Recent Advancements in KRH-Interpolative-Type Contractions

Abstract

The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus Ćirić interpolative and Hardy–Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville–Caputo fractional derivatives and fractional differential equations is provided in this paper.

1. Introduction

The concept of b-metric space, introduced by Czerwik et al. in 1993 [1], is a generalization of the traditional metric space. The b-metric spaces relax the triangle inequality property of metrics, allowing for more flexibility in the distance function. There have been several recent developments in the study of b-metric spaces. References [2,3,4] provide some of the latest results in this field, offering further insights into the properties and applications of b-metric spaces.

In 2017, Kamran et al. [5] introduced an extension of b-metric spaces by replacing the coefficient of the b-metric space with a binary function. This new concept is known as an extended b-metric space. Some results related to extended b-metric spaces can be found in references [6,7,8,9,10,11], which delve into various aspects and properties of this space.

Furthermore, Mlaiki et al. [12] recently introduced the concept of a controlled metric space, which serves as a further extension of the extended b-metric space. Controlled metric spaces likely incorporate additional structures or properties that allow for more control over the behavior of the distance function. These developments indicate the ongoing exploration and expansion of the theory of generalized metric spaces, offering alternative frameworks and tools to study various mathematical structures and phenomena.

Recently, the principles of interpolative contraction involve the multiplication of distances with exponents that meet certain conditions. The concept of “Interpolative Contraction” was coined by Karapinar, a well-known mathematician, in his 2018 paper [13]. The structure of Interpolative contraction can be defined as follows:

A self-mapping $\mathsf{\ell }:X\to X$ is said to be an Interpolative contraction on a traditional metric space $(X,\mathsf{\pounds })$ if it satisfies the following inequalities:

$$\left\{\begin{array}{c}\mathsf{\pounds }(\mathsf{\ell }x,\aleph \widehat{c})\le k{\left(\mathsf{\pounds }(x,\mathsf{\ell }x)\right)}^a.{\left(\mathsf{\pounds }(\widehat{c},\aleph \widehat{c})\right)}^{1-a};\hfill \\ \mathsf{\pounds }(\mathsf{\ell }x,\aleph \widehat{c})\le k{\left(\mathsf{\pounds }(x,\widehat{c})\right)}^a.{\left(\mathsf{\pounds }(x,\mathsf{\ell }x)\right)}^{\beta }.{\left(\mathsf{\pounds }(\widehat{c},\aleph \widehat{c})\right)}^{1-a-\beta },a+\beta <1;\hfill \\ \mathsf{\pounds }(\mathsf{\ell }x,\aleph \widehat{c})\le k{\left(\mathsf{\pounds }(x,\widehat{c})\right)}^a.{\left(\mathsf{\pounds }(x,\mathsf{\ell }x)\right)}^{\beta }.{\left(\mathsf{\pounds }(\widehat{c},\aleph \widehat{c})\right)}^{\gamma }.{\left(\frac{1}{2}(\mathsf{\pounds }(x,\aleph \widehat{c})+\mathsf{\pounds }(\widehat{c},\mathsf{\ell }x))\right)}^{1-a-\beta -\gamma }.\hfill \end{array}\right.$$

For all $x,\widehat{c}\in X,$ then ℓ and ℵ are called Interpolative Kannan contraction, Interpolative Reich-Rus-Ćirić contraction, and Interpolative Hardy–Rogers contraction, respectively. See more detail in [14,15,16].

2. Preliminaries

Further, a few fundamental principles of need that are applied to the primary outcomes:

Definition 1

([1]). Given a nonempty set X and a constant $s\ge 1$, the function $\xi :X\times X\to [0,\infty )$ is called a b-metric type if the following conditions hold true:

($b_i$)

if $\xi \left(x,\widehat{c}\right)=0$ iff $x=\widehat{c}$;

($b_{ii}$)

$\xi \left(x,\widehat{c}\right)=\xi \left(\widehat{c},x\right)$;

($b_{iii}$)

$\xi \left(x,\widehat{c}\right)\le s[\xi \left(x,r\right)+\xi \left(r,\widehat{c}\right)]$.

For all $x,\widehat{c},r\in X$, the pair $\left(X,\xi \right)$ is called b-metric space.

After, in 2017 Kamran et al. [5] presented the notion of extended b-metric space.

Definition 2

([5]). Given a nonempty set X and $\theta :$$X\times$$X\to [1,\infty ).$ The function ${\xi }_e:$$X\times$$X\to [0,\infty )$ is called extended b-metric type if the following conditions hold true:

($e_i$)

if ${\xi }_e\left(x,\widehat{c}\right)=0$ iff $x=\widehat{c}$;

($e_{ii}$)

${\xi }_e\left(x,\widehat{c}\right)={\xi }_e\left(\widehat{c},x\right)$;

($e_{iii}$)

${\xi }_e\left(x,\widehat{c}\right)\le \theta \left(x,\widehat{c}\right)[{\xi }_e\left(x,r\right)+{\xi }_e\left(r,\widehat{c}\right)]$.

For all $x,\widehat{c},r\in X$, the pair $\left(X,{\xi }_e\right)$ is called extended b-metric space.

Further, in 2018 Mlaiki et al. [12] presented a new kind of metric space called controlled metric space.

Definition 3

([12]). Given a nonempty set X and $\theta :$$X\times$$X\to [1,\infty ).$ The function $\mathsf{\pounds }:$$X\times$$X\to [0,\infty )$ is called a Controlled metric type if the following conditions hold true:

($c_i$)

if $\mathsf{\pounds }\left(x,\widehat{c}\right)=0$ iff $x=\widehat{c}$;

($c_{ii}$)

$\mathsf{\pounds }\left(x,\widehat{c}\right)=\mathsf{\pounds }\left(\widehat{c},x\right)$;

($c_{iii}$)

$\mathsf{\pounds }\left(x,r\right)\le \theta \left(x,\widehat{c}\right)\left[\mathsf{\pounds }\left(x,r\right)\right]+\theta \left(\widehat{c},r\right)\left[\mathsf{\pounds }\left(r,\widehat{c}\right)\right]$.

For all $x,\widehat{c},r\in X$, the pair $\left(X,\mathsf{\pounds }\right)$ is called Controlled metric space.

Example 1.

Let $X=\left\{0,1,2\right\}$. Consider the function $\mathsf{\pounds }:$$X\times$$X\to [0,\infty )$ is given by $\mathsf{\pounds }\left(x,\widehat{c}\right)=0$ iff $x=\widehat{c}$, and

$$\mathsf{\pounds }\left(0,1\right)=\frac{2}{5},\mathsf{\pounds }\left(0,2\right)=\frac{1}{2},\mathsf{\pounds }\left(1,2\right)=1.$$

Here, $\theta :$$X\times$$X⟶[1,\infty )$ is symmetric such that

$$\theta \left(0,0\right)=\theta \left(1,1\right)=\theta \left(2,2\right)=\theta \left(1,2\right)=1,\theta \left(0,1\right)=\theta \left(0,2\right)=\frac{3}{2}.$$

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(0,0\right)& =& \mathsf{\pounds }\left(1,1\right)=\mathsf{\pounds }\left(2,2\right)=0\hfill \\ \hfill \mathsf{\pounds }\left(1,2\right)& =& \mathsf{\pounds }\left(2,1\right)\Rightarrow 1=1\hfill \\ \hfill \mathsf{\pounds }\left(1,2\right)& \le & \theta \left(1,0\right)\mathsf{\pounds }\left(1,0\right)+\theta \left(0,2\right)\mathsf{\pounds }\left(0,2\right)\hfill \\ \hfill 1& \le & \left(\frac{3}{2}\right)\left(\frac{2}{5}\right)+\left(\frac{3}{2}\right)\left(\frac{1}{2}\right)=1.35\hfill \end{array}$$

Hence, $\mathsf{\pounds }$ is controlled metric space. However, since

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(1,2\right)& =& 1\nleqq \left(1\right)\left[\mathsf{\pounds }\left(1,0\right)+\mathsf{\pounds }\left(0,2\right)\right]=0.9\hfill \\ \hfill \mathsf{\pounds }\left(1,2\right)& =& 1\nleqq \theta \left(1,2\right)\left[\mathsf{\pounds }\left(1,0\right)+\mathsf{\pounds }\left(0,2\right)\right]=0.9,\hfill \end{array}$$

the pair $\left(X,\mathsf{\pounds }\right)$ is not a b-metric space [1] and nor an extended b-metric space [5].

Definition 4

([17]). Let $\left(X,\mathsf{\pounds }\right)$ be a Controlled metric space and ${\left\{x_{\acute{r}}\right\}}_{\acute{r}\ge 0}$ be a sequence in $X.$ Then,

• $\left(i\right)$ the sequence $\left\{x_{\acute{r}}\right\}$ converges to $x\in$ X if for each $ϵ>0$ such that $\exists {\acute{r}}_0\in \mathbb{N}$ then

  $$\mathsf{\pounds }\left(x_{\acute{r}},x\right)<ϵ,\forall \acute{r}\ge {\acute{r}}_0\mathrm{implies}\underset{\acute{r}\to \infty }{lim}{x}_{\acute{r}}=x.$$

  $\left(ii\right)$ the sequence $\left\{x_{\acute{r}}\right\}$ is Cauchy controlled if for each $ϵ>0$ such that $\mathrm{\exists }{\acute{r}}_0\in \mathbb{N}$ then

  $$\mathsf{\pounds }\left(x_{\acute{r}},x_m\right)<ϵ,\forall \acute{r},m\ge {\acute{r}}_0.$$

  $\left(iii\right)$ a sequence is called complete if every Cauchy sequence is convergent.

Let $x\in X$ and $ϵ>0,$ the open ball $B\left(x,ϵ\right)$ is

$$B\left(x,ϵ\right)=\left\{y\in X:{\delta }_{\varsigma }\left(x,y\right)<ϵ\right\}.$$

The mapping $Y:X\to X$ is continuous at $x\in X$ if for each $ϵ>0,$ there is $\alpha >0$ so that

$$Y\left(B\left(x,\alpha \right)\right)\subseteq B\left(Yx,ϵ\right).$$

Owing to the above proposition, we clearly say that if Y is continuous at $x\in X,$ then for $x_i\to x$, we have $Y{x}_i\to Yx$ as $i\to \infty .$

3. Main Results

Let us demonstrate some new fixed point theorems utilizing the concept of Interpolative contraction in a Controlled metric space.

Definition 5.

Let $\left(X,\mathsf{\pounds }\right)$ be a Controlled metric space, and let $\mathsf{\ell },\aleph :$$X\to$X be known as Interpolative Kannan contraction mappings, if for all $x,\widehat{c}\in X$ such that $\mathsf{\ell }x\ne x$ whenever $\aleph \widehat{c}\ne \widehat{c}$ with $k\in \left(0,1\right)$, and $a\in \left(0,1\right),$ such that

$$\mathsf{\pounds }\left(\mathsf{\ell }x,\aleph \widehat{c}\right)\le k{\left[\mathsf{\pounds }\left(x,\mathsf{\ell }x\right)\right]}^a.{\left[\mathsf{\pounds }\left(\widehat{c},\aleph \widehat{c}\right)\right]}^{1-a}.$$

(1)

Theorem 1.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space, and $\mathsf{\ell },\aleph :$$X\to$X be an Interpolative Kannan contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

(2)

Assume that for every $x\in$$X,$

$$\underset{\acute{r}\to \infty }{lim}\theta \left(x_{\acute{r},}x\right)\mathrm{and}\underset{\acute{r}\to \infty }{lim}\theta \left(x_{,}{x}_{\acute{r}}\right)$$

exist and are finite. Then, the sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X,$ then $x^{*}$ is a common unique fixed point of ℓ and ℵ.

Proof. 

Let $x_0\in$X and we define the class of iterative sequences $\left\{x_{\acute{r}}\right\}$ such that $x_{\acute{r}+1}=\mathsf{\ell }{x}_{\acute{r}},$$x_{\acute{r}+2}$$=\aleph x_{\acute{r}+1}$ for all $\acute{r}\in \mathbb{N}$. Without loss of generality, we assume that $x_{\acute{r}+2}$$\ne \aleph x_{\acute{r}+1}$ for each nonnegative integer $\acute{r}$. Indeed, if there exist a nonnegative integer ${\acute{r}}_0$ such that $x_{{\acute{r}}_0+2}$$=\aleph x_{{\acute{r}}_0+1}$, then our proof of the Theorem proceeds as follows. Thus, we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)& =& \mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}+1}\right)\hfill \\ & \le & k{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}+1},\aleph x_{\acute{r}+1})\right]}^{1-a}\hfill \\ & =& k{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{1-a}\hfill \\ \hfill \left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]a& \le & k{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^a.\hfill \end{array}$$

From this, we can write

$$\begin{array}{ccc}\hfill \mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})& \le & k\mathsf{\pounds }(x_{\acute{r},},x_{\acute{r}+1})\hfill \\ & \le & k^2\mathsf{\pounds }\left(x_{\acute{r}-1,}{x}_{\acute{r}}\right)\le \cdots \le k^{\acute{r}+1}\mathsf{\pounds }(x_0,x_1).\hfill \end{array}$$

$$\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\le k^{\acute{r}+1}\mathsf{\pounds }(x_0,x_1).$$

(3)

On the other hand, one writes

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)& =& \mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}-1}\right)\hfill \\ & \le & k{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}-1},\aleph x_{\acute{r}-1})\right]}^{1-a}\hfill \\ & =& k{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{1-a}\hfill \\ \hfill [\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}}){]}^{1-a}& \le & k{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{1-a}.\hfill \end{array}$$

which yields that,

$$\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\le k\mathsf{\pounds }\left(x_{\acute{r}-1,}{x}_{\acute{r}}\right)\le k^2\mathsf{\pounds }\left(x_{\acute{r}-2,}{x}_{\acute{r}-1}\right)\le \cdots \le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

$$\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

(4)

By appealing to (3) and (4), we find that

$$\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

(5)

As a next step, by using (5), we will prove that the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy sequence based on the triangle inequality. More precisely, we have for all natural numbers $\acute{r}<m,$

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)& \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\theta \left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\hfill \\ & \le & \cdots \hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)\mathsf{\pounds }\left(x_{i,}{x}_{i+1}\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{k,}{x}_m\right)\mathsf{\pounds }\left(x_{m-1,}{x}_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)k^{m-1}\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +\left({\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)\right)k^{m-1}\theta \left(x_{m-1,}{x}_m\right)\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & =& \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right).\hfill \end{array}$$

In light of $\theta \left(x,\widehat{c}\right)\ge 1$, we have

$${\aleph }_l={\Sigma }_{i=0}^l\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i.$$

Which yield,

$$\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)\le \mathsf{\pounds }\left(x_{0,}{x}_1\right)\left[k^{\acute{r}}\theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\left({\aleph }_{m-1}-{\aleph }_{\acute{r}}\right)\right].$$

(6)

So, we conclude that ${lim}_{\acute{r}\to \infty }{\aleph }_{\acute{r}}$ exist and we find that the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy. Therefore, if we take the limit of the inequality (6) as $\acute{r},m\to \infty$ we conclude that

$$\underset{\acute{r},m\to \infty }{lim}\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)=0.$$

(7)

Hence, the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy in the complete controlled metric space $\left(X,\mathsf{\pounds }\right).$ So, there is some $x^{*}\in$X. Further, we show that $x^{*}$ is the fixed point of ℓ and ℵ.

$$\mathsf{\pounds }\left(x_{\acute{r}+2},\mathsf{\ell }{x}^{*}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}+1},\mathsf{\ell }{x}^{*}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)\right]}^{1-a}.$$

Upon letting $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)=0,$ which implies that $\mathsf{\ell }{x}^{*}=x^{*}.$ On the other hand

$$\mathsf{\pounds }\left(x_{\acute{r}+1},\aleph x^{*}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x^{*}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)\right]}^{1-a}.$$

Upon letting $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)=0,$ which implies that $\aleph x^{*}=x^{*}.$ Let $\mathsf{\pounds }$ have two fixed points $x^{*}$ and q

$$\begin{array}{ccc}\hfill \mathsf{\pounds }(x^{*},q)& =& \mathsf{\pounds }(\mathsf{\ell }{x}^{*},\aleph q)\le k{\left[\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)\right]}^a.{\left[\mathsf{\pounds }\left(q,\aleph q\right)\right]}^{1-a}\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x^{*},x^{*}\right)\right]}^a.{\left[\mathsf{\pounds }\left(q,q\right)\right]}^{1-a}=0.\hfill \end{array}$$

This implies that $x^{*}=q.$ So, we conclude that $x^{*}$ has a unique fixed point of ℓ and ℵ. □

Corollary 1.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space, and $\mathsf{\ell },\aleph :$$X\to$X be a continuous Interpolative Kannan contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X,$ then $x^{*}$ is a common fixed point of ℓ and ℵ. Also, the sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}$ in $X,$ if ${lim}_{\acute{r}\to \infty }\mathsf{\pounds }\left(x_{\acute{r}},x\right)=\mathsf{\pounds }\left(x,x\right).$ In this case, we write ${lim}_{\acute{r}\to \infty }{x}_{\acute{r}}=x^{*}$, and $x^{*}$ is a unique fixed point of ℓ and ℵ.

In the view of $\theta \left(x,\widehat{c}\right)=\theta \left(\widehat{c},r\right)=1,$ we easily conclude the result of Karapinar [13]:

Corollary 2.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete metric space, and $\mathsf{\ell },\aleph :$$X\to$X be an Interpolative Kannan contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X,$ then $x^{*}$ is a common fixed point of ℓ and ℵ. Moreover, the sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}$ in $X,$ if ${lim}_{\acute{r}\to \infty }\mathsf{\pounds }\left(x_{\acute{r}},x\right)=\mathsf{\pounds }\left(x,x\right).$ In this case, we write ${lim}_{\acute{r}\to \infty }{x}_{\acute{r}}=x^{*}$, and $x^{*}$ is a unique fixed point of ℓ and ℵ.

Example 2.

Let $X=\left[0,2\right].$ Consider the function $\mathsf{\pounds }:$$X\times$$X\to [0,\infty )$ define by

$$\mathsf{\pounds }\left(x,\widehat{c}\right)=\left|x-\widehat{c}\right|$$

Here, $\theta :$$X\times$$X⟶[1,\infty )$ is symmetric such that

$$\theta \left(x,\widehat{c}\right)=1+\left|x-\widehat{c}\right|.$$

Then, clearly $\left(X,\mathsf{\pounds }\right)$ is a Controlled metric space. Let $\mathsf{\ell },\aleph :$$X\to$X be a self mapping defined as

$$\mathsf{\ell }\left(x\right)=\left\{\begin{array}{c}0\mathit{if}x=0\\ \left|\frac{x-2}{2}\right|\mathit{if}x\in (0,2]\end{array}\right.$$

and

$$\aleph \left(x\right)=\left\{\begin{array}{c}0\mathit{if}x=0\\ \left|\frac{x-2}{3}\right|\mathit{if}x\in (0,2]\end{array}\right.\forall x,\widehat{c}\in X$$

Clearly, ℓ, and ℵ are not a Kannan contraction with $k\in [0,\frac{1}{3})$.

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(1\right),\aleph \left(0\right)\right)=\mathsf{\pounds }\left(\frac{1}{2},0\right)=\frac{1}{2}$$

$$\nleqq k\left[\mathsf{\pounds }\left(1,\mathsf{\ell }\left(1\right)\right)+\mathsf{\pounds }\left(0,\aleph \left(0\right)\right)\right]=k[\mathsf{\pounds }\left(1,\frac{1}{2}\right)+\mathsf{\pounds }\left(0,0\right)]=k\left(\frac{1}{2}\right)$$

Otherwise, by taking $a=\frac{1}{2}$ with $a+\beta <1$, and $k=\frac{99}{100}$ $\forall x,\widehat{c}\in$$X,$ such that $\mathsf{\ell }\left(x\right)\ne x$ whenever $\aleph \left(\widehat{c}\right)\ne \widehat{c},$ then (1) is satisfied.

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(1\right),\aleph \left(\frac{3}{2}\right)\right)=\mathsf{\pounds }\left(\frac{1}{2},\frac{1}{6}\right)=\frac{1}{3}=0.33$$

$$\le k{\left[\mathsf{\pounds }\left(1,\mathsf{\ell }\left(1\right)\right)\right]}^{0.5}{\left[\mathsf{\pounds }\left(\frac{3}{2},\aleph \left(\frac{3}{2}\right)\right)\right]}^{0.5}=k{\left[\mathsf{\pounds }\left(1,\frac{1}{2}\right)\right]}^{0.5}{\left[\mathsf{\pounds }\left(\frac{3}{2},\frac{1}{6}\right)\right]}^{0.5}=0.8$$

On the other hand, by taking $x_0=0$ and $x_{\acute{r}}=1$ $\forall \acute{r}\ge 1$, then (2) is hold.

$$\frac{\theta \left(1,1\right)\theta \left(1,1\right)}{\theta \left(0,1\right)}=\frac{\left(1\right)\left(1\right)}{\left(2\right)}=\frac{1}{2}<1.01=\frac{1}{k},\mathit{where}k=0.99\in \left(0,1\right).$$

So all conditions in Theorem 1 are fulfilled and there is a unique common fixed point, which is $x^{*}=0.$

4. Riech Rus Ćirić Interpolative Contraction

Definition 6.

Let $\left(X,\mathsf{\pounds }\right)$ be a Controlled metric space, and $\mathsf{\ell },\aleph :$$X\to$X be an Interpolation Riech-Rus-Ćirić contraction mappings. If for all $x,\widehat{c}\in X$ such that $\mathsf{\ell }x\ne x$ whenever $\aleph \widehat{c}\ne \widehat{c}$ with $k\in \left(0,1\right)$ and $a,\beta \in \left(0,1\right)$ where $a+\beta <1,$ such that

$$\mathsf{\pounds }\left(\mathsf{\ell }x,\aleph \widehat{c}\right)\le k{\left[\mathsf{\pounds }\left(x,\widehat{c}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x,\mathsf{\ell }x\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(\widehat{c},\aleph \widehat{c}\right)\right]}^{1-a-\beta }.$$

(8)

Theorem 2.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space and let $\mathsf{\ell },\aleph :$$X\to$X be an interpolation Riech-Rus-Ćirić contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

(9)

Additionally, assume that for every $x\in$$X,$

$$\underset{\acute{r}\to \infty }{lim}\theta \left(x_{\acute{r},}x\right)\mathit{and}\underset{\acute{r}\to \infty }{lim}\theta \left(x_{,}{x}_{\acute{r}}\right)$$

exist and are finite. Then, the sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X.$ Then, $x^{*}$ is a common unique fixed point of ℓ and ℵ.

Proof. 

Let $x_0\in$X, and we define the family of iterative sequences $\left\{x_{\acute{r}}\right\}$ such that $x_{\acute{r}+1}=\mathsf{\ell }{x}_{\acute{r}},$$x_{\acute{r}+2}$$=\aleph x_{\acute{r}+1}$ for all $\acute{r}\in \mathbb{N}$. Without loss of generality, we assume that $x_{\acute{r}+2}$$\ne \aleph x_{\acute{r}+1}$ for each nonnegative integer $\acute{r}$. Indeed, if there exist a nonnegative integer ${\acute{r}}_0$ such that $x_{{\acute{r}}_0+2}$$=\aleph x_{{\acute{r}}_0+1}$, then our proof of Theorem proceeds as follows. Thus, by using (8), we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)& =& \mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}+1}\right)\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},\aleph x_{\acute{r}+1})\right]}^{1-a-\beta }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{1-a-\beta }\hfill \\ \hfill {\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{a+\beta }& \le & k{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{a+\beta }\hfill \\ \hfill \left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]& \le & k\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right].\hfill \end{array}$$

Which implies

$$\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\le k\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\le k^2\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\le \cdots \le k^{\acute{r}+1}\mathsf{\pounds }(x_0,x_1).$$

$$\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\le k^{\acute{r}+1}\mathsf{\pounds }(x_0,x_1).$$

(10)

On the other hand, we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)& =& \mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}-1}\right)\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},\aleph x_{\acute{r}-1})\right]}^{1-a-\beta }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{1-a-\beta }\hfill \\ \hfill {\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\right]}^{1-\beta }& \le & k{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{1-\beta }\hfill \\ \hfill \left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\right]& \le & k\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right].\hfill \end{array}$$

This yields

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\le k\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\le k^2\mathsf{\pounds }(x_{\acute{r}-2},x_{\acute{r}-1})\le \cdots \le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

(11)

By (10) and (11), we have

$$\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\le k^{\acute{r}}\mathsf{\pounds }(x_0,x_1).$$

(12)

Now, by using Equation (12), we will prove that the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy sequence. For all natural numbers $\acute{r}<m$, we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)& \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\theta \left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\hfill \\ & \le & \cdots \hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)\mathsf{\pounds }\left(x_{i,}{x}_{i+1}\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{k,}{x}_m\right)\mathsf{\pounds }\left(x_{m-1,}{x}_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)k^{m-1}\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +\left({\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)\right)k^{m-1}\theta \left(x_{m-1,}{x}_m\right)\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & =& \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right).\hfill \end{array}$$

Taking $\theta \left(x,\widehat{c}\right)\ge 1,$ we have

$${\aleph }_l={\Sigma }_{i=0}^l\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i.$$

We obtain,

$$\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)\le \mathsf{\pounds }\left(x_{0,}{x}_1\right)\left[k^{\acute{r}}\theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\left({\aleph }_{m-1}-{\aleph }_{\acute{r}}\right)\right].$$

(13)

We have ${lim}_{\acute{r}\to \infty }{\aleph }_{\acute{r}}$ exists, and the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy. Therefore, if we take limit in the inequality (13), as $\acute{r},m\to \infty$, we conclude that

$$\underset{\acute{r},m\to \infty }{lim}\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)=0.$$

(14)

Thus, the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy in the complete Controlled metric space $\left(X,\mathsf{\pounds }\right).$ So, there is some $x^{*}\in$X. Next, we show that $x^{*}$ is the fixed point of ℓ and ℵ.

$$\mathsf{\pounds }\left(\mathsf{\ell }{x}^{*},x_{\acute{r}+2}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}^{*},\aleph x_{\acute{r}+1}\right)\le k{\left[\mathsf{\pounds }\left(x^{*},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},\aleph x_{\acute{r}+1}\right)\right]}^{1-a-\beta }.$$

Taking $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)=0$, which implies that $\mathsf{\ell }{x}^{*}=x^{*}.$ On the other hand,

$$\mathsf{\pounds }\left(x_{\acute{r}+2},\aleph x^{*}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}+1},\aleph x^{*}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x^{*}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}_{\acute{r}+1}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)\right]}^{1-a-\beta }.$$

Taking $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)=0$, which implies that $\aleph x^{*}=x^{*}.$ Hence, $x^{*}$ is the common fixed point of ℓ and $\aleph .$ □

Corollary 3.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space and $\mathsf{\ell },\aleph :$$X\to$X be a continuous Interpolation the Riech-Rus-Ćirić contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X.$ Then, $x^{*}$ is a common unique fixed point of ℓ and ℵ.

By virtue of $\theta \left(x,\widehat{c}\right)=\theta \left(\widehat{c},r\right)=1,$ we easily conclude the result of Karapinar [13]:

Corollary 4.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete metric space and $\mathsf{\ell },\aleph :$$X\to$X be an Interpolation Riech-Rus-Ćirić contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$$X.$ Then, $x^{*}$ is a common unique fixed point of ℓ and ℵ.

Example 3.

Let $X=\left\{0,1,2\right\}.$ Consider the function £ given as,

$$\mathsf{\pounds }\left(0,0\right)=\mathsf{\pounds }\left(1,1\right)=\mathsf{\pounds }\left(2,2\right)=0,$$

and

$$\mathsf{\pounds }\left(0,1\right)=\frac{2}{5},\mathsf{\pounds }\left(0,2\right)=\frac{1}{2},\mathsf{\pounds }\left(1,2\right)=1.$$

Here, $\theta :$$X\times$$X⟶[1,\infty )$ is symmetric such that

$$\theta \left(0,0\right)=\theta \left(1,1\right)=\theta \left(2,2\right)=\theta \left(1,2\right)=1,\theta \left(0,1\right)=\theta \left(0,2\right)=\frac{3}{2}.$$

Then, clearly $\left(X,\mathsf{\pounds }\right)$ is a Controlled metric space. Let $\mathsf{\ell },\aleph :$$X\to$X be a self mapping defined as

$$\mathsf{\ell }:\left(\begin{array}{c}012\\ 020\end{array}\right)\aleph :\left(\begin{array}{c}012\\ 010\end{array}\right)$$

Then, $\mathsf{\ell },\aleph$ is not a Riech-Rus-Ćirić contraction with $k\in [0,\frac{1}{2})$,

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(0\right),\aleph \left(1\right)\right)=\mathsf{\pounds }\left(0,1\right)=\frac{2}{5}$$

$$\nleqq k\left[\mathsf{\pounds }\left(0,1\right)+\mathsf{\pounds }\left(0,\mathsf{\ell }\left(0\right)\right)+\mathsf{\pounds }\left(1,\aleph \left(1\right)\right)\right]=k\left[\mathsf{\pounds }\left(0,1\right)+\mathsf{\pounds }\left(0,0\right)+\mathsf{\pounds }\left(1,1\right)\right]=k\left(\frac{2}{5}\right).$$

While, by taking $a=\beta =\frac{2}{5},k=\frac{9}{10}$, then $\forall x,\widehat{c}\in$ X such that $\mathsf{\ell }\left(x\right)\ne x$ whenever $\aleph \left(\widehat{c}\right)\ne \widehat{c}$; thus, (8) is satisfied.

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(1\right),\aleph \left(2\right)\right)=\mathsf{\pounds }\left(2,0\right)=\frac{1}{2}$$

$$\le \frac{9}{10}{\left[\mathsf{\pounds }\left(1,2\right)\right]}^{0.4}.{\left[\mathsf{\pounds }\left(1,\mathsf{\ell }\left(1\right)\right){]}^{0.4}.\left[\mathsf{\pounds }\right(2,\aleph \left(2\right)\right]}^{0.2}=\frac{9}{10}{\left[\mathsf{\pounds }\left(1,2\right)\right]}^{0.4}.{\left[\mathsf{\pounds }\left(1,2\right)\right]}^{0.4}.{\left[\mathsf{\pounds }\left(2,0\right)\right]}^{0.2}=0.78$$

On the other hand, by taking $x_0=0$ and $x_{\acute{r}}=1$ $\forall \acute{r}\ge 1$, then (9) holds.

$$\frac{\theta \left(1,1\right).\theta \left(1,1\right)}{\theta \left(0,1\right)}=\frac{\left(1\right)\left(1\right)}{\left(\frac{3}{2}\right)}=\frac{2}{3}<\frac{10}{9}=\frac{1}{k},\mathrm{where}k=0.9\in \left(0,1\right).$$

Hence, all conditions of Theorem 2 are fulfilled. So, ℵ and ℓ is a common unique fixed point, which is $x^{*}=0.$

5. Hardy–Roger Interpolative Contractions

Definition 7.

Let $\left(X,\mathsf{\pounds }\right)$ be a Controlled metric space, and the two mapping $\mathsf{\ell },\aleph :X\to X$ be an Interpolative Hardy–Roger contraction with $a=\beta =\frac{2}{5},\gamma =\frac{1}{10}$ and $k\in \left(0,1\right)$ where $a+\beta +\gamma <1$ with $\forall x,\widehat{c}\in X$ with $x\ne \mathsf{\ell }x$ and $\widehat{c}\ne \aleph \widehat{c},$ such that

$$\mathsf{\pounds }\left(\mathsf{\ell }x,\aleph \widehat{c}\right)\le k{\left[\mathsf{\pounds }\left(x,\widehat{c}\right)\right]}^a{\left[\mathsf{\pounds }\left(x,\mathsf{\ell }x\right)\right]}^{\beta }{\left[\mathsf{\pounds }\left(\widehat{c},\aleph \widehat{c}\right)\right]}^{\gamma }{\left[\frac{1}{2}(\mathsf{\pounds }\left(x,\aleph \widehat{c}\right)+\mathsf{\pounds }\left(\widehat{c},\mathsf{\ell }x\right))\right]}^{1-a-\beta -\gamma }.$$

(15)

Theorem 3.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space. Let $\mathsf{\ell },\aleph :X\to X$ be an interpolative Hardy–Roger contraction. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

(16)

Assume that for every $x\in$$X,$

$$\underset{\acute{r}\to \infty }{lim}\theta \left(x_{\acute{r},}x\right)and\underset{\acute{r}\to \infty }{lim}\theta \left(x_{,}{x}_{\acute{r}}\right)$$

exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$ X. Then, $x^{*}$ is a common unique fixed point of ℓ and ℵ.

Proof. 

Let $x_0\in$X and we define the class of iterative sequences $\left\{x_{\acute{r}}\right\}$ such that $x_{\acute{r}+1}=\mathsf{\ell }{x}_{\acute{r}},$$x_{\acute{r}+2}$$=\aleph x_{\acute{r}+1}$ for all $\acute{r}\in \mathbb{N}$. Without loss of generality, we assume that $x_{\acute{r}+2}$$\ne \aleph x_{\acute{r}+1}$ for each nonnegative integer $\acute{r}$. Indeed, if there exists a nonnegative integer ${\acute{r}}_0$ such that $x_{{\acute{r}}_0+2}=\aleph x_{{\acute{r}}_0+1}$, then our proof of the Theorem proceeds as follows. Thus, by using (15), we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)& =& \mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}+1}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},\aleph x_{\acute{r}+1})\right]}^{\gamma }.\hfill \\ & & {\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},\aleph x_{\acute{r}+1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}_{\acute{r}}\right)\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{\gamma }\hfill \\ & & {\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+2}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+1}\right)\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{\gamma }\hfill \\ & & {\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]}^{1-a-\beta -\gamma }.\hfill \end{array}$$

Let $\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\le$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)$ for some $\acute{r}\ge 1$, then

$$\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right))\le \mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right).$$

By using (15), we have

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{\gamma }.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]}^{1-a-\beta -\gamma }$$

$$\begin{array}{ccc}\hfill \mathsf{\pounds }{\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)}^{a+\beta }& \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^{a+\beta }\hfill \\ \hfill \mathrm{or}\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]& \le & k\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right].\hfill \end{array}$$

This refers to a contradiction, so we have

$$\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right))\le \mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right).$$

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}+1},x_{\acute{r}+2})\right]}^{\gamma }.{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^{1-a-\beta -\gamma }$$

$$\begin{array}{ccc}\hfill {\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]}^{1-\gamma }& \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right]}^{1-\gamma }\hfill \\ \hfill \mathrm{or}\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]& \le & k\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\right].\hfill \end{array}$$

We conclude that

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\le k\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}+1}\right)\le k^2\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right)\le \cdots \le k^{\acute{r}+1}\mathsf{\pounds }\left(x_0,x_1\right).$$

(17)

On the other hand, we have

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}},\aleph x_{\acute{r}-1}\right)$$

$$\begin{array}{ccc}& \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},\mathsf{\ell }{x}_{\acute{r}})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},\aleph x_{\acute{r}-1})\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},\aleph x_{\acute{r}-1}\right)+\mathsf{\pounds }\left(x_{\acute{r}-1},\mathsf{\ell }{x}_{\acute{r}}\right)\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}}\right)+\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}+1}\right)\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right))\right]}^{1-a-\beta -\gamma }.\hfill \end{array}$$

From (13), we have

$$\frac{1}{2}\left(\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\right)\le \mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right).$$

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\le k{\left[\mathsf{\pounds }\left(x_{\acute{r}},x_{\acute{r}-1}\right)\right]}^a.{\left[\mathsf{\pounds }(x_{\acute{r}},x_{\acute{r}+1})\right]}^{\beta }.{\left[\mathsf{\pounds }(x_{\acute{r}-1},x_{\acute{r}})\right]}^{\gamma }.{\left[\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right)\right]}^{1-a-\beta -\gamma }$$

$$\begin{array}{ccc}\hfill {\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\right]}^{1-\beta }& \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right)\right]}^{1-\beta }\hfill \\ \hfill \mathrm{or}\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\right]& \le & k\left[\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right)\right].\hfill \end{array}$$

This implies

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\le k\mathsf{\pounds }\left(x_{\acute{r}-1},x_{\acute{r}}\right)\le k^2\mathsf{\pounds }\left(x_{\acute{r}-2},x_{\acute{r}-1}\right)\le \cdots \le k^{\acute{r}}\mathsf{\pounds }\left(x_0,x_1\right).$$

(18)

From (17) and (18), one writes

$$\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}}\right)\le k^{\acute{r}}\mathsf{\pounds }\left(x_0,x_1\right).$$

Now, we will prove that the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy sequence. For all natural numbers $\acute{r}<m,$ we have

$$\begin{array}{ccc}\hfill \mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)& \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\hfill \\ & & +\theta \left(x_{\acute{r}+1},x_m\right)\theta \left(x_{\acute{r}+2},x_m\right)\theta \left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\mathsf{\pounds }\left(x_{\acute{r}+2},x_{\acute{r}+3}\right)\hfill \\ & \le & \cdots \hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)\mathsf{\pounds }\left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)\mathsf{\pounds }\left(x_{i,}{x}_{i+1}\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{k,}{x}_m\right)\mathsf{\pounds }\left(x_{m-1,}{x}_m\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +{\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)k^{m-1}\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-2}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & & +\left({\Pi }_{k=\acute{r}+1}^{m-1}\theta \left(x_{i,}{x}_m\right)\right)k^{m-1}\theta \left(x_{m-1,}{x}_m\right)\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & =& \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=\acute{r}+1}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right)\hfill \\ & \le & \theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)k^{\acute{r}}\mathsf{\pounds }\left(x_{0,}{x}_1\right)+{\Sigma }_{i=\acute{r}+1}^{m-1}\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i\mathsf{\pounds }\left(x_{0,}{x}_1\right).\hfill \end{array}$$

Taking $\theta \left(x,\widehat{c}\right)\ge 1,$ we write

$${\aleph }_l={\Sigma }_{i=0}^l\left({\Pi }_{j=0}^i\theta \left(x_{j,}{x}_m\right)\right)\theta \left(x_{i,}{x}_{i+1}\right)k^i.$$

Thus, we obtain

$$\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)\le \mathsf{\pounds }\left(x_{0,}{x}_1\right)\left[k^{\acute{r}}\theta \left(x_{\acute{r},}{x}_{\acute{r}+1}\right)+\left({\aleph }_{m-1}-{\aleph }_{\acute{r}}\right)\right].$$

(19)

We have that ${lim}_{\acute{r}\to \infty }{\aleph }_{\acute{r}}$ exist and that the sequence $\left\{x_{\acute{r}}\right\}$ is Cauchy. Therefore, if we take the limit in the inequality (19) as $\acute{r},m\to \infty$, we get

$$\underset{\acute{r},m\to \infty }{lim}\mathsf{\pounds }\left(x_{\acute{r},}{x}_m\right)=0.$$

(20)

Thus, the sequence $\left\{x_{\acute{r}}\right\}$ is a Cauchy sequence in complete Controlled metric space $\left(X,\mathsf{\pounds }\right).$ So, there is some $x^{*}\in$X such that ${lim}_{\acute{r}\to \infty }\mathsf{\pounds }\left(x_{\acute{r},}{x}^{*}\right)=0.$ Now, we show that $x^{*}$ is the unique fixed point of ℵ and $\mathsf{\ell }.$

$$\mathsf{\pounds }\left(\mathsf{\ell }{x}^{*},x_{\acute{r}+2}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}^{*},\aleph x_{\acute{r}+1}\right)$$

$$\begin{array}{ccc}& \le & k{\left[\mathsf{\pounds }\left(x^{*},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},\aleph x_{\acute{r}+1}\right)\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x^{*},\aleph x_{\acute{r}+1}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}^{*}\right))\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x^{*},x_{\acute{r}+1}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x^{*},x_{\acute{r}+2}\right)+\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}^{*}\right))\right]}^{1-a-\beta -\gamma }.\hfill \end{array}$$

taking $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}^{*}\right)=0$, which implies that $\mathsf{\ell }{x}^{*}=x^{*}.$ On the other hand

$$\mathsf{\pounds }\left(x_{\acute{r}+2},\aleph x^{*}\right)=\mathsf{\pounds }\left(\mathsf{\ell }{x}_{\acute{r}+1},\aleph x^{*}\right)$$

$$\begin{array}{ccc}& \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x^{*}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},\mathsf{\ell }{x}_{\acute{r}+1}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}+1},\aleph x^{*}\right)+\mathsf{\pounds }\left(x^{*},\mathsf{\ell }{x}_{\acute{r}+1}\right))\right]}^{1-a-\beta -\gamma }\hfill \\ & \le & k{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x^{*}\right)\right]}^a.{\left[\mathsf{\pounds }\left(x_{\acute{r}+1},x_{\acute{r}+2}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)\right]}^{\gamma }.{\left[\frac{1}{2}(\mathsf{\pounds }\left(x_{\acute{r}+1},\aleph x^{*}\right)+\mathsf{\pounds }\left(x^{*},x_{\acute{r}+2}\right))\right]}^{1-a-\beta -\gamma }.\hfill \end{array}$$

Taking $\acute{r}\to \infty$, we derive $\mathsf{\pounds }\left(x^{*},\aleph x^{*}\right)=0$, which implies that $\aleph x^{*}=x^{*}.$ Hence, $x^{*}$ is the common fixed point of ℓ and $\aleph .$ □

Corollary 5.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete Controlled metric space. Let $\mathsf{\ell },\aleph :X\to X$ be a continuous Interpolative Hardy–Roger contraction mapping. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$ X then ℓ and ℵ is a common unique fixed point in $X.$

By virtue of $\theta \left(x,\widehat{c}\right)=\theta \left(\widehat{c},r\right)=1,$ we easily conclude the result of Karapinar [13]:

Corollary 6.

Let $\left(X,\mathsf{\pounds }\right)$ be a complete metric space. Let $\mathsf{\ell },\aleph :X\to X$ be a continous Interpolative Hardy–Roger contraction mapping. Suppose that

$$\underset{m\ge 1}{sup}\underset{i\to \infty }{lim}\frac{\theta \left(x_{i+1,}{x}_{i+2}\right)\theta \left(x_{i+1,}{x}_m\right)}{\theta \left(x_i,x_{i+1,}\right)}<\frac{1}{k}.$$

Assume that for every $x\in$$X,$${lim}_{\acute{r}\to \infty }\theta \left(x_{\acute{r},}x\right)$ and ${lim}_{\acute{r}\to \infty }\theta \left(x_{,}{x}_{\acute{r}}\right)$ exist and are finite. The sequence $\left\{x_{\acute{r}}\right\}$ converges to some $x^{*}\in$ X then ℓ and ℵ is a common fixed point in $X.$

Example 4.

Let $X=\left\{0,1,2,3\right\}.$ Consider $\mathsf{\pounds }\left(x,\widehat{c}\right)=0$ if $x=\widehat{c}$, and

$$\mathsf{\pounds }\left(0,1\right)=2,\mathsf{\pounds }\left(0,2\right)=\mathsf{\pounds }\left(0,3\right)=\mathsf{\pounds }\left(2,3\right)=\frac{2}{5},\mathsf{\pounds }\left(1,2\right)=\mathsf{\pounds }\left(1,3\right)=\frac{1}{2}.$$

Here, $\theta :$$X\times$$X⟶[1,\infty )$ is given as $\theta \left(x,\widehat{c}\right)=1+\left|x-\widehat{c}\right|$. Then, $\left(X,\mathsf{\pounds }\right)$ is a Controlled-metric space. Let $\mathsf{\ell },\aleph :$$X\to$X be a self mapping defined as

$$\begin{array}{ccc}\hfill \mathsf{\ell }\left(0\right)& =& 0,\mathsf{\ell }\left(1\right)=3,\mathsf{\ell }\left(2\right)=2,\mathsf{\ell }\left(3\right)=3\hfill \\ \hfill \aleph \left(0\right)& =& 0,\aleph \left(1\right)=1,\aleph \left(2\right)=3,\aleph \left(3\right)=2.\hfill \end{array}$$

Thus, ℓ, and ℵ are not Hardy–Roger contractions with $k\in [0,\frac{1}{4})$,

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(0\right),\aleph \left(1\right)\right)=\mathsf{\pounds }\left(0,1\right)=2$$

$$\nleqq k\left[\mathsf{\pounds }\left(0,1\right)+\mathsf{\pounds }\left(0,\mathsf{\ell }\left(0\right)\right)+\mathsf{\pounds }\left(1,\aleph \left(1\right)\right)+\frac{1}{2}\left(\mathsf{\pounds }\left(0,\aleph \left(1\right)\right)+\mathsf{\pounds }(1,\mathsf{\ell }\left(0\right)\right)\right]=k\left(4\right).$$

While, by taking $a=\beta =\frac{2}{5},\gamma =\frac{1}{10}$ and $k=\frac{9}{10}$, then $\forall x,\widehat{c}\in$X such that $\mathsf{\ell }\left(x\right)\ne x$ whenever $\aleph \left(\widehat{c}\right)\ne \widehat{c}$. Thus, (15) is satisfied.

$$\mathsf{\pounds }\left(\mathsf{\ell }\left(1\right),\aleph \left(3\right)\right)=\mathsf{\pounds }\left(3,2\right)=\frac{2}{5}$$

$$\le \frac{99}{100}{\left[\mathsf{\pounds }\left(1,3\right)\right]}^{.4}.{\left[\mathsf{\pounds }\left(1,\mathsf{\ell }\left(1\right)\right){]}^{.4}.\left[\mathsf{\pounds }\right(3,\aleph \left(3\right)\right]}^{.1}.{\left[\frac{1}{2}\left(\mathsf{\pounds }\right(1,\aleph \left(3\right)+\mathsf{\pounds }\left(3,\mathsf{\ell }\left(1\right)\right)\right]}^{.1}=0.45$$

On the other hand, by taking $x_0=0$ and $x_{\acute{r}}=1$ $\forall \acute{r}\ge 1$, then (16) holds.

$$\frac{\theta \left(1,1\right).\theta \left(1,1\right)}{\theta \left(0,1\right)}=\frac{\left(1\right)\left(1\right)}{\left(2\right)}=\frac{1}{2}<1.01=\frac{1}{k},\mathrm{where}k=0.99\in \left(0,1\right).$$

Hence, all conditions of theorem (3.8) are fulfilled. So, ℓ and ℵ is a common unique fixed point which is $x^{*}=0.$

6. Application

Many recent developments on fractional calculus and fixed point theory are investigated in [18,19], and also in the references therein.

Consider the Liouville–Caputo fractional differential equations viewed on order $\xi$$\left({\check{D}}_{\left(\widehat{c},\xi \right)}\right)$ given as

$${\check{D}}_{\left(\widehat{c},\xi \right)}\left(\omega \left(a\right)\right)=\frac{1}{\Gamma \left(j-\xi \right)}{\int }_0^a{\left(a-x\right)}^{j-\xi -1}{\omega }^{\left(j\right)}\left(x\right)dx$$

(21)

where $j-1<\xi <j,$$j=\left[\omega \right]+1,$$\omega \in C^j\left(\left[0,+\infty \right]\right)$, the collection $\left[\gamma \right]$ corresponds to a positive real number and $\Gamma$ is the Gamma function. Let the complete Controlled-metric space ${\mathsf{\pounds }}_{\theta }:C\left(I\right)\times C\left(I\right)\to R^{+}$ be given as

$${\mathsf{\pounds }}_{\theta }({\zeta }_{i-1},{\zeta }_i)={∥{\left({\zeta }_1-{\zeta }_2\right)}^2∥}_{\infty }=\underset{a\in I}{sup}{\left|{\zeta }_1\left(a\right)-{\zeta }_2\left(a\right)\right|}^2$$

(22)

with setting $\theta \left({\zeta }_1,{\zeta }_2\right)=\theta \left({\zeta }_2,{\zeta }_3\right)=2.$ Now, consider the following fashion of Liouville–Caputo fractional derivative

$${\check{D}}_{\left(\widehat{c},\xi \right)}\left(\Omega \left(x\right)\right)={\acute{L}}_f\left(x,\Omega \left(x\right)\right),$$

(23)

where $x\in \left(0,1\right)$ and $\xi \in \left(1,2\right]$ with

$$\left\{\begin{array}{c}\Omega \left(0\right)=0,\hfill \\ \Omega \left(1\right)={\int }_0^{\vartheta }\Omega \left(x\right)dx,\vartheta \in \left(0,1\right),\hfill \end{array}\right.$$

(24)

where $I=\left[0,1\right],$ $\Omega \in C\left(I,R\right)$ and $\acute{L}:I\times R\to R$ is a continuous function. Take $P:\Lambda \to \Lambda$ as

$$\breve{O}v\left(h\right)=\left\{\begin{array}{c}\frac{1}{\Gamma \left(\xi \right)}{\int }_0^w{\left(w-u\right)}_f^{\xi -1}\acute{L}\left(u,v\left(u\right)\right)du\\ -\frac{2u}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^1{\left(1-u\right)}_f^{\gamma -1}\acute{L}\left(u,u\left(u\right)\right)du\\ +\frac{2u}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^{\vartheta }\left({\int }_0^{w_1}{\left(w_1-u_1\right)}_f^{\xi -1}\acute{L}\left(u_1,v\left(u_1\right)\right)d{u}_1\right)du\end{array}\right.$$

(25)

for $v\in \Lambda$ and $w\in \left[0,1\right].$ Now, we state the main result.

Theorem 4.

Assume that $\acute{L}$ is non-decreasing on its second variable and there is ${\zeta }_{i-1},{\zeta }_i\in {\check{D}}_{\theta }(\xi ,{\zeta }_0)$ and $u\in \left[0,1\right]$ such that

$$\left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|\le \Psi \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{x\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2},$$

(26)

where $\Psi =\frac{\left(2\xi -1\right)\left(\Gamma \left(\xi +1\right)\right)}{2\left(5\xi +2\right)}$ and

$$\Delta ({\xi }_{i-1},{\xi }_i)\left(r\right)=k{\left[\mathsf{\pounds }\left(x,\mathsf{\ell }x\right)\right]}^a.{\left[\mathsf{\pounds }\left(\widehat{c},\aleph \widehat{c}\right)\right]}^{1-a}.$$

Then, Equations (23) and (24) have at least one solution, i.e., say ${\zeta }^{*}\in \Psi .$

Proof. 

For each $x\in I$, consider

$$\begin{array}{ccc}& & \left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|\hfill \\ & =& \left|\left(\frac{1}{\Gamma \left(\xi \right)}{\int }_0^w{\left(w-u\right)}^{\xi -1}{\acute{L}}_f\left(u,{\zeta }_{i-1}\left(u\right)\right)du\right.\right.\hfill \\ & & -\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^1{\left(1-u\right)}^{\xi -1}{\acute{L}}_f\left(u,{\zeta }_{i-1}\left(u\right)\right)du\hfill \\ & & \left.+\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^{\vartheta }\left({\int }_0^{w_1}{\left(w_1-u_1\right)}_f^{\xi -1}{\acute{L}}_f\left(u_1,{\zeta }_{i-1}\left(u_1\right)\right)d{u}_1\right)du\right)\hfill \\ & & -\left(\left(\frac{1}{\Gamma \left(\xi \right)}{\int }_0^w{\left(w-u\right)}^{\xi -1}{\acute{L}}_f\left(u,{\zeta }_i\left(u\right)\right)du\right.\right.\hfill \\ & & -\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^1{\left(1-u\right)}^{\xi -1}{\acute{L}}_f\left(u,{\zeta }_i\left(u\right)\right)du\hfill \\ & & \left.\left.+\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^{\vartheta }\left({\int }_0^{w_1}{\left(w_1-u_1\right)}^{\xi -1}{\acute{L}}_f\left(u_1,{\zeta }_i\left(u_1\right)\right)d{u}_1\right)du\right)\right|\hfill \\ & \le & \frac{1}{\Gamma \left(\xi \right)}{\int }_0^w{\left(w-u\right)}^{\xi -1}\left|\acute{L}\left(u,{\zeta }_{i-1}\left(u\right)\right)-{\acute{L}}_f\left(u,{\zeta }_i\left(u\right)\right)\right|du\hfill \\ & & +\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^1{\left(1-u\right)}^{\xi -1}\left|{\acute{L}}_f\left(u,{\zeta }_{i-1}\left(u\right)\right)-{\acute{L}}_f\left(u,{\zeta }_i\left(u\right)\right)\right|du\hfill \\ & & +\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^{\vartheta }\left|{\int }_0^{w_1}{\left(w_1-u_1\right)}^{\xi -1}\left({\acute{L}}_f\left(u_1,{\zeta }_{i-1}\left(u_1\right)\right)-{\acute{L}}_f\left(u,{\zeta }_i\left(u\right)\right)\right)d{u}_1\right|du.\hfill \end{array}$$

Now, we have

$$\begin{array}{ccc}& & \left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|\hfill \\ & \le & \frac{1}{\Gamma \left(\xi \right)}{\int }_0^w{\left(w-u\right)}^{\xi -1}\Psi \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}du\hfill \\ & & +\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^1{\left(1-u\right)}^{\xi -1}\Psi \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{x\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}du\hfill \\ & & +\frac{2w}{\left(2-{\vartheta }^2\right)\Gamma \left(\xi \right)}{\int }_0^{\vartheta }{\int }_0^{w_1}{\left(w_1-u_1\right)}^{\xi -1}\Psi \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}d{u}_{1}du\hfill \\ & \le & \frac{\Psi \Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{\Gamma \left(\xi \right){\left(1+\sqrt{{max}_{x\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}\left\{\begin{array}{c}{\int }_0^w{\left(w-u\right)}^{\xi -1}du\\ +\frac{2w}{\left(2-{\vartheta }^2\right)}{\int }_0^1{\left(1-u\right)}^{\xi -1}du\\ +\frac{2x}{\left(2-{\vartheta }^2\right)}{\int }_0^{\vartheta }{\int }_0^{w_1}{\left(w_1-u_1\right)}^{\xi -1}d{u}_{1}du\end{array}\right\}.\hfill \end{array}$$

This yields that

$$\begin{array}{ccc}\hfill \left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|& \le & \frac{\Psi \Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{\Gamma \left(\xi \right){\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}\hfill \\ & & \left\{\frac{w^{\xi }}{\xi }+\frac{2w}{\left(2-{\vartheta }^2\right)}\frac{1}{\xi }+\frac{2w}{\left(2-{\vartheta }^2\right)}\frac{{\vartheta }^{\xi +1}}{\xi \left(\xi +1\right)}\right\}\hfill \\ & \le & \frac{\Omega \Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}\hfill \\ & & \underset{w\in \left(0,1\right)}{sup}\left\{w^{\xi }+\frac{2w}{\left(2-{\vartheta }^2\right)}+\frac{2w}{\left(2-{\vartheta }^2\right)}\frac{{\vartheta }^{\xi +1}}{\left(\xi +1\right)}\right\}\hfill \\ & =& \frac{\left(2\xi -1\right)}{2\left(5\xi +2\right)}\frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{x\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}\hfill \\ & & \underset{w\in \left(0,1\right)}{sup}\left\{w^{\xi }+\frac{2w}{\left(2-{\vartheta }^2\right)}+\frac{2w}{\left(2-{\vartheta }^2\right)}\frac{{\vartheta }^{\xi +1}}{\left(\xi +1\right)}\right\}\hfill \\ & =& \frac{\left(2\xi -1\right)}{2\left(5\xi +2\right)}\frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{x\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}.\hfill \end{array}$$

It implies that

$$\left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|\le \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}.$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathsf{\pounds }}_{\theta }(\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right))& =& \underset{a\in I}{sup}{\left|\breve{O}{\zeta }_{i-1}\left(r\right)-\breve{O}{\zeta }_i\left(r\right)\right|}^2\hfill \\ & \le & \frac{\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}{{\left(1+\sqrt{{max}_{w\in I}\Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right)}\right)}^2}\hfill \\ & <& \Delta ({\zeta }_{i-1},{\zeta }_i)\left(r\right).\hfill \end{array}$$

So, in view of contractive condition (1), we have

$$\mathsf{\pounds }\left(\mathsf{\ell }{\zeta }_{i-1},\aleph {\zeta }_i\right)\le k{\left[\mathsf{\pounds }\left({\zeta }_{i-1},{\zeta }_i\right)\right]}^a.{\left[\mathsf{\pounds }\left({\zeta }_{i-1},\mathsf{\ell }{\zeta }_{i-1}\right)\right]}^{\beta }.{\left[\mathsf{\pounds }\left({\zeta }_i,\aleph {\zeta }_i\right)\right]}^{1-a-\beta },$$

for all $i\in \mathbb{N}$, ${\zeta }_i\in \Psi$. Thus, all the required hypotheses of Theorem 1 are satisfied and we ensure that the Equations (23) and (24) have at least one solution in $\breve{O}.$ □

Example 5.

Consider the Liouville–Caputo fractional differential equations based on order ξ $\left({\check{D}}_{\left(\widehat{c},\xi \right)}\right)$

$${\check{D}}_{\left(\widehat{c},\frac{3}{2}\right)}\left(\Omega \left(w\right)\right)=\frac{1}{{\left(w+3\right)}^2}\frac{\left|\Omega \left(w\right)\right|}{1+\left|\Omega \left(w\right)\right|},$$

(27)

and its integral boundary valued problem:

$$\left\{\begin{array}{c}\Omega \left(0\right)=0,\hfill \\ \Omega \left(1\right)={\int }_0^{\frac{3}{4}}\Omega \left(w\right)dw,\vartheta \in \left(0,1\right),\hfill \end{array}\right.$$

(28)

where $\xi =\frac{3}{2},$$\vartheta =\frac{3}{4}$ and $\acute{L}\left(w,v\left(w\right)\right)=\frac{1}{{\left(w+3\right)}^2}\frac{\left|\Omega \left(w\right)\right|}{1+\left|\Omega \left(w\right)\right|}.$ So, the above setting is an example of Equations (23) and (24). Hence, the Equations (27) and (28) have at least one solution.

7. Conclusions

In our present investigation, the paper has conducted a comprehensive analysis of Interpolative contraction, expanding upon the initial ideas introduced by Karapinar in 2018. Our study has extended the concept of Interpolative contraction mappings to include non-linear Kannan Interpolative, Riech Rus Ćirić interpolative, and Hardy–Roger Interpolative contraction mappings based on Controlled functions. Through the exploration of Controlled metric spaces, we have established several fixed point results, thereby advancing the current understanding of this analysis. Additionally, we have presented a concrete example that exemplifies the motivation behind our investigations. Lastly, we have showcased the application of the proposed non-linear Interpolative contractions to Liouville–Caputo fractional derivatives and fractional differential equations. Overall, this research contributes to the field by providing new insights and potential applications in the study of Interpolative contractions. In the future, these findings can be extended to obtain fixed point results for single and multi-valued mappings within the framework of double Controlled-metric space and triple Controlled-metric spaces.