This paper proves fixed point theorems for relation-theoretic $F$ -interpolative mapping endowed with binary relation in Branciari Distance. Henceforth, the results obtained will be verified with the help of illustrative examples. Also, we demonstrate the results with an application in matrix equations.

Mathematics Subject Classification 2010: 47H10, 54H25.

Keywords: Fixed point, interpolative mapping, $F$ -contraction, metric space, binary relation, matrix equation.

Fixed point theory is a fascinating area of research for the researchers studying non-linear phenomena. It has many applications for non-linear functional analysis, Approximation theory, Optimization Theory (Saddle function), Variation inequalities, Game theory (Nash equilibrium) and Economics (Black Scholes theorem). Fixed point theory is quite and sequel to the existing theory of Differential, Integral, Partial, Fractional differential, functional equations and matrix equations. Fixed point theory as well as Banach contraction principle have been studied and generalized in different spaces and various fixed point theorems are developed. In 1968 Kannan¹ introduced a discontinuity of contraction mappings that can possess a fixed point on a complete metric space by filling the gap created by Banach for more than thirty years. Reich proved the fixed point theorem using three metric points by combining the concept of Banach and Kanann on complete metric space. Dass-Gupta proved the results of the fixed point theorem of the rational type operator by using contraction mapping in metric space.

In 2000, Branciari² introduced a class of generalized metric spaces by replacing triangular inequality with similar ones which involve four or more points instead of three and improved Banach contraction mapping principle. In 2008, Azam and Arshad³ using the concept of Branciari investigated the mappings given by Kannan by applying the rectangular properties in a generalized metric space. In 2011, Moradi and Alimohammadi⁴ generalized Kannan’s results, by using the sequentially convergent mappings and rectangular properties in metric space. Furthermore, Morandi and Alimohammadi investigated and extended Kannan’s mapping by using the ideal due to Branciari. Since then, several authors involved in investigations of Banach’s contraction mappings using rectangular properties in different spaces.

In 2004, Ran and Reurings⁵ proved an order-theoretic analogue of Banach contraction principle which marks the beginning of a vigorous research activity. This result was discovered while investigating the solutions to some special matrix equations.. In continuation of Ran and Reurings, Nieto and Rodríguez-López⁶ who proved two very useful results and used them to solve some differential equations.

In 2012, Wardowski⁷ initiated the study of fixed points of a new type of contractive mappings in complete metric spaces. In 2014, Wardowski and Dung,⁸ proved fixed points of $F$ -weak contractions on complete metric spaces. Acar et al.⁹ and Altun et al.¹⁰ gave Generalized multivalued $F$ -contractions on complete metric spaces. In 2014, Minak et al.,¹¹ proved Ćirić type generalized $F$ -contractions on complete metric spaces and fixed point results. Paesano and Vetro¹² gave the proof on Multi-valued $F-$ contractions in 0-complete partial metric spaces with application to Volterra type integral equation. Piri and Kumam¹³ proved some fixed point theorems concerning $F-$ contraction in complete metric spaces. Sawangsup and Sintunavarat¹⁴ proved the fixed point theorems for $F_{\Re }$ contractions with applications to the solution of non-linear matrix equations, Tomar and Sharma¹⁵ proved some coincidence and common fixed point theorems concerning $F-$ contraction and applications and Bashir¹⁶ proved the fixed point results of a generalized reversed $F-$ contraction mapping and its application.

On the other hand, Alam and Imdad¹⁷ gave a generalizatiFor more res on of the Banach contraction principle in a complete metric space equipped with binary relation. Their results show that the contraction condition holds only for those elements linked with the binary relation, not for every pair of elements. Recently, Kannan’s and Reich’s¹⁸ fixed point theorems have been studied and extended in several directions, Karapinar¹⁹ modified the classical Kannan contraction phenomena to an interpolative Kannan contraction one to maximize the rate of convergence of an operator to a unique fixed point. However, by giving a counter-example, Karapinar and Agarwal²⁰ pointed out a gap in the paper about the assumption of the fixed point being unique and came up with a corrected version. They provided a counter-example to verify that the fixed point need not be unique and invalidate the assumption of a unique fixed point. Since then, several results for variants of interpolative mapping proved for single and multivalued in various abstract spaces.

Further, Karapinar and Agarwal²¹ proved interpolative Rus-Reich-Ćirić type contractions via simulation functions. Errai et al.²² gave some new results of interpolative Hardy-Rogers and Reich-Rus-Ćirić type contraction in-metric spaces to prove the existence of the coincidence point. Mishra et al.²³ proved the common fixed point theorems for interpolative Hardy-Rogers and Reich-Rus-Ćirić type contraction on quasi partial -metric space. Aydi et al.²⁴ proved -interpolative Reich-Rus-Ćirić type contractions on metric spaces. Aydi et al.²⁵ proved an interpolative Ćirić-Reich-Rus type contractions via the Branciari distance. Gautam et al.²⁶ proved the fixed point of interpolative Rus-Reich-Ćirić contraction mapping on rectangular quasi-partial -metric space.

This manuscripts prove a fixed points theorem for relation-theoretic -contraction mappings via an arbitrary binary relation concept in Branciari distance metric space. In particular, we improve and extend the works due to Alam and Imdad,²⁷ Ahmadullah et al.,²⁸ Ahmadullah et al.,²⁹ Eke et al.,³⁰ Sawangsup and Sintunavarat,¹⁴ Aydi et al.³¹ and Karapinar et al.²⁰ In doing so, we will generalize several other works in the literature having the same setting.

This section introduces some definitions, theorems and preliminary results, which will help develop the main result.

The concept of a Branciari distance space has been introduced by Brianciari where the triangular inequality is replaced by a quadrilateral one, which states as follows:

Definition 1². Let $ℳ$  be a non-empty set. Suppose that the mapping $d:ℳ\times ℳ\to \left[0,\infty \right)$ be a function for all and all distinct points $z,w\in ℳ$ , each distinct from $\mu$ and $\nu$ .

(i) $d\left(\mu ,\nu \right)\ge 0$ and $d\left(\mu ,\nu \right)=0$ if and only if $\mu =\nu$ ;

(ii) $d\left(\mu ,\nu \right)=d\left(\mu ,\nu \right)$ ;

(iii) $d\left(\mu ,\nu \right)\le d\left(\mu ,w\right)+d\left(w,z\right)+d\left(z,\nu \right).$

Then is called a Branciari distance and the pair $\left(ℳ,d\right)$  is called a Branciari distance space.

Definition 2². Let $\left(ℳ,d\right)$  be a metric space. A mapping $\Gamma :ℳ\to ℳ$ is said to be sequentially convergent if we have, for every sequence $\left\{{\nu }_n\right\}$ , if $\left\{\Gamma {\nu }_n\right\}$ is convergence then $\left\{{\nu }_n\right\}$ also is convergence.$ℳ$ is said to be subsequentially convergent if we have, for every sequence $\left\{{\nu }_n\right\}$ , if $\left\{\Gamma {\nu }_n\right\}$  is convergence then $\left\{{\nu }_n\right\}$  has a convergent subsequence.

Definition 3³⁶. Let $( ℳ,d)$ be a Branciari distance space and $\left\{{\mu }_n\right\}$  be a sequence in $ℳ$ .

(i)A sequence $\left\{{\mu }_n\right\}$ is converges to point ${\mu }^{\star }\in ℳ$ if ${\text{lim}}_{n\to \infty }d\left({\mu }_n,{\mu }^{\star }\right)=0$

(ii) A sequence $\left\{{\mu }_n\right\}$  is said to be Cauchy if for every $ϵ>0$ , there exists a positive integer $\mathbb{N}=\mathbb{N}\left(ϵ\right)$ such that $d\left({\mu }_n,{\mu }_m\right)<ϵ,$  for all $n,m>\mathbb{N}$ .

(iii) We say that $\left(ℳ,d\right)$  is complete if each Cauchy sequence in $ℳ$ is convergent.

Lemma 1³⁶. Let $\left(ℳ,d\right)$  be a Branciari distance space. A mapping $\Gamma :ℳ\to ℳ$ is continuous at ${\mu }^{\star }\in ℳ$ , if we have $\Gamma {\mu }_n\to \Gamma {\mu }^{\star }$ or ${\text{lim}}_{n\to \infty }d\left(\Gamma {\mu }_n,\Gamma {\mu }^{\star }\right)=0$ , for any sequence $\left\{{\mu }_n\right\}$  in $ℳ$  converges to ${\mu }^{\star }\in ℳ$ , that is ${\mu }_n\to {\mu }^{\star }$ .

Proposition 1³⁷. Suppose ${\mu }_n$ is a Cauchy sequence in a Branciari distance space such that $\underset{n\to \infty }{\text{lim}}d\left({\mu }_n,{\mu }^{\star }\right)=d\left({\mu }_n,w^{\star }\right)=0,$  where ${\mu }^{\star },w^{\star }\in ℳ$ . Then ${\mu }^{\star }=w^{\star }$ .

Another noted attempt to extend the Banach contraction principle is essentially due to Wardowski.

The following explanations for developing the -contraction definition was obtained from Wardowski,⁷ Wardowski and Van Dung,⁸ and Cosentino et al.³²

Let $F:{ℝ}^{+}\to ℝ$ be a mapping satisfying:

(F1) $F$  is strictly increasing, i.e. for all $\mathcal{J},\mathcal{K}\in {ℝ}^{+},\mathcal{J}<\mathcal{K}$ implies $F\left(\mathcal{J}\right)<F\left(\mathcal{K}\right)$ ;

(F2) For each sequence ${\left\{{\mathcal{J}}_n\right\}}_{n\in \mathbb{N}}$ of positive numbers,${\text{lim}}_{n\to \infty }{\mathcal{J}}_n=0$ if and only if ${\text{lim}}_{n\to \infty }F\left({\mathcal{J}}_n\right)=-\infty$ ;

(F3) There exists $z\in \left(0,1\right)$ satisfying ${\text{lim}}_{{\mathcal{J}}_n\to 0^{+}}{\mathcal{J}}_n^{z}F\left({\mathcal{J}}_n\right)=0$

We denote the family of all functions $F$ satisfying conditions $\left(F1-F3\right)$ by $ℱ$ . Some examples of functions $F\in ℱ$ are:

$F_1\left(c\right)=\text{ln}c\Rightarrow \frac{d\left(\Gamma \mu ,\Gamma \nu \right)}{d\left(\mu ,\nu \right)}\le e^{-\eta };$

$F_2\left(c\right)=c+\text{ln}c\Rightarrow \frac{d\left(\Gamma \mu ,\Gamma \nu \right)}{d\left(\mu ,\nu \right)}\le e^{-\eta +d\left(\mu ,\nu \right)-d\left(\Gamma \mu ,\Gamma \nu \right)}$ ;

$F_3\left(c\right)=-\frac{1}{\sqrt{c}}\Rightarrow \frac{d\left(\Gamma \mu ,\Gamma \nu \right)}{d\left(\mu ,\nu \right)}\le \frac{1}{{\left(1+\eta \sqrt{d\left(\mu ,\nu \right)}\right)}^2}$ ;

$F_4\left(c\right)=\text{ln}\left(c^2+c\right)\Rightarrow \frac{d\left(\Gamma \mu ,\Gamma \nu \right)\left(1+d\left(\Gamma \mu ,\Gamma \nu \right)\right)}{d\left(\mu ,\nu \right)\left(1+d\left(\mu ,\nu \right)\right)}\le e^{-\eta }$ .

Wardowski introduced a generalization of the Banach contraction principle in metric spaces as follows:

Definition 4⁷.   Let $\left(ℳ,d\right)$  be a metric space. A self-mapping $\Gamma$ on $ℳ$ is called an $F$ -contraction mapping if there exists $F\in ℱ$ and $\eta \in {ℝ}^{+}$ such that for all $\mu ,\nu \in ℳ$ ,$\begin{array}{rrr}\hfill d\left(\Gamma \mu ,\Gamma \nu \right) >0\Rightarrow \eta +F\left(d\left(\Gamma \mu ,\Gamma \nu \right)\right)& \hfill \le & \hfill F\left(d\left(\mu ,\nu \right)\right).\end{array}$

Wardowski⁷ proved the following fixed point theorem:

Theorem 1⁷. Let $\left(ℳ,d\right)$  be a complete metric space and $\Gamma :ℳ\to ℳ$ be a $F$ -contraction mapping. If there exist $\eta >0$ such that for all $\mu ,\nu \in ℳ,d\left(\Gamma \mu ,\Gamma \nu \right) >0$ , implies $\eta +F\left(d\left(\Gamma \mu ,\Gamma \nu \right)\right)\le F\left(d\left(\mu ,\nu \right)\right),$  then $\Gamma$ has a unique fixed point.

Kannan¹ proved the following theorem:

Theorem 2¹. Let $\left(ℳ,d\right)$ be a complete metric space and a self-mapping $\Gamma :ℳ\to ℳ$ be a mapping such that $d\left(\Gamma \mu ,\Gamma \nu \right)\le \eta \left\{d\left(\mu ,\Gamma \mu \right)+d\left(\nu ,\Gamma \nu \right)\right\},$ for all $\mu ,\nu \in ℳ$ and $0\le \eta \le \frac{1}{2}.$  The $\Gamma$  has a unique fixed point $\delta \in ℳ$ and for any $\mu \in ℳ$ the sequence of iterate $\left\{{\Gamma }^n\mu \right\}$  converges to $\delta$ .

The following results for interpolative Kannan contraction have been proved in as follows:

Definition 5¹⁹. Let $\left(ℳ,d\right)$ be a metric space, the mapping $\Gamma :ℳ\to ℳ$ is said to be interpolative Kannan contraction mappings if $\begin{array}{rrr}\hfill d\left(\Gamma \mu ,\Gamma \nu \right)& \hfill \le & \hfill \eta {\left[d\left(\mu ,\Gamma \mu \right)\right]}^{\delta }.{\left[d\left(\nu ,\Gamma \nu \right)\right]}^{1-\delta },\end{array}$ for all $\mu ,\nu \in ℳ$ with $\mu \ne \Gamma \mu$ , where $\eta \in \left[0,1\right)$ and $\delta \in \left(0,1\right)$ .

Theorem 3¹⁹. Let $\left(ℳ,d\right)$ be a complete metric space and $\Gamma$ be an interpolative Kannan type contraction. Then $\Gamma$ has a unique fixed point in $ℳ$ .

In 2018, Karapinar et al.²¹ proved an interpolative Reich-Rus-Ćirić type contractions fixed point result on partial metric space as follows.

Theorem 4²¹. Let $\left(ℳ,d\right)$ be a complete metric space $\Gamma :ℳ\to ℳ$ . be a mapping such that $\begin{array}{rrr}\hfill p\left(\Gamma \mu ,\Gamma \nu \right)& \hfill \le & \hfill \eta {\left[p\left(\mu ,\nu \right)\right]}^{\delta }.{\left[p\left(\mu ,\Gamma \mu \right)\right]}^{\alpha }.{\left[p\left(\nu ,\Gamma \nu \right)\right]}^{1-\alpha -\delta },\end{array}$ for all $\mu ,\nu \in ℳ\setminus Fix\left(\Gamma \right)$ where $Fix\left(\Gamma \right)=\left\{\mu \in ℳ,\Gamma \mu =\mu \right\}$ . Then $\Gamma$  has a fixed point in $ℳ$ .

Binary relation-theoretic in metric spaces

In this part, we will recall some definitions of relation theoretic notion related to binary relation with relevant relation-theoretical variants of some metrical concepts such as completeness and continuity which will be useful in developing our main results.

In the following discussion $ℛ$ stands for a nonempty binary relation while ${\mathbb{N}}_0$ denotes the set of whole numbers, i.e., $\mathbb{N}\cup \left\{0\right\}$ .

Definition 6³⁸. A binary relation on a non-empty set $ℳ$ is defined as a subset of $ℳ\times ℳ$ , which will be denoted by $ℛ$ . We say that $\mu$ relates to $\nu$ under $ℛ$ iff $\left(\mu ,\nu \right)\in ℛ$ .

Definition 7¹⁷. Let $ℛ$ be a binary relation defined on a non-empty set $ℳ$ and $\mu ,\nu \in ℳ$ . We say that $\mu$ and $\nu$ are $ℛ$ -comparative if either $\left(\mu ,\nu \right)\in ℛ$ or $\left(\nu ,\mu \right)\in ℛ$ . We denote it by $\left[\mu ,\nu \right]\in ℛ$ .

Definition 8³⁸. Let $ℛ$  be a binary relation defined on a non-empty set $ℳ$ . Then the symmetric closure of $ℛ$ is defined as the smallest symmetric relation containing $ℛ\left(i.e.{ℛ}^s:=ℛ\cup {ℛ}^{-1}\right)$ , where ${ℛ}^{-1}=\left\{\left(\mu ,\nu \right)\in {ℳ}^2:\left(\nu ,\mu \right)\in ℛ\right\}$ .

Proposition 2¹⁷. If $ℛ$ is a binary relation defined on a non-empty set $ℳ$ , then $\begin{array}{rrr}\hfill \left(\mu ,\nu \right)\in {ℛ}^s& \hfill \iff & \hfill \left[\mu ,\nu \right]\in ℛ.\end{array}$

Definition 9¹⁷. Let $ℛ$ be a binary relation defined on a non-empty set $ℳ$ . Then a sequence $\left\{{\mu }_n\right\}\subset ℳ$ is called $ℛ$ -preserving if $\left({\mu }_n,{\mu }_{n+1}\right)\in ℛ, \forall n\in {\mathbb{N}}_0$ .

Definition 10¹⁷. Let $ℳ$ be a non-empty set and $\Gamma$ a self-mapping on$ℳ$ . A binary relation $ℛ$ on $X$ is called $\Gamma$ -closed if for any $\mu ,\nu \in ℳ$ , $\begin{array}{rrr}\hfill \left(\mu ,\nu \right)\in ℛ& \hfill \Rightarrow & \hfill \left(\Gamma \mu ,\Gamma \nu \right)\in ℛ.\end{array}$

Definition 11²⁷. Let $\left(ℳ,d\right)$ be a metric space and $ℛ$ a binary relation on $ℳ$ . We say that $\left(ℳ,d\right)$ is $ℛ$ -complete if every $ℛ$ -preserving Cauchy sequence in$ℳ$ converges.

Definition 12¹⁷. Let $\left(ℳ,d\right)$  be a metric space. A binary relation $ℛ$ defined on$ℳ$ is called d-self closed if whenever $\left\{{\mu }_n\right\}$  is an $ℛ$ -preserving sequence and ${\mu }_n\stackrel{d}{\to }\mu$ , then there is a sub sequence $\left\{{\mu }_{n_k}\right\}$ of $\left\{{\mu }_n\right\}$  with $\left[{\mu }_{n_k},\mu \right]\in ℛ$ for all $k\in {\mathbb{N}}_0$ .

Definition 13¹⁷. Let$ℳ$ be a non-empty set and $ℛ$ a binary relation on$ℳ$ . A subset $D$ of$ℳ$ is called $ℛ$ -directed if for each $\mu ,\nu \in D$ , there exists $z$  in$ℳ$ such that $\left(\mu ,z\right)\in ℛ$ and $\left(\nu ,z\right)\in ℛ$ .

Definition 14³⁹. Let$ℳ$  be a non-empty set and $ℛ$ be a binary relation defined on a non-empty set$ℳ$ . Let $k$ be a natural number, a path $ℛ$ from $\mu$ to is a finite sequence $\left\{z_0,z_1,z_2,\mathrm{...},z_k\right\}\in ℳ$ which satisfies the following conditions:

1. $z_0=\mu$ and $z_1=\mu$ ;
2. $\left[z_i,z_{i+1}\right]\in ℛ$ for each $i\in \left\{0,1,2,3,\mathrm{...}k-1\right\}$ for all $\mu ,\nu \in ℳ)$ ;
3. $ℳ\left(\Gamma ,ℛ\right)$ : the collection of all points $\mu \in ℳ$ such that $\left(\mu ,\Gamma \mu \right)\in ℛ$ ;

(vi) Let us denote $\gamma \left(\mu ,\nu ,ℛ\right)$ : the collection of all paths $\left\{z_0,z_1,z_2,\mathrm{...},z_k\right\}$  joining $\mu$ to $\nu$ in $ℛ$ such that $\left[z_i,\Gamma z_i\right]\in ℛ$ for each $i\in \left\{1,2,3,\mathrm{...}k-1\right\}$ .

Further, we state some preliminary results which will be helpful to develop our main results.

Ahmadullah et al.³³ proved the results in metric-like spaces as well as partial metric spaces equipped with a binary relation. Sawangsup and Sintunavarat¹⁴ by combining the concepts of Wardowski and proved the fixed point theorems for $F_{\Re }$ -contractions in metric space with applications to the solution of non-linear matrix equations with binary relation as follows:

Theorem 5³⁴. Let $\left(ℳ,d\right)$ be a complete metric space, $ℛ$ a binary relation on $ℳ$ and let $\Gamma$ be a self-mapping on $ℳ$ . Suppose that the following conditions hold:

(i) $ℳ\left(\Gamma ,ℛ\right)$ is non-empty,

(ii)$ℛ$ is $\Gamma$ -closed,

(iii) either $\Gamma$ is continuous or $ℛ$ is G -self-closed,

(iv)there exists $F\in ℱ$ and $\eta \in {ℝ}^{+}$ such that for all $\mu ,\nu \in ℳ$ with $(\mu ,\nu \in ℛ)$ ,$\begin{array}{rrr}\hfill d\left(\Gamma \mu ,\Gamma \nu \right)>0\Rightarrow \eta +F\left(d\left(\Gamma \mu ,\Gamma \nu \right)\right)& \hfill \le & \hfill F\left(d\left(\mu ,\nu \right)\right).\end{array}$

Then $\Gamma$ has a fixed point. Moreover, for each $x_0\in ℳ\left(\Gamma ,ℛ\right)$ the Picard sequence $\left\{{\Gamma }^n{x}_0\right\}$ is convergent to the fixed point $\Gamma$ .

Now, we prove the main results using interpolative Reich-Rus-Ćirić- -contraction mapping concepts via binary relation in generalized metric spaces.

Theorem 6. Let $\left(ℳ,d\right)$  be a complete metric space, $ℛ$ a binary relation on $ℳ$ and let $\Gamma$ be an interpolative Reich-Rus-Ćirić type contractions mapping on $ℳ$ . Suppose that the following conditions hold:

(i)$\left(ℳ,d\right)$ is $\Gamma$ complete,

(ii)$ℳ\left(\Gamma ,ℛ\right)$ is non-empty,

(iii)$ℛ$ is $\Gamma$ -closed,

(iv) the sequence $\left\{{\mu }_n\right\}$  is $ℛ$ -preserving,

(v) either $\Gamma$ is continuous or $ℛ$ is d-self closed,

(vi) there exists a constant $\eta >0$ such that $\forall \mu ,\nu \in ℳ$ with $(\mu ,\nu \in ℛ)$ $\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma \mu ,\Gamma \nu \right)\right)& \hfill \le & \hfill F\left({ℳ}_{ℛ}\left(\mu ,\nu \right)\right),\end{array}$ $\mu ,\nu \in ℳ\setminus Fix\left(\Gamma \right)$ where for all where $Fix\left(\Gamma \right)=\left\{\mu \in ℳ,\Gamma \mu =\mu \right\}$ . Then $\Gamma$ has a fixed point. Also, if $\Gamma$ is subsequentially convergent then for every ${\mu }_{n-1}\in ℳ$ the sequence of iterate $\left\{{\Gamma }^n{\mu }_{n-1}\right\}$  converges to this fixed point. Moreover, if

(vii)$\gamma \left(\mu ,\nu ,{ℛ}^s\right)$ is non-empty, for each $\mu ,\nu \in ℳ$ . Then $\Gamma$ has a unique fixed point.

Proof. Assume $x_0$ be an arbitrary point in $ℳ\left(\Gamma ,ℛ\right)$ . We construct a sequence $\left\{{\mu }_n\right\}$ of Picard iterates such that ${\mu }_n={\Gamma }^n{\mu }_0=\Gamma {\mu }_{n-1}$ for all $n\in \mathbb{N}$ . By condition $\left(iii\right)$  of Theorem 6, we have $\left({\mu }_0,\Gamma {\mu }_0\right)\in ℛ$ and $ℛ$ is $\Gamma$ -closed, therefore

$\left(\Gamma {\mu }_{n-1},{\Gamma }^{n+1}{\mu }_{n-1}\right),\left({\Gamma }^{n+1}{\mu }_{n-1},{\Gamma }^{n+2}{\mu }_{n-1}\right),\dots ,\left({\Gamma }^n{\mu }_{n-1},{\Gamma }^{n+2}{\mu }_{n-1}\right)$ .

Using $($\ref{Eqt 3.3}$)$, we note that

$\begin{array}{r}\hfill \left({\Gamma }^n{\mu }_{n-1},{\Gamma }^{n+1}{\mu }_{n-1}\right)\in ℛ,\end{array}$

$\begin{array}{r}\hfill \forall n\in {\mathbb{N}}_0.\end{array}$  Therefore the sequence $\left\{{\mu }_n\right\}$ is $ℛ$ -preserving.

If there exists $n$ such that ${\mu }_n={\mu }_{n+1}$ , then ${\mu }_n$ is a fixed pint of $\Gamma$ . The proof is completed. For that case, we assume that ${\mu }_n\ne {\mu }_{n+1}$ for each $n\ge 0$ . Therefore

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}d\left({\mu }_n,{\mu }_{n+1}\right)& \hfill =& \hfill 0.\end{array}$

To show this, let $\mu ={\mu }_{n-1}$ and $\nu ={\mu }_n$ , using $($\ref{eqt 3.1}$)$ for all $n\in {\mathbb{N}}_0$ , we deduce that

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }_{n-1},\Gamma {\mu }_n\right)\right)& \hfill \le & \hfill F\left({ℳ}_{ℛ}\left({\mu }_{n-1},{\mu }_n\right)\right),\end{array}$

where

$\begin{array}{rrr}\hfill {ℳ}_{ℛ}\left({\mu }_{n-1},{\mu }_n\right)& \hfill =& \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\delta }.{\left[d\left({\mu }_{n-1},\Gamma {\mu }_{n-1}\right)\right]}^{\alpha }.{\left[d\left({\mu }_n,\Gamma {\mu }_n\right)\right]}^{1-\alpha -\delta },\\ \hfill & \hfill \le & \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\delta }.{\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha }.{\left[d\left({\mu }_n,{\mu }_{n+1}\right)\right]}^{1-\alpha -\delta },\\ \hfill & \hfill =& \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_n,{\mu }_{n+1}\right)\right]}^{1-\alpha -\delta }.\end{array}$  

Taking $($\ref{Eqt 3.7}$)$ into $($\ref{Eqt 3.6}$)$, we obtain

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }_{n-1},\Gamma {\mu }_n\right)\right)& \hfill \le & \hfill F\left({\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_n,{\mu }_{n+1}\right)\right]}^{1-\alpha -\delta }\right).\end{array}$

By the continuity property of , and $($\ref{Eqt 3.8}$)$, we get

$\begin{array}{rrr}\hfill d\left(\Gamma {\mu }_{n-1},\Gamma {\mu }_n\right)& \hfill \le & \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_n,{\mu }_{n+1}\right)\right]}^{1-\alpha -\delta },\\ \hfill {\left(d\left({\mu }_n,{\mu }_{n+1}\right)\right)}^{1-\left(1-\alpha -\delta \right)}& \hfill \le & \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha +\delta },\\ \hfill {\left(d\left({\mu }_n,{\mu }_{n+1}\right)\right)}^{\alpha +\delta }& \hfill \le & \hfill {\left[d\left({\mu }_{n-1},{\mu }_n\right)\right]}^{\alpha +\delta }.\end{array}$

So, we conclude that

$\begin{array}{rrr}\hfill d\left({\mu }_n,{\mu }_{n+1}\right)& \hfill \le & \hfill d\left({\mu }_{n-1},{\mu }_n\right),\end{array}$

for all $n\ge 1$ .

Consequently, we have

$\begin{array}{rrr}\hfill \eta +F\left(d\left({\mu }_n,{\mu }_{n+1}\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }_{n-1},{\mu }_n\right)\right).\end{array}$

Equivalent to

$\begin{array}{rrr}\hfill F\left(d\left({\mu }_n,{\mu }_{n+1}\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }_{n-1},{\mu }_n\right)\right)-\eta .\end{array}$

Similar, let $\mu ={\mu }_n$ , $\nu ={\mu }_{n+1}$ , using $($\ref{eqt 3.1}$)$ and $($\ref{Eqt 3.9}$)$ for all $n\in {\mathbb{N}}_0$ , we get

$\begin{array}{rrr}\hfill F\left(d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }_{n-1},{\mu }_n\right)\right)-2\eta .\end{array}$

Proceeding this way, by induction we deduce

$\begin{array}{rrr}\hfill F\left(d\left({\mu }_n,{\mu }_{n+1}\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }_{n-1},{\mu }_n\right)\right)-n\eta ,\forall n\ge 1.\end{array}$

That is $d\left({\mu }_{n-1},{\mu }_n\right)$ is non-increasing sequence with non-negative terms. We denote ${\mathcal{J}}_n=d\left({\mu }_n,{\mu }_{n+1}\right)$ , for all $n\in {\mathbb{N}}_0$ . Since $\Gamma$ is an $F$ - $ℛ$ -interpolative contraction mapping.

From $($\ref{Eqt 3.11}$)$, we obtain

$\begin{array}{rrr}\hfill F\left({\mathcal{J}}_n\right)& \hfill \le & \hfill F\left({\mathcal{J}}_{n-1}\right)-\eta \le F\left({\mathcal{J}}_{n-2}\right)-2\eta \le \dots \le F\left({\mathcal{J}}_0\right)-n\eta ,\end{array}$

for all $n\in {\mathbb{N}}_0$ .

By $\left(F2\right)$ , we have

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}{\mathcal{J}}_n& \hfill =& \hfill 0.\end{array}$

If and only if

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}F\left({\mathcal{J}}_n\right)& \hfill =& \hfill -\infty .\end{array}$

From $\left(F3\right)$ and $($\ref{Eqt 3.12}$)$, there exists $z\in \left(0,1\right)$ such that

$\begin{array}{rrr}\hfill {\mathcal{J}}_n^{z}F\left({\mathcal{J}}_n\right)& \hfill \le & \hfill {\mathcal{J}}_n^z\left(F\left({\mathcal{J}}_{n-1}\right)-\eta \right)\le \dots \le {\mathcal{J}}_n^z\left(F\left({\mathcal{J}}_0\right)-n\eta \right),\\ \hfill {\mathcal{J}}_n^{z}F\left({\mathcal{J}}_n\right)& \hfill \le & \hfill {\mathcal{J}}_n^z\left(F\left({\mathcal{J}}_0\right)-n\eta \right)\le 0,\\ \hfill {\mathcal{J}}_n^{z}F\left({\mathcal{J}}_n\right)& \hfill \le & \hfill {\mathcal{J}}_n^{z}F\left({\mathcal{J}}_0\right)-{\mathcal{J}}_n^{z}n\eta )\le 0,\\ \hfill {\mathcal{J}}_n^{z}F\left({\mathcal{J}}_n\right)-{\mathcal{J}}_n^{z}F\left({\mathcal{J}}_0\right)& \hfill \le & \hfill -{\mathcal{J}}_n^{z}n\eta \le 0,\\ \hfill {\mathcal{J}}_n^z\left(F\left({\mathcal{J}}_n\right)-F\left({\mathcal{J}}_0\right)\right)& \hfill \le & \hfill -{\mathcal{J}}_n^{z}n\eta \le 0.\end{array}$

Letting $n\to \infty$ in $($\ref{Eqt 3.15}$)$, we obtain that

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}{\mathcal{J}}_n^{z}n& \hfill =& \hfill 0.\end{array}$

Now, from $($\ref{Eqt 3.16}$)$ there exist $n_1\in {\mathbb{N}}_0$ such that ${\mathcal{J}}_n^{z}n\le 1$ for all $n\ge n_1$ .

Consequently, we have that

$\begin{array}{rrr}\hfill {\mathcal{J}}_n^{z}n& \hfill \le & \hfill 1,\\ \hfill {\mathcal{J}}_n^z& \hfill \le & \hfill \frac{1}{n},\\ \hfill {\mathcal{J}}_n& \hfill \le & \hfill \frac{1}{n^{\frac{1}{z}}},\\ \hfill {\mathcal{J}}_n& \hfill \le & \hfill n^{-\frac{1}{z}}.\end{array}$

Therefore, ${\displaystyle \sum }_{n=0}^{\infty }d\left({\mu }_n,{\mu }_{n+1}\right)=0$ converges.

Next, we claim that $\left\{{\mu }_n\right\}$ is Cauchy sequence, that is, ${\text{lim}}_{n\to \infty }d\left({\mu }_n,{\mu }_m\right)=0$ $\forall n,m\in {\mathbb{N}}_0$  such that $m\ge n$ , by using the rectangular property we have

$\begin{array}{rrr}\hfill d\left({\mu }_n,{\mu }_m\right)& \hfill \le & \hfill d\left({\mu }_n,{\mu }_{n+1}\right)+d\left({\mu }_{n+1},{\mu }_{n+2}\right)+\dots +d\left({\mu }_{m-1},{\mu }_m\right),\\ \hfill & \hfill \le & \hfill {\mathcal{J}}_n+{\mathcal{J}}_{n+1}+{\mathcal{J}}_{n+2}+\dots +{\mathcal{J}}_{m-1},\\ \hfill & \hfill =& \hfill {\displaystyle \sum }_{i=n}^{m-1}{\mathcal{J}}_i,\\ \hfill & \hfill \le & \hfill {\displaystyle \sum }_{i=n}^{m-1}{n}^{-\frac{1}{z}}.\end{array}$   

Since ${\displaystyle \sum }_{i=n}^{m-1}{n}^{-\frac{1}{z}}<\infty$ , we get that $\left\{{\mu }_n\right\}$ is a Cauchy sequence in $ℳ$ . Since $\left(ℳ,d\right)$ is complete, there exists ${\mu }^{\star }\in ℳ$ such that

$\begin{array}{rrr}\hfill d\left({\mu }_n,{\mu }^{\star }\right)& \hfill =& \hfill \underset{n\to \infty }{\text{lim}}d\left({\mu }_n,{\mu }^{\star }\right)=0.\end{array}$   

Now, by the continuity of $\Gamma$ , we get $\Gamma {\mu }^{\star }={\mu }^{\star }$ . We show that ${\mu }^{\star }$ is a fixed point of $\Gamma$ . Assume that $\Gamma {\mu }^{\star }\ne {\mu }^{\star }$ such that $\Gamma {\mu }_n\ne {\mu }_n$ $\forall n\ge {\mathbb{N}}_0$ . By letting $\mu ={\mu }_n$ and $\nu ={\mu }^{\star }$ in $($\ref{eqt 3.1}$)$, we obtain

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }_n,\Gamma {\mu }^{\star }\right)\right)& \hfill \le & \hfill F\left({ℳ}_{ℛ}\left({\mu }_n,{\mu }^{\star }\right)\right),\end{array}$   

where

$\begin{array}{rrr}\hfill {ℳ}_{ℛ}\left({\mu }_n,{\mu }^{\star }\right)& \hfill =& \hfill {\left[d\left({\mu }_n,{\mu }^{\star }\right)\right]}^{\delta }.{\left[d\left({\mu }_n,\Gamma {\mu }_n\right)\right]}^{\alpha }.{\left[d\left({\mu }^{\star },\Gamma {\mu }^{\star }\right)\right]}^{1-\alpha -\delta },\\ \hfill & \hfill \le & \hfill {\left[d\left({\mu }^{\star },{\mu }^{\star }\right)\right]}^{\delta }.{\left[d\left({\mu }^{\star },{\mu }^{\star }\right)\right]}^{\alpha }.{\left[d\left({\mu }^{\star },{\mu }^{\star }\right)\right]}^{1-\alpha -\delta },\\ \hfill & \hfill =& \hfill {\left[d\left({\mu }^{\star },{\mu }^{\star }\right)\right]}^{\left(\alpha +\delta \right)+\left(1-\alpha -\delta \right)},\\ \hfill & \hfill =& \hfill d\left({\mu }^{\star },{\mu }^{\star }\right).\end{array}$   

Taking $($\ref{Eqt 3.20}$)$ into $($\ref{Eqt 3.19}$)$, we get

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }^{\star },\Gamma {\mu }^{\star }\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }^{\star },{\mu }^{\star }\right)\right),\\ \hfill \eta +F\left(0\right)& \hfill \le & \hfill F\left(0\right),\\ \hfill \eta & \hfill \le & \hfill 0,\end{array}$  

which is a contradiction. Hence, $d\left({\mu }^{\star },\Gamma {\mu }^{\star }\right)=0$ therefore ${\mu }^{\star }=\Gamma {\mu }^{\star }$ , which shows that ${\mu }^{\star }$ is a fixed point of $\Gamma$ . Also $\Gamma$ is subsequentially convergent on $ℳ$ . To observe this, let $\mu ={\mu }_{n_{k-1}}$ and $\nu ={\mu }_{n_k}$ , using $($\ref{eqt 3.1}$)$ we obtain

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }_{n_{k-1}},\Gamma {\mu }_{n_k}\right)\right)& \hfill \le & \hfill F\left({ℳ}_{ℛ}\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right),\end{array}$

where

${ℳ}_{ℛ}\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)={\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\delta }.{\left[d\left({\mu }_{n_{k-1}},\Gamma {\mu }_{n_{k-1}}\right)\right]}^{\alpha }.{\left[d\left({\mu }_{n_k},\Gamma {\mu }_{n_k}\right)\right]}^{1-\alpha -\delta },$   

$\le {\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\delta }.{\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha }.{\left[d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right]}^{1-\alpha -\delta },$   

$= {\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right]}^{1-\alpha -\delta }.$   

Using $($\ref{Eqt 3.22}$)$ in $($\ref{Eqt 3.21}$)$, we get

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma {\mu }_{n_{k-1}},\Gamma {\mu }_{n_k}\right)\right)& \hfill \le & \hfill F\left({\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right]}^{1-\alpha -\delta }\right).\end{array}$   

By the property of F and F1 with $($\ref{Eqt 3.23}$)$, we get

$\begin{array}{rrr}\hfill d\left(\Gamma {\mu }_{n_{k-1}},\Gamma {\mu }_{n_k}\right)& \hfill \le & \hfill {\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha +\delta }.{\left[d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right]}^{1-\alpha -\delta },\\ \hfill {\left(d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right)}^{1-\left(1-\alpha -\delta \right)}& \hfill \le & \hfill {\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha +\delta },\\ \hfill {\left(d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right)}^{\alpha +\delta }& \hfill \le & \hfill {\left[d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right]}^{\alpha +\delta },\end{array}$

which is equivalent to

$\begin{array}{rrr}\hfill F\left(d\left({\mu }_{n_k},{\mu }_{n_{k+1}}\right)\right)& \hfill \le & \hfill F\left(d\left({\mu }_{n_{k-1}},{\mu }_{n_k}\right)\right)-\eta .\end{array}$   

Due to continuity of $\Gamma$ , it implies that

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}\Gamma {\mu }_{n_k}& \hfill =& \hfill \Gamma {\mu }^{\star }={\mu }^{\star }.\end{array}$   

This shows that $\Gamma$ is subsequentially convergent.

Consider the hypothesis in Theorem 6, we prove assertion $\left(vii\right)$  as follows: we observe that $ℳ\left(\Gamma ,ℛ\right)$ is non-empty, so let us take a pair of elements say $\left({\mu }^{\star },w^{\star }\right)$ in $ℳ\left(\Gamma ,ℛ\right)$ such that

$\begin{array}{r}\hfill \Gamma \mu ={\mu }^{\star },\\ \hfill \Gamma \nu =w^{\star }.\end{array}$   

Next, we claim that ${\mu }^{\star }\ne w^{\star }$ . By the above equalities, there exists a S-path (say,$z_0,z_1,z_2,\mathrm{...},z_l)$ of length in ${ℛ}^s$ from $\Gamma \mu$ to $\Gamma \nu$ , with

$\begin{array}{r}\hfill \Gamma z_0=\Gamma \mu ,\\ \hfill \Gamma z_l=\Gamma \nu ,\end{array}$   

such that

$\left[\Gamma z_i,\Gamma z_{i+1}\right]\in {ℛ}^s\subseteq ℛ,$   

for all $i\in 0,1,2,3,\mathrm{...}l-1$ .

Define two constant sequences such that

$z_n^0=\mu and z_n^l=\nu$   

By using ([equation 4.20]), for all $n\in \mathbb{N}$ , we have

$\begin{array}{rrr}\hfill \Gamma z_n^0& \hfill =& \hfill \Gamma \mu ={\mu }^{\star },\\ \hfill \Gamma z_n^l& \hfill =& \hfill \Gamma \nu =w^{\star }.\end{array}$   

By usual substitution for $z_0^i=z_i$ for each $i\in 0,1,2,\mathrm{...}l$ , that is

$\begin{array}{rrr}\hfill z_0^1& \hfill =& \hfill z_1,\\ \hfill z_0^2& \hfill =& \hfill z_2,\\ \hfill z_0^3& \hfill =& \hfill z_3,\\ \hfill z_0^4& \hfill =& \hfill z_4,\\ \hfill z_0^{l-1}& \hfill =& \hfill z_{l-1}.\end{array}$   

Thus we construct a sequence

$\left\{z_n^1\right\},\left\{z_n^2\right\},\left\{z_n^3\right\},\dots ,\left\{z_n^i\right\}\in ℳ.$   

Corresponding to each $z_i$ , we have $\left[\Gamma z_0^i,\Gamma z_1^i\right]\in ℛ$ from ([equation 4.20]), ([equation 4.21]) and $ℛ$ is $\Gamma$ -closed, we get

$\begin{array}{rrr}\hfill \underset{n\to \infty }{\text{lim}}d\left(\Gamma z_n^i,\Gamma z_n^{i+1}\right)& \hfill =& \hfill 0,\end{array}$   

for each $i\in 1,2,3,\mathrm{...}l-1$ and for all $n\in \mathbb{N}$ .

Define $d_n^i=d\left(\Gamma z_n^i,\Gamma z_n^{i+1}\right)$  for each $i\in 0,1,2,3,\mathrm{...}l-1$ and for all $n\in \mathbb{N}$ . We assert that, ${\text{lim}}_{n\to \infty }{d}_n^i>0$ .

Since $\left[\Gamma z_n^i,\Gamma z_n^{i+1}\right]\in ℛ$ , either $\left[\Gamma z_n^i,\Gamma z_n^{i+1}\right]\in ℛ$ or $\left[\Gamma z_n^{i+1},\Gamma z_n^i\right]\in ℛ$ .

If $\left[\Gamma z_n^i,\Gamma z_n^{i+1}\right]\in ℛ$ , for $\mu =z_n^i$ and $\nu =z_n^{i+1}$ . Then applying the condition $($\ref{eqt 3.1}$)$, we have

$\begin{array}{rrr}\hfill \eta +F\left(d\left(\Gamma z_n^i,\Gamma z_n^{i+1}\right)\right)& \hfill \le & \hfill F\left({ℳ}_{ℛ}\left(z_n^i,z_n^{i+1}\right)\right),\end{array}$   

where

${ℳ}_{ℛ}\left(z_n^i,z_n^{i+1}\right) = {\left[d\left(z_n^i,z_n^{i+1}\right)\right]}^{\delta }.{\left[d\left(z_n^i,\Gamma z_n^i\right)\right]}^{\alpha }.{\left[d\left(z_n^{i+1},\Gamma z_n^{i+1}\right)\right]}^{1-\alpha -\delta },$   

$\le {\left[d\left(z_n^i,z_n^{i+1}\right)\right]}^{\delta }.{\left[d\left(z_n^i,z_n^{i+1}\right)\right]}^{\alpha }.{\left[d\left(z_n^{i+1},z_n^{i+2}\right)\right]}^{1-\alpha -\delta },$