On the Fixed Points of Large Enriched Contractions in Convex Metric Space with an Application

Abstract

This paper investigates the fixed points of large enriched contractions in a convex metric space as well as in a convex G-metric space. We establish the sufficient conditions for the existence and uniqueness of fixed points for these mappings. We use the Kransnoselskij-type iterative procedure for the approximation of these fixed points in complete convex metric spaces. We demonstrate that the Kransnoselskij-type iterative approach converges to the unique fixed point associated with large enriched contractions. Our results extend and generalize classical fixed point results by introducing this novel contraction mapping. Some illustrative examples are presented to demonstrate the applicability of our theorems. In the last section, we study the existence of a solution of nonlinear equations as a practical application of our principle findings.

1. Introduction and Preliminaries

The study of fixed points in metric spaces has advanced significantly, leading to the development of various contraction principles. Among these, the Banach contraction principle [1] remains a foundational result, ensuring the existence and uniqueness of a fixed point under certain contractive conditions. However, one limitation of the Banach contraction principle is that it requires the mapping to be continuous throughout its domain. In 1968, Kannan [2] introduced a new condition that also guarantees the existence of a unique fixed point, similar to the Banach contraction principle. Unlike the Banach condition, Kannan demonstrated that there are mappings that have a discontinuity point in their domain but still have a fixed point, although such mappings are continuous at their fixed point. In 1973, motivated by the work of Kannan, Hardy and Rogers [3] proposed new contractive conditions for ensuring the existence and uniqueness of fixed points. Subsequently, in 2012, Samet et al. [4] introduced the concept of $\alpha$-$\psi$ contraction mapping, which generalizes several earlier contraction mappings. In 2012, Wardowski [5] enhanced the concept of F-contraction mapping. This contraction mapping introduces a more generalized framework that can address a wider variety of nonlinear mappings, thus broadening the applicability of fixed point results. In 2018, Karapinar [6] proposed the concept of an interpolative-type contraction mapping by merging the frameworks of metric fixed point theory and interpolation theory. Further, Karapinar et al. [7] proposed the notion of interpolative Hardy–Rogers-type contraction by combining the concepts of interpolative contraction and Hardy–Rogers-type contraction mapping.

Investigating various abstract spaces with a more general structure than a metric space is another possible way to improve the Banach contraction principle. In 1989, Bakhtin [8] introduced a new axiom that is weaker than the classical triangular inequality, leading to the definition of a b-metric space. Matthews [9] explored the concept of a partial metric space to provide a better framework for certain computer science applications where a classical metric is not sufficient. Later, in 2000, Hitzler and Seda [10] introduced the notion of a dislocated metric space in which the self-distance of a point need not be equal to zero. Mustafa and Sims [11] enhanced the concept of classical metric spaces by introducing a more generalized form of distance measurement known as a G-metric space. Furthermore, in 2019, Jain and Kaur [12] introduced the concept of a $G_b$-metric space by combining the concepts of a b-metric space and G-metric space. Recently, Jleli and Samet [13] introduced the notion of a perturbed metric space to address real-world problems where slight deviations in measurements are unavoidable.

In 1970, Takahashi [14] proposed the concept of convex structure in a metric space, known as a convex metric space. This concept of convexity has been used as a fundamental tool for proving various fixed point results. In 1988, Ding [15] studied the convergence of the Ishikawa iterative scheme for quasi-contractive and quasi-nonexpansive mappings in a convex metric space. Beg [16] explored the convergence of asymptotically nonexpansive mappings using the Mann-type iterative scheme in a uniformly convex metric space. Later, in 2016, Fukhar-ud-din and Berinde [17] introduced the modified Noor iterative method in a convex metric space for the class of quasi-contractive-type operators. Chen et al. [18], in 2020, established the notion of the convex b-metric space and generalized Mann iterative scheme for this newly defined space. Further, in 2022, Li et al. [19] explored the concept of the rectangular b-metric space and generalize Mann iterative algorithm and proved fixed point results in this space.

Many authors have studied applications of fixed point results for solving differential equations, integral equations, systems of linear equations, etc., by making use of various mappings. In 1996, Burton [20] developed the idea of large contraction mappings to solve integral equations. These mappings are weaker than Banach contraction mappings; that is, every Banach contraction mapping is a large contraction, but the converse is not true. We now present some definitions from the literature.

In 1970, Takahashi defined the convex metric space as follows.

Definition 1

([14]). Let $(H,d)$ be a metric space. A continuous function $W:H\times H\times [0,1]\to H$ is said to have a convex structure if there exist $v_1,v_2\in H$ and $\mu \in [0,1]$ such that

$$d(v_3,W(v_1,v_2,\mu ))\le \mu d(v_3,v_1)+(1-\mu )d(v_3,v_2)$$

(1)

for any $v_3\in H.$

A metric space $(H,d)$ endowed with a convex structure W is called a convex metric space and is denoted by $(H,d,W).$

In 1996, Burton developed the idea of large contraction mapping as follows.

Definition 2

([20]). Let $(H,d)$ be a metric space. A mapping $R:H\to H$ is said to be a large contraction if for $v_1,v_2\in H$ with $v_1\ne v_2$, we have $d(R{v}_1,R{v}_2)<d(v_1,v_2)$ and for all $ϵ>0,$ there exists $0<\delta <1$ such that

$$[d(v_1,v_2)\ge ϵ]\Rightarrow d(R{v}_1,R{v}_2)\le \delta d(v_1,v_2).$$

Remark 1

([20]). Every Banach contraction is a large contraction, but the converse need not to be true.

Example 1.

Let $R:[0,1]\to [0,1]$ be a function defined as $Ru=u-{\displaystyle \frac{1}{3}}{u}^3,$ for all $u\in [0,1].$ Assume that $d(u_1,u_2)=|u_1-u_2|.$ Then, the mapping R is a large contraction, but not a Banach contraction.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R{u}_1,R{u}_2)& =|u_1-{\displaystyle \frac{1}{3}}{u}_1^3-u_2+{\displaystyle \frac{1}{3}}{u}_2^3|\hfill \\ \hfill & =|(u_1-u_2)-{\displaystyle \frac{1}{3}}(u_1-u_2)(u_1^2+u_1{u}_2+u_2^2)|\hfill \\ \hfill & =|(u_1-u_2)-{\displaystyle \frac{1}{3}}(u_1-u_2)\left(u_1^2+u_2^2+{\displaystyle \frac{u_1^2+u_2^2-{(u_1-u_2)}^2}{2}}\right)|.\hfill \end{array}\end{array}$$

Using the inequality $|u_1-u_2{|}^2\le 2(u_1^2+u_2^2)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R{u}_1,R{u}_2)& =|u_1-u_2|\left|1-{\displaystyle \frac{1}{2}}(u_1^2+u_2^2)+{\displaystyle \frac{|u_1-u_2{|}^2}{6}}\right|\hfill \\ \hfill & \le |u_1-u_2||1-{\displaystyle \frac{1}{4}}{|u_1-u_2|}^2+{\displaystyle \frac{|u_1-u_2{|}^2}{6}}|\hfill \\ \hfill & =|u_1-u_2||1-{\displaystyle \frac{1}{12}}{|u_1-u_2|}^2|.\hfill \end{array}\end{array}$$

If $|u_1-u_2|>ϵ,$ then we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R{u}_1,R{u}_2)& \le |u_1-u_2||1-{\displaystyle \frac{1}{12}}{ϵ}^2|\hfill \\ \hfill & =\delta |u_1-u_2|.\hfill \end{array}\end{array}$$

Thus, R is a large contraction mapping. Assume $u_2=0$ and $u_1\to 0.$ Then,

$${\displaystyle \underset{u_1\to 0}{lim}\frac{d(R{u}_1,R{u}_2)}{d(u_1,u_2)}=\underset{u_1\to 0}{lim}\frac{u_1-\frac{1}{3}{u}_1^3}{u_1}=1.}$$

Hence, R is not a Banach contraction.

Recently, Berinde and Păcurar [21] introduced the notion of enriched contraction in the framework of a convex metric space.

Definition 3

([21]). Let $(H,d,W)$ be a convex metric space. A mapping $R:H\to H$ is called an enriched contraction if there exist $r,\mu \in [0,1)$ such that

$$d(W(u_1,R{u}_1,\mu ),W(u_2,R{u}_2,\mu ))\le rd(u_1,u_2)$$

(2)

for all distinct $u_1,u_2\in H.$

Remark 2.

Large contraction mappings and enriched contraction mappings are independent mappings.

Example 2.

Let $R:[0,1]\to [0,1]$ be a mapping defined as $Ru=u-{\displaystyle \frac{u^4}{8}},$ for all $u\in [0,1].$ Assume that $d(u_1,u_2)=|u_1-u_2|.$ Then, the mapping R is a large contraction but not an enriched contraction.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R{u}_1,R{u}_2)& =|u_1-{\displaystyle \frac{u_1^4}{8}}-u_2+{\displaystyle \frac{u_2^4}{8}}|\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R{u}_1,R{u}_2)& =|u_1-u_2||1-{\displaystyle \frac{1}{8}}(u_1+u_2)(u_1^2+u_2^2)|\hfill \\ \hfill & \le |u_1-u_2|\left(1-{\displaystyle \frac{1}{16}}{|u_1-u_2|}^3\right)\hfill \\ \hfill & \le \left(1-{\displaystyle \frac{1}{16}}{ϵ}^3\right)|u_1-u_2|\hfill \\ \hfill & =\delta \left(ϵ\right)|u_1-u_2|.\hfill \end{array}\end{array}$$

Thus, R is a large contraction but not an enriched contraction. If we take $W(u_1,u_2,\mu )=\mu u_1+(1-\mu )u_2$, $u_2=0$ and $u_1\to 0,$ then

$${\displaystyle \underset{u_1\to 0}{lim}\frac{d(W(u_1,R{u}_1,\mu ),W(u_2,R{u}_2,\mu ))}{d(u_1,u_2)}=\underset{u_1\to 0}{lim}\left|\frac{\mu u_1+(1-\mu )\left(u_1-\frac{u_1^4}{8}\right)}{u_1}\right|=1.}$$

Example 3.

Let $H:\mathbb{R}\to \mathbb{R}$ be a mapping defined as $Hu=au+b,$ where a and b are constants such that $a<-1$ and $u\in \mathbb{R}.$ Assume that $d(u_1,u_2)=|u_1-u_2|$ and $W(u_1,u_2,\mu )=\mu u_1+(1-\mu )u_2,$ where ${\displaystyle \mu \in \left(\frac{-a-1}{1-a},1\right).}$ Then, the mapping H is an enriched contraction but not a Banach contraction.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(W(u_1,H{u}_1,\mu ),W(u_2,H{u}_2,\mu ))& =\left|\right(W(u_1,H{u}_1,\mu )-W(u_2,H{u}_2,\mu )|\hfill \\ \hfill & =|\mu +a(1-\mu )||u_1-u_2|.\hfill \end{array}\end{array}$$

It is clear that if $\left|a\right|>1$, then $d(H{u}_1,H{u}_2)\ge d(u_1,u_2).$ But if $a>1,$ then we cannot have $|\mu +a(1-\mu \left)\right|<1.$ We need $-1<\mu +a(1-\mu )<1,$ which is satisfied whenever

$$\mu \in \left({\displaystyle \frac{-a-1}{1-a}},1\right)\in (0,1);\mathrm{where}a<-1.$$

In order to generalize the concept of large contractions, Özyurt [22] defined the concept of extended large contraction mapping as follows.

Definition 4

([22]). Let $(H,d)$ be a metric space. A mapping $R:H\to H$ is said to be an extended large contraction if for $u_1,u_2\in H$ with $u_1\ne u_2,$ we have $d(R{u}_1,R{u}_2)<d(u_1,u_2)$ and for all $ϵ>0$, there exists $\psi \in \Psi$ such that

$$[d(u_1,u_2)\ge ϵ]\Rightarrow d(R{u}_1,R{u}_2)\le \psi \left(d(u_1,u_2)\right),$$

where Ψ is the family of functions $\psi :[0,\infty )\to [0,\infty )$ satisfying the following conditions:

(i) 

ψ is nondecreasing;

(ii) 

${\displaystyle \sum _{n=1}^{\infty }{\psi }^n\left(t\right)<\infty }$, for all $t>0,$ where ${\psi }^n=\psi$oψo…o$\psi .$

Clearly, conditions (i) and (ii) of Definition 4 imply $\psi \left(t\right)<t$ for every $t>0.$

In 2006, Mustafa and Sims [13] defined the G-metric space as follows.

Definition 5

([13]). Let H be a nonempty set. Assume that $G:H\times H\times H\to [0,+\infty )$ is a mapping satisfying the following conditions:

(i) 

$G(u_1,u_2,u_3)=0$, if $u_1=u_2=u_3;$

(ii) 

$0<G(u_1,u_1,u_2)$, for all $u_1,u_2\in H$ with $u_1\ne u_2;$

(iii) 

$G(u_1,u_1,u_2)\le G(u_1,u_2,u_3)$ for all $u_1,u_2,u_3\in H$ with $u_2\ne u_3;$

(iv) 

$G(u_1,u_2,u_3)=G(u_1,u_3,u_2)=G(u_2,u_3,u_1)=\dots$ (symmetry in all three variables);

(v) 

$G(u_1,u_2,u_3)\le G(u_1,u,u)+G(u,u_2,u_3)$, for all $u_1,u_2,u_3,u\in H.$

Then, G is called G-metric on H and $(H,G)$ is called a G-metric space.

Ji et al. [23] introduced the notion of a convex G-metric space as follows.

Definition 6

([23]). Let $(H,G)$ be a G-metric space. A mapping $W:H\times H\times [0,1]\to H$ is said to be a convex structure on H if for each $u_1,u_2,u_3,u_4\in H$ and $\mu \in [0,1],$

$$G(u_1,u_2,W(u_3,u_4,\mu ))\le \mu G(u_1,u_2,u_3)+(1-\mu )G(u_1,u_2,u_4)$$

(3)

is satisfied. Then, the triplet $(H,G,W)$ is called a convex G-metric space.

Lemma 1

([23]). Let $(H,G,W)$ be a convex G-metric space. If $\mu \in [0,1]$, then the convex G-metric space is symmetric.

In recent years, researchers have explored various generalizations and extensions of classical fixed point results by considering enriched contractions in larger settings, such as convex metric spaces. In order to generalize the enriched contraction, Berinde and Păcurar [24] introduced the notion of enriched Ćirić–Reich–Rus contractions in Banach spaces as well as in convex metric spaces. In 2022, Panicker and Shukla [25] obtained stability results of fixed point sets associated with a sequence of enriched contraction mappings in the setting of convex metric spaces. Further, Rawat et al. [26] defined and studied interpolative enriched contractions of the Kannan type, Hardy–Rogers type, and Matkowski type within the setting of convex metric spaces. Anjali et al. [27] introduced enriched Ćirić-type and enriched Hardy–Rogers contractions for which they established fixed point theorems in the Banach space and convex metric space. They showed that Ćirić-type and Hardy–Rogers contractions are unsaturated classes of mappings. Recently, in 2024, Rani et al. [28] proposed the theory of enriched contraction mapping by introducing a new type of contraction mappings known as hybrid enriched contractions in convex metric spaces.

The aim of this paper is to generalize the notions of large contraction, extended large contraction, and enriched contraction and to study the existence and uniqueness of fixed points in the framework of convex metric spaces as well as in convex G-metric spaces. This paper is divided into four sections. Section 1 is introductory, Section 2 introduces the concept of large enriched contractions in the framework of convex metric spaces, Section 3 extends the idea of large enriched contractions in convex G-metric spaces, and an application is studied in Section 4 to explore the existence of solutions of nonlinear equations.

2. Large Contractions in Convex Metric Space

In this section, we define the concept of large enriched contraction in the framework of a convex metric space.

Definition 7.

Let $(H,d,W)$ be a convex metric space. A mapping $R:H\to H$ is said to be a large enriched contraction if for $u_1,u_2\in H$ with $u_1\ne u_2$ and $\mu \in [0,1),$ we have

$$d(R_{\mu }{u}_1,R_{\mu }{u}_2)<d(u_1,u_2),\mathrm{where}{R}_{\mu }u=W(u,Ru,\mu ),$$

and for all $ϵ>0,$ there exists $0<\delta <1$ such that

$$[d(u_1,u_2)\ge ϵ]\Rightarrow d(R_{\mu }{u}_1,R_{\mu }{u}_2)\le \delta d(u_1,u_2).$$

(4)

Definition 8.

Let $(H,d,W)$ be a convex metric space. A mapping $R:H\to H$ is said to be an extended large enriched contraction if for $u_1,u_2\in H$ with $u_1\ne u_2$ and $\mu \in [0,1),$ we have

$$d(R_{\mu }{u}_1,R_{\mu }{u}_2)<d(u_1,u_2)\mathrm{where}{R}_{\mu }u=W(u,Ru,\mu ),$$

and for all $ϵ>0,$ there exists $\psi \in \Psi$ such that

$$[d(u_1,u_2)\ge ϵ]\Rightarrow d(R_{\mu }{u}_1,R_{\mu }{u}_2)\le \psi \left(d(u_1,u_2)\right).$$

(5)

Remark 3.

Every enriched contraction is a large enriched contraction, but the converse need not to be true.

Example 4.

Let $R:[0,1]\to [0,1]$ be a mapping defined as $Ru=u-u^4,$ for all $u\in [0,1].$ Assume that $d(u_1,u_2)=|u_1-u_2|$ and $W(u_1,u_2,\mu )=\mu u_1+(1-\mu )u_2$. Then, the mapping R is a large enriched contraction but not an enriched contraction.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R_{\mu }{u}_1,R_{\mu }{u}_2)& =|\mu u_1+(1-\mu )(u_1-u_1^4)-\mu u_2-(1-\mu )(u_2-u_2^4)|\hfill \\ \hfill & =|u_1-u_2||1-(1-\mu )(u_1^2+u_2^2)(u_1+u_2)|.\hfill \end{array}\end{array}$$

Using the inequalities $|u_1-u_2{|}^2\le 2(u_1^2+u_2^2)$ and $|u_1-u_2|\le |u_1+u_2|$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R_{\mu }{u}_1,R_{\mu }{u}_2)& \le |u_1-u_2||1-(1-\mu )|u_1-u_2|.{\displaystyle \frac{|u_1-u_2{|}^2}{2}}|\hfill \\ \hfill & =|u_1-u_2||1-(1-\mu ){\displaystyle \frac{|u_1-u_2{|}^3}{2}}|\hfill \\ \hfill & \le |u_1-u_2||1-(1-\mu ){\displaystyle \frac{{ϵ}^3}{2}}|\hfill \\ \hfill & =\delta \left(ϵ\right)|u_1-u_2|.\hfill \end{array}\end{array}$$

Thus, R is a large enriched contraction, but R is not an enriched contraction. If we take $u_2=0,$ and $u_1$ is very small, then

$${\displaystyle \underset{u_1\to 0}{lim}\left|\frac{d(R_{\mu }{u}_1,R_{\mu }{u}_2)}{d(u_1,u_2)}\right|=\underset{u_1\to 0}{lim}\left|\frac{\mu u_1+(1-\mu )(u_1-u_1^4)}{u_1}\right|=1.}$$

Example 5.

Let $R:[-1,1]\to [-1,1]$ be a mapping defined as $Ru=-u^3$, for all $u\in [-1,1]$. Assume that $d(u_1,u_2)=|u_1-u_2|$ and $W(u_1,u_2,\mu )=\mu u_1+(1-\mu )u_2.$ Then, the mapping R is a large enriched contraction, but not a large contraction.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(R_{\mu }{u}_1,R_{\mu }{u}_2)& =|\mu u_1+(1-\mu )(-u_1^3)-\mu u_2-(1-\mu )(-u_2^3)|\hfill \\ \hfill & =|\mu (u_1-u_2)-(1-\mu )(u_1-u_2)(u_1^2+u_2^2+u_1{u}_2)|\hfill \\ \hfill & =|u_1-u_2||\mu -(1-\mu ){\displaystyle \frac{3}{2}}(u_1^2+u_2^2)+{\displaystyle \frac{(1-\mu )}{2}}{|u_1-u_2|}^2|\hfill \\ \hfill & \le |u_1-u_2||\mu -(1-\mu ){\displaystyle \frac{3}{4}}|u_1-u_2{|}^2+{\displaystyle \frac{(1-\mu )}{2}}{|u_1-u_2|}^2|\hfill \\ \hfill d(R_{\mu }{u}_1,R_{\mu }{u}_2)& =|u_1-u_2||\mu -(1-\mu ){\displaystyle \frac{1}{4}}{|u_1-u_2|}^2|\hfill \\ \hfill & =|u_1-u_2||\mu -(1-\mu ){\displaystyle \frac{1}{4}}{ϵ}^2|\hfill \\ \hfill & =\delta \left(ϵ\right)|u_1-u_2|.\hfill \end{array}\end{array}$$

Thus, R is a large enriched contraction.

For $u_1=1$ and $u_2={\displaystyle \frac{1}{2}}$, we have

$$d(H{u}_1,H{u}_2)={\displaystyle \frac{9}{8}}\ge {\displaystyle \frac{1}{2}}=d(u_1,u_2).$$

Thus, R is not a large contraction.

Figure 1 represents the relationship between various contraction mappings.

Figure 1. Relationship between various contraction mappings.

We now present the results concerning the existence and uniqueness of a fixed point associated with these two categories of large enriched contractions in the framework of a convex metric space.

Theorem 1.

Let $(H,d,W)$ be a complete convex metric space and $R:H\to H$ be a large enriched contraction map. Assume that there exist a $u_0\in H$ and a constant $L>0$ such that $d(u_0,R_{\mu }^n{u}_0)\le L$ for all $n\ge 1$. Then, R has a unique fixed point in H.

Proof.

The proof of this theorem is mainly divided into four steps:

Step 1.

Initially, we shall prove that the sequence $\left\{{\xi }_n\right\}=\left\{d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\right\}$ is a decreasing sequence.

Using the definition of a large enriched contraction, we have

$$d(R_{\mu }{u}_1,R_{\mu }{u}_2)<d(u_1,u_2).$$

(6)

Setting $u_1=R_{\mu }^n{u}_0$ and $u_2=R_{\mu }^{n-1}{u}_0$ in (6), we obtain

$$d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)<d(R_{\mu }^n{u}_0,R_{\mu }^{n-1}{u}_0)<\cdots <d(R_{\mu }{u}_0,u_0).$$

(7)

Step 2.

In the second step, our aim is to prove that ${\displaystyle \underset{n\to \infty }{lim}{\xi }_n=0.}$

Since the sequence $\left\{{\xi }_n\right\}=\left\{d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\right\}$ is strictly decreasing, ${\displaystyle \underset{n\to \infty }{lim}{\xi }_n=r\ge 0.}$

If possible, let $r>0$. Then, for all $n\ge 1$, we have

$$d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\ge r.$$

(8)

As a result, there exists $0<\delta <1$ such that

$$\begin{array}{ll}d(R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+1}{u}_0)& =d(R_{\mu }\left(R_{\mu }^{n+1}{u}_0\right),R_{\mu }\left(R_{\mu }^n{u}_0\right))\\ & \le \delta d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\\ & \le {\delta }^{2}d(R_{\mu }^n{u}_0,R_{\mu }^{n-1}{u}_0)\\ & .\\ & .\\ d(R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+1}{u}_0)& \le {\delta }^{n+1}d(u_0,R_{\mu }{u}_0)\\ & \le {\delta }^{n+1}L.\end{array}$$

(9)

Using (9), we have ${\displaystyle \underset{n\to \infty }{lim}d(R_{\mu }^n{u}_0,R_{\mu }^{n+1}{u}_0)=0}$, which is a contradiction. Hence, $r=0.$

Step 3.

The objective of the next step is to prove that the sequence $\left\{u_n\right\}$ given by $R_{\mu }^n{u}_0=u_{n+1}$ is a Cauchy sequence.

By contradiction, assume that $\left\{u_n\right\}$ is not a Cauchy sequence. Therefore, there exist an $ϵ>0$ and subsequences of integers $m_k$, $n_k$, and $N_k$ such that

$$d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)\ge ϵ,\mathrm{for}\mathrm{some}{m}_k>n_k>N_k.$$

As R is a large enriched contraction, there exists a $\delta \in [0,1)$ such that

$$ϵ\le d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)\le \delta d(R_{\mu }^{m_k-1}{u}_0,R_{\mu }^{n_k-1}{u}_0)<\cdots <{\delta }^{n_k}d(u_0,R_{\mu }^{m_k-n_k}{u}_0)\le {\delta }^{n_k}L.$$

By taking $k\to \infty$ in the above equation, we obtain ${\displaystyle \underset{k\to \infty }{lim}d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)=0}$, which is not possible. Hence, $\left\{u_n\right\}$ is a Cauchy sequence in $H.$

Step 4.

In the last phase, we shall prove the existence and uniqueness of a fixed point.

Because $(H,d,W)$ is a complete metric space, there exists a $u\in H$ such that ${\displaystyle \underset{n\to \infty }{lim}{R}_{\mu }^n{u}_0=u}$. The continuity of $R_{\mu }$ implies that $R_{\mu }u=u.$

Let $u^{\prime }$ be another fixed point of $R_{\mu }$ such that $u^{\prime }\ne u;$ as a consequence, there exists an $ϵ>0$ such that $d(u,u^{\prime })\ge ϵ.$ Using the contractive condition, we have

$$d(u,u^{\prime })=d(R_{\mu }u,R_{\mu }{u}^{\prime })\le \delta d(u,u^{\prime }),$$

which contradicts $u\ne u^{\prime }$. Hence, $R_{\mu }$ has a unique fixed point. Using Lemma 4 of [21], we obtain $Ru=u.$ □

Theorem 2.

Let $(H,d,W)$ be a complete convex metric space, and $R:H\to H$ be an extended large enriched contraction map. Assume that there exists $u_0\in H$ such that the sequence $\left\{{\left(d(u_0,R_{\mu }^n{u}_0)\right)}_{n\ge 1}\right\}$ is bounded. Then, R has a unique fixed point in H.

Proof.

Define ${\xi }_n=d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)$. Using assumptions of the theorem and following Step 1 of Theorem 1, we obtain that the sequence $\left\{{\xi }_n\right\}=\left\{d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\right\}$ is strictly decreasing; therefore, ${\displaystyle \underset{n\to \infty }{lim}{\xi }_n=r\ge 0.}$

Now, let $r>0$. Then, for all $n\ge 1$, we have

$$d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\ge r.$$

(10)

Hence, there exists $\psi \in \Psi$ such that

$$\begin{array}{ll}d(R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+1}{u}_0)& =d(R_{\mu }\left(R_{\mu }^{n+1}{u}_0\right),R_{\mu }\left(R_{\mu }^n{u}_0\right))\\ & \le \psi \left(d(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0)\right)\\ & \le {\psi }^2\left(d(R_{\mu }^n{u}_0,R_{\mu }^{n-1}{u}_0)\right)\\ & .\\ & .\\ & .\\ d(R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+1}{u}_0)& \le {\psi }^{n+1}\left(d(u_0,R_{\mu }{u}_0)\right)\\ & \le {\psi }^{n+1}\left(L\right).\end{array}$$

(11)

Using (11), we have ${\displaystyle \underset{n\to \infty }{lim}d(R_{\mu }^n{u}_0,R_{\mu }^{n+1}{u}_0)=0}$, which is a contradiction. Thus, $r=0.$

Now, assume that $\left\{u_n\right\}$ is not a Cauchy sequence. So, there exist an $ϵ>0$ and subsequences of integers $m_k$, $n_k$, and $N_k$ such that

$$d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)\ge ϵ,\mathrm{for}\mathrm{some}{m}_k>n_k>N_k.$$

Further, as R is an extended large enriched contraction,

$$ϵ\le d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)\le \psi \left(d(R_{\mu }^{m_k-1}{u}_0,R_{\mu }^{n_k-1}{u}_0)\right)<\cdots <{\psi }^{n_k}\left(d(u_0,R_{\mu }^{m_k-n_k}{u}_0)\right)\le {\psi }^{n_k}\left(L\right).$$

By considering $k\to \infty$ in the above equation, we find that ${\displaystyle \underset{k\to \infty }{lim}d(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0)=0}$, which is not possible. Thus, $\left\{u_n\right\}$ is a Cauchy sequence in $H.$ Because $(H,d,W)$ is a complete metric space, there exists a $u\in H$ such that ${\displaystyle \underset{n\to \infty }{lim}{R}_{\mu }^n{u}_0=u}$. Moreover, the continuity of $R_{\mu }$ implies that $R_{\mu }u=u.$

Let $u^{\prime }\ne u$ be another fixed point of $R_{\mu }$. Therefore, there exists an $ϵ>0$ such that $d(u,u^{\prime })\ge ϵ$. Using the contractive condition, we obtain

$$d(u,u^{\prime })=d(R_{\mu }u,R_{\mu }{u}^{\prime })\le \psi \left(d(u,u^{\prime })\right)<d(u,u^{\prime }),$$

which is a contradiction. Hence, $R_{\mu }$ has a unique fixed point. By making use of Lemma 4 of [21], we obtain $Ru=u$. □

Now, we present the consequences of Theorems 1 and 2.

Corollary 1.

Let $(R,d)$ be a complete metric space. Let $R:H\to H$ be a large contraction mapping. Assume that there exist a $u_0\in H$ and a constant $L>0$ such that $d(u_0,R^n{u}_0)\le L$ for all $n\ge 1$. Then, R has a unique fixed point.

Proof.

By substituting $\mu =0$ in Theorem 1, we obtain the desired result. □

Corollary 2.

Let $(R,d)$ be a complete metric space. Let $R:H\to H$ be an extended large contraction mapping. Assume that there exist a $u_0\in H$ and a constant $L>0$ such that $d(u_0,R^n{u}_0)\le L$ for all $n\ge 1$. Then, R has a unique fixed point.

Proof.

Put $\mu =0$ in Theorem 2, we obtain the result. □

3. Large Contractions in Convex G-Metric Space

Lemma 2.

Let $(H,G,W)$ be a convex G-metric space. For all $u_1,u_2\in H$ and $\mu \in [0,1],$ the following equality holds:

$$G(u_1,u_2,u_2)=G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))+G(W(u_1,u_2,\mu ),u_2,u_2).$$

Proof.

Using the rectangular inequality and Lemma 1,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G({\mu }_1,u_2,u_2)& \le G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))+G(W(u_1,u_2,\mu ),u_2,u_2)\hfill \\ \hfill & =G(u_1,u_1,W(u_1,u_2,\mu ))+G(W(u_1,u_2,\mu ),u_2,u_2)\hfill \\ \hfill & \le \mu G(u_1,u_1,u_1)+(1-\mu )G(u_1,u_1,u_2)+\mu G(u_1,u_2,u_2)+(1-\mu )G(u_2,u_2,u_2)\hfill \\ \hfill & =\mu G(u_1,u_2,u_2)+(1-\mu )G(u_1,u_2,u_2)\hfill \\ \hfill & =G(u_1,u_2,u_2).\hfill \end{array}\end{array}$$

□

Lemma 3.

Let $(H,G,W)$ be a convex G-metric space. For all $u_1$, $u_2$$\in H$ and $\mu \in [0,1]$, we have

$$G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))=(1-\mu )G(u_1,u_2,u_2)\mathrm{and}$$

$$G(W(u_1,u_2,\mu ),u_2,u_2)=\mu G(u_1,u_2,u_2).$$

Proof.

Using the rectangular inequality and Lemma 1,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))& =G(u_1,u_1,W(u_1,u_2,\mu ))\hfill \\ \hfill & \le (1-\mu )G(u_1,u_1,u_2)\hfill \\ \hfill & =(1-\mu )G(u_1,u_2,u_2).\hfill \end{array}\end{array}$$

If $G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))<(1-\mu )G(u_1,u_2,u_2)$, then by Lemma 2,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G(u_1,u_2,u_2)& =G(u_1,W(u_1,u_2,\mu ),W(u_1,u_2,\mu ))+G(W(u_1,u_2,\mu ),u_2,u_2)\hfill \\ \hfill & <(1-\mu )G(u_1,u_2,u_2)+\mu G(u_1,u_2,u_2)\hfill \\ \hfill & =G(u_1,u_2,u_2)\hfill \end{array}\end{array}$$

which is a contradiction. Similarly, $G(W(u_1,u_2,\mu ),u_2,u_2)=\mu G(u_1,u_2,u_2)$. □

Lemma 4.

Let $(H,G,W)$ be a convex G-metric space and $R:H\to H$ be a mapping. Define $R_{\mu }:H\to H$ as $R_{\mu }u=W(u,Ru,\mu )$, $u\in H$. Then, for $\mu \in [0,1)$, $Fix\left(R\right)=Fix\left(R_{\mu }\right)$.

Proof.

For $\mu =0$, $R_{\mu }=R$. Let $\mu \in (0,1)$ and $u\in Fix\left(R\right)$. Therefore,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G(u,R_{\mu }u,R_{\mu }u)& =G(u,W(u,Ru,\mu ),W(u,Ru,\mu \left)\right)\hfill \\ \hfill & =G(u,u,W(u,Ru,\mu \left)\right)\hfill \\ \hfill & \le \mu G(u,u,u)+(1-\mu )G(u,u,Ru)\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$

Thus, $u\in Fix\left(R_{\mu }\right)$.

Conversely, let $u\in Fix\left(R_{\mu }\right)$. So, $G(u,R_{\mu }u,R_{\mu }u)=G(u,W(u,Ru,\mu ),W(u,Ru,\mu ))=0$.

Using Lemma 3, we obtain

$$(1-\mu )G(u,Ru,Ru)=G(u,W(u,Ru,\mu ),W(u,Ru,\mu \left)\right)=0.$$

Hence, $Ru=R_{\mu }u.$ □

Definition 9.

Let $(H,G,W)$ be a convex G-metric space. A function $R:H\to H$ is said to be a large enriched contraction if for distinct $u_1,u_2,u_3\in H$,

$$G(R_{\mu }{u}_1,R_{\mu }{u}_2,R_{\mu }{u}_3)<G(u_1,u_2,u_3)$$

and for all $ϵ>0,$ there exists $0<\delta <1$ such that

$$[G(u_1,u_2,u_3)\ge ϵ]\Rightarrow G(R_{\mu }{u}_1,R_{\mu }{u}_2,R_{\mu }{u}_3)\le \delta G(u_1,u_2,u_3).$$

(12)

Definition 10.

Let $(H,G,W)$ be a convex G-metric space. A function $R:H\to H$ is said to be an extended large enriched contraction if for distinct $u_1,u_2,u_3\in H$,

$$G(R_{\mu }{u}_1,R_{\mu }{u}_2,R_{\mu }{u}_3)<G(u_1,u_2,u_3)$$

and for all $ϵ>0,$ there exists $\psi \in \Psi$ such that

$$[G(u_1,u_2,u_3)\ge ϵ]\Rightarrow G(R_{\mu }{u}_1,R_{\mu }{u}_2,R_{\mu }{u}_3)\le \psi \left(G(u_1,u_2,u_3)\right).$$

(13)

Theorem 3.

Let $(H,G,W)$ be a complete convex G-metric space and $R:H\to H$ be a large enriched contraction map. Assume that there exist a $u_0\in H$ and a constant $L>0$ such that $G(u_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)\le L$ for all $n\ge 1.$ Then, R has a unique fixed point in $H.$

Proof.

The proof of this theorem is mainly divided into three steps:

Step 1.

Firstly, we shall prove that the sequence $\left\{{\xi }_n\right\}=\left\{G(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)\right\}$ is a decreasing sequence and ${\displaystyle \underset{n\to \infty }{lim}{\xi }_n=r\ge 0.}$

Using the assumptions of the theorem,

$$G(R_{\mu }{u}_1,R_{\mu }{u}_2,R_{\mu }{u}_3)<G(u_1,u_2,u_3).$$

(14)

Put $u_1=R_{\mu }^{n-1}{u}_0$, $u_2=R_{\mu }^n{u}_0,$ and $u_3=R_{\mu }^n{u}_0$ in (14) to obtain

$$G(R_{\mu }^n{u}_0,R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+1}{u}_0)<G(R_{\mu }^{n-1}{u}_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)<\cdots <G(u_0,R_{\mu }{u}_0,R_{\mu }{u}_0).$$

(15)

Since the sequence $\left\{{\xi }_n\right\}=\left\{G(R_{\mu }^{n+1}{u}_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)\right\}$ is strictly decreasing, ${\displaystyle \underset{n\to \infty }{lim}{\xi }_n=r\ge 0.}$

If possible, let $r>0$. Then, for all $n\ge 1$, we obtain

$$G(R_{\mu }^n{u}_0,R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+1}{u}_0)\ge r.$$

(16)

As a result, there exists a $0<\delta <1$ such that

$$\begin{array}{ll}G(R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+2}{u}_0)& =G\left(R_{\mu }\right(R_{\mu }^n{u}_0),R_{\mu }(R_{\mu }^{n+1}{u}_0),R_{\mu }(R_{\mu }^{n+1}{u}_0\left)\right)\\ & \le \delta G(R_{\mu }^n{u}_0,R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+1}{u}_0)\\ G(R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+2}{u}_0)& \le {\delta }^{2}G(R_{\mu }^{n-1}{u}_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)\\ & .\\ & .\\ & .\\ G(R_{\mu }^{n+1}{u}_0,R_{\mu }^{n+2}{u}_0,R_{\mu }^{n+2}{u}_0)& \le {\delta }^{n+1}G(u_0,R_{\mu }{u}_0,R_{\mu }{u}_0)\\ & \le {\delta }^{n+1}L.\end{array}$$

(17)

We obtain a contradiction from Equation (17). Hence, $r=0.$

Step 2:

In this step, we will show that the sequence $\left\{u_n\right\}$ is a Cauchy sequence.

Define $u_n=R_{\mu }^{n-1}{u}_0,$ for $n\ge 1$ and $u_0\in H.$

Assume that $\left\{u_n\right\}$ is not a Cauchy sequence. Thus, there exist an $ϵ>0,$ and subsequences of integers $N_k$, $m_k$, and $n_k$ such that

$$G(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0,R_{\mu }^{n_k}{u}_0)\ge ϵ,\mathrm{for}\mathrm{some}{m}_k>n_k>N_k.$$

Since R is a large enriched contraction, there exists a $\delta \in [0,1)$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G(R_{\mu }^{m_k}{u}_0,R_{\mu }^{n_k}{u}_0,R_{\mu }^{n_k}{u}_0)\le \delta G(R_{\mu }^{m_k-1}{u}_0,R_{\mu }^{n_k-1}{u}_0,R_{\mu }^{n_k-1}{u}_0)<...& <{\delta }^{n_k}G(u_0,R_{\mu }^{m_k-n_k}{u}_0,R_{\mu }^{m_k-n_k}{u}_0)\hfill \\ \hfill & \le {\delta }^{n_k}L.\hfill \end{array}\end{array}$$

We observe that our assumption is wrong. Thus, $\left\{u_n\right\}$ is a Cauchy sequence.

Step 3.

Next, we will prove the existence and uniqueness of the fixed point.

By the completeness of $(H,G,W)$ and continuity of $R_{\mu },$ we obtain $R_{\mu }u=u.$

Let $u^{\prime }$ be another fixed point of $R_{\mu }$ such that $u^{\prime }\ne u.$ Therefore, there exists an $ϵ>0$ such that $G(u,u^{\prime },u^{\prime })\ge ϵ.$ Using the contractive conditions, we have

$$G(u,u^{\prime },u^{\prime })=G(R_{\mu }u,R_{\mu }{u}^{\prime },R_{\mu }{u}^{\prime })\le \delta G(u,u^{\prime },u^{\prime }),$$

which is not possible. Hence, $R_{\mu }$ has a unique fixed point. Further, we obtain $Ru=u$ through the use of Lemma 4. □

Theorem 4.

Let $(H,G,W)$ be a complete convex G-metric space, and $R:H\to H$ be an extended large enriched contraction map. Assume that there exists a $u_0\in H$ and a constant $L>0$ such that $G(u_0,R_{\mu }^n{u}_0,R_{\mu }^n{u}_0)\le L$ for all $n\ge 1.$ Then, R has a unique fixed point in $H.$

Proof.

The proof of Theorem 4 follows a similar approach to that of Theorems 2 and 3, using the same key steps and logical structure.

We now present the following corollaries, which are direct consequences of Theorems 3 and 4. □

Corollary 3.

Let $(H,G)$ be a complete G-metric space. Let $R:H\to H$ be a mapping such that the following hold:

(i) 

$G(R{u}_1,R{u}_2,R{u}_3)<G(u_1,u_2,u_3);$

(ii) 

For $ϵ>0$, there exists a $0<\delta <1$ such that

$$[G(u_1,u_2,u_3)\ge ϵ]\Rightarrow G(R{u}_1,R{u}_2,R{u}_3)\le \delta G(u_1,u_2,u_3),$$

(iii) 

There exist $u_0\in H$ and a constant $L>0$ such that $G(u_0,R{u}_0,R{u}_0)\le L.$

Then, R has a unique fixed point.

Corollary 4.

Let $(H,G)$ be a complete G-metric space. Let $R:H\to H$ be a mapping such that the following hold:

(i) 

$G(R{u}_1,R{u}_2,R{u}_3)<G(u_1,u_2,u_3);$

(ii) 

For $ϵ>0$, there exists a $\psi \in \Psi$ such that

$$[G(u_1,u_2,u_3)\ge ϵ]\Rightarrow G(R{u}_1,R{u}_2,R{u}_3)\le \psi \left(G(u_1,u_2,u_3)\right),$$

(iii) 

There exist $u_0\in H$ and a constant $L>0$ such that $G(u_0,R{u}_0,R{u}_0)\le L.$

Then, R has a unique fixed point.

4. Application

The aim of this section is to check the existence of a solution to nonlinear equations using large enriched contractions.

Example 6.

Suppose we want to solve nonlinear equation $u^3-12u+6=0$. The solution of this nonlinear equation will be the fixed point of mapping $R:[0,1]\to [0,1],$ where $Ru={\displaystyle \frac{u^3}{6}}-u+1.$

First, we will show that R satisfies all the assumptions of Theorem 1. Take $W(u,Ru,\mu )=\mu u+(1-\mu )Ru$ and $d(u_1,u_2)=|u_1-u_2|$.

Consider

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(W(u_1,R{u}_1,\mu ),W(u_2,R{u}_2,\mu ))& =|\mu u_1+(1-\mu )\left({\displaystyle \frac{u_1^3}{6}}-u_1+1\right)-\mu u_2-(1-\mu )\left({\displaystyle \frac{u_2^3}{6}}-u_2+1\right)|\hfill \\ \hfill & =|\mu (u_1-u_2)+(1-\mu )\left({\displaystyle \frac{u_1^3}{6}}-u_1-{\displaystyle \frac{u_2^3}{6}}+u_2\right)|\hfill \\ \hfill & =|\mu (u_1-u_2)+(1-\mu )\left({\displaystyle \frac{(u_1-u_2)(u_1^2+u_1{u}_2+u_2^2)}{6}}-u_1+u_2\right)|\hfill \\ \hfill & =|u_1-u_2||\mu +(1-\mu )\left({\displaystyle \frac{(u_1^2+u_1{u}_2+u_2^2)}{6}}-1\right)|\hfill \\ \hfill & =|u_1-u_2||\mu -(1-\mu )\left(1-{\displaystyle \frac{3}{12}}(u_1^2+u_2^2)+{\displaystyle \frac{|u_1-u_2{|}^2}{12}}\right)|\hfill \\ \hfill & \le |u_1-u_2||\mu -(1-\mu )\left(1-{\displaystyle \frac{1}{24}}{|u_1-u_2|}^2\right)|\hfill \\ \hfill & =|u_1-u_2||\mu -(1-\mu )\left(1-{\displaystyle \frac{1}{24}}{ϵ}^2\right)|\hfill \\ \hfill & =\delta \left(ϵ\right)|u_1-u_2|.\hfill \end{array}\end{array}$$

Therefore, by Theorem 1, $Ru=u$, where $u=0.511128$ is the unique fixed point of mapping $R.$

Further, we can also find the fixed point of mapping R using the convergence of iterative sequence $\left\{W(u_n,R{u}_n,\mu )\right\}$ (as taken in the proof of Theorem 1).

The Figure 2 represents the convergence of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{2}}\right)$ with different starting points.

Figure 2. Convergence of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{2}}\right)$ with different starting points for Example 6.

It is clear from the Table 1 that $\left\{W\left(u_n,R{u}_n,{\displaystyle \frac{1}{2}}\right)\right\}$ converges to the point 0.5111 irrespective of the initial guess. Thus, 0.5111 is the fixed point of mapping R up to four decimal places (the solution of the nonlinear equation) in [0,1].

Table 1. Values of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{2}}\right)$ with different starting points for Example 6.

$\left\{\mathit{W}\left({\mathit{u}}_{\mathit{n}},{\mathbf{Ru}}_{\mathit{n}},\frac{1}{2}\right)\right\}$
	${\mathbf{u}}_{\mathbf{0}}$ = 0	${\mathbf{u}}_{\mathbf{0}}$  = 0.5	${\mathbf{u}}_{\mathbf{0}}$  = 0.8	${\mathbf{u}}_{\mathbf{0}}$  = 1
1	0.5000	0.5104	0.5427	0.5833
2	0.5104	0.5111	0.5133	0.5165
3	0.5111	0.5111	0.5113	0.5115
4	0.5111	0.5111	0.5111	0.5112
5	0.5111	0.5111	0.5111	0.5111

Example 7.

Consider the nonlinear equation $f\left(u\right)=u^2-2u+1=0$. The solution of this nonlinear equation corresponds to the fixed points of mapping $R:[0,1]\to [0,1],$ where $Ru=u^2-u+1.$

First, we will demonstrate that R satisfies all the requirements of Theorem 1. Take $W(u,Ru,\mu )=\mu u+(1-\mu )Ru$ and $d(u_1,u_2)=|u_1-u_2|$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(W(u_1,R{u}_1,\mu ),W(u_2,R{u}_2,\mu ))& =|\mu (u_1-u_2)+(1-\mu )(u_1^2-u_1-u_2^2+u_2)|\hfill \\ \hfill & =|u_1-u_2\left|\right|\mu -(1-\mu )(1-(u_1+u_2))|\hfill \\ \hfill & \le |u_1-u_2\left|\right|\mu -(1-\mu )(1-ϵ)|\hfill \\ \hfill & =\delta \left(ϵ\right)|u_1-u_2|.\hfill \end{array}\end{array}$$

Therefore, by Theorem 1, $Ru=u$, where $u=1$ is the unique fixed point of mapping $R.$ Since $f\left(u\right)>0$ for all $u\in \mathbb{R}$, the existence of a solution for $f\left(u\right)=0$ cannot be checked using Bolzano’s theorem.

Moreover, the fixed point of mapping R can also be found through the convergence of the iterative sequence $\left\{W(u_n,R{u}_n,\mu )\right\}$, as outlined in the proof of Theorem 1.

The Figure 3 shows the convergence of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{5}}\right)$ with different starting points.

Figure 3. Convergence of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{5}}\right)$ with different starting points for Example 7.

The Table 2 clearly indicates that the sequence $\left\{W\left(u_n,R{u}_n,{\displaystyle \frac{1}{5}}\right)\right\}$ converges to the point 0.9996, regardless of the initial guess. Consequently, 0.9996 serves as the fixed point of mapping R (the solution of the nonlinear equation) up to four decimal places within the interval [0,1].

Table 2. Values of $W\left(u_n,R{u}_n,{\displaystyle \frac{1}{5}}\right)$ with different starting points for Example 7.

$\left\{\mathit{W}\left({\mathit{u}}_{\mathit{n}},{\mathbf{Ru}}_{\mathit{n}},\frac{1}{5}\right)\right\}$
	${\mathbf{u}}_0$  = 0.7	${\mathbf{u}}_0$  = 0.8
1	0.7720	0.8320
2	0.8136	0.8546
3	0.8414	0.8715
.	.	.
.	.	.
.	.	.
2999	0.9996	0.9996
3000	0.9996	0.9996

5. Conclusions and Future Scope

In this study, we examined the existence and approximation of fixed points for large enriched contractions in the framework of a convex metric space and convex G-metric space. We demonstrate that a Kransnoselskij-type iterative procedure can be used to approximate the unique fixed point of large enriched contractions in a complete convex metric space. Several significant related findings that are already available in the literature are generalized in our study.

In the future, we can expand the applicability of our findings by exploring different abstract spaces such as quasi-metric spaces, partial metric spaces, and fuzzy metric spaces. This could lead to novel applications in diverse fields including image processing and optimization.