infty )$ defined by

$$\alpha (ð,\hslash )=1+{∥ð∥}_{\infty }^2,\mu (ð,\hslash )=1+{∥\hslash ∥}_{\infty }^2.$$

Define the operator $\Re :G⟶G$ by

$$\Re ð\left(s\right)=f\left(s\right)+\varrho {\int }_i^{j}K(s,\tau )ð\left(\tau \right)d\tau ,ð\in G.$$

For ${\delta }_n\subset [0,1),$ consider the Mann iteration process

$${ð}_{n+1}=\omega \left({ð}_n,\Re {ð}_n;{\delta }_n\right)={\delta }_n{ð}_n+\left(1-{\delta }_n\right)\Re {ð}_n,n\in \mathbb{N}.$$

Theorem 4.

Consider the linear integral in Equation (6). Let $M={max}_{s,\tau \in \left[i,j\right]}\left|K(s,\tau )\right|$. Suppose that the sequence $\left\{{ð}_n\right\}$ is bounded in $C\left(\left[i,j\right]\right)$. If

$$\left|\varrho \right|M\left(j-i\right)<1,$$

then the Fredholm integral in Equation (6) has a unique solution ${ð}^{\ast }\in G$, and the sequence $\left\{{ð}_n\right\}$ converges to ${ð}^{\ast }.$

Proof. 

For any $ð,\hslash \in G=C\left(\left[i,j\right]\right)$ and $s\in \left[i,j\right]$, we have

$$\begin{array}{ccc}\hfill d_c\left(\Re ð,\Re \hslash \right)& =& \underset{s,\tau \in \left[i,j\right]}{max}{\left|\Re ð\left(s\right)-\Re \hslash \left(s\right)\right|}^2\hfill \\ & =& \underset{s,\tau \in \left[i,j\right]}{max}{\left|\varrho {\int }_i^{j}K(s,\tau )\left(ð\left(\tau \right)-\hslash \left(\tau \right)\right)d\tau \right|}^2\hfill \\ & \le & \underset{s,\tau \in \left[i,j\right]}{max}{\left|\varrho \right|}^2{\left|{\int }_i^j\left|K(s,\tau )\right|\left|ð\left(\tau \right)-\hslash \left(\tau \right)\right|d\tau \right|}^2\hfill \\ & \le & {\varrho }^2{M}^2{\left(j-i\right)}^2\underset{s,\tau \in \left[i,j\right]}{max}\left|ð\left(\tau \right)-\hslash \left(\tau \right)\right|\hfill \\ & =& {\varrho }^2{M}^2{\left(j-i\right)}^2{d}_c\left(ð,\hslash \right).\hfill \end{array}$$

Since $\left|\varrho \right|M\left(j-i\right)<1,$ it follows that $\beta ={\varrho }^2{M}^2{\left(j-i\right)}^2<1$, and hence

$$d_c\left(\Re ð,\Re \hslash \right)\le \beta d_c\left(ð,\hslash \right).$$

Therefore, the operator ℜ is a contraction.

Thus, all the hypotheses of Theorem 1 are satisfied. Therefore, ℜ has a unique fixed point ${ð}^{\ast }\in G$, and the Mann iteration sequence $\left\{{ð}_n\right\}$ converges to ${ð}^{\ast }$ in the sense that ${lim}_{n\to \infty }{d}_c({ð}_n,{ð}^{\ast })=0$ as $n⟶\infty .$

Finally, for each $s,\tau \in \left[i,j\right],$ we find that

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}sup\left|{\int }_i^{j}K(s,\tau )\left({ð}_n\left(\tau \right)-{ð}^{\ast }\left(\tau \right)\right)d\tau \right|& \le & \underset{n\to \infty }{lim}sup{\int }_i^j\left|K(s,\tau )\right|\left|{ð}_n\left(\tau \right)-{ð}^{\ast }\left(\tau \right)\right|d\tau \hfill \\ & \le & M\left(j-i\right)\underset{n\to \infty }{lim}supmax\left|{ð}_n\left(\tau \right)-{ð}^{\ast }\left(\tau \right)\right|d\tau \hfill \\ & =& M\left(j-i\right)\underset{n\to \infty }{lim}sup{d}_c{({ð}_n\left(s\right),{ð}^{\ast }\left(s\right))}^{1/2}.\hfill \end{array}$$

Since ${lim}_{n\to \infty }{d}_c({ð}_n,{ð}^{\ast })=0$ as $n⟶\infty ,$ we have

$$\underset{n\to \infty }{lim}{\int }_i^{j}K(s,\tau ){ð}_n\left(\tau \right)d\tau ={\int }_i^{j}K(s,\tau ){ð}^{\ast }\left(\tau \right)d_c\tau .$$

Substituting this equality into ${ð}^{\ast }\left(s\right)=\Re {ð}^{\ast }\left(s\right)$ shows that ${ð}^{\ast }\left(s\right)$ satisfies the Fredholm integral in Equation (6).

Hence, ${ð}^{\ast }\left(s\right)$ is the unique solution. □

5. Conclusions

In this paper, we introduced the concept of convex double-controlled metric-type spaces and investigated the convergence of the Mann iteration process for contractive mappings in this framework. Under suitable boundedness conditions on the control functions, we established the existence and uniqueness of fixed points and proved that the Mann iterative sequence converges to the fixed point.

Furthermore, we studied the stability behavior of the Mann iteration process and obtained a weak ℜ stability result. A data dependence theorem was also established to describe the relationship between the fixed points of an operator and its approximate operator. Finally, illustrative examples were provided to demonstrate the applicability of the obtained results.

These results extend several existing works on iterative methods for generalized metric-type spaces and contribute to the study of fixed point theory in double-controlled metric-type spaces equipped with convex structures.