(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

Abstract

In this paper, we introduce the $(𝒞,{\Psi }^{*},G)$ class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and prove common fixed point theorems for such mappings in a metric space endowed with a graph. Our results unify and generalize many important fixed point results existing in literature. As an application of our main result, we have derived fixed point theorems for a pair of $\alpha$-admissible set valued mappings in a metric space.

1. Introduction and Preliminaries

In [1], Ran and Reurings proved the existence of fixed points for single-valued mappings in partially ordered metric spaces, and their results were extended by Neito and Lopez [2]. However, it became clear that the concept of a graph gives a better vision of fixed points instead of partial ordering, and the first attempt in this direction was done by Jachymsky [3]. He defined the Banach G- contraction for single-valued mapping, which was later extended by Beg et al. [4] for the multivalued mappings. After these, there was a lot of work done in the direction of fixed points in metric spaces endowed with graphs, see [5,6,7,8,9,10,11,12,13,14].

In 1973, Geraghty [15] defined $\Theta$ as the class of functions $\theta :[0,\infty )\to [0,1)$ such that

$$\begin{array}{cc}\hfill \theta \left(t_n\right)\to 1& \Rightarrow t_n\to 0,\hfill \end{array}$$

(1)

and also showed generalizations of the Banach-Neumann contractive mapping principle.

We now recall the following class of functions:

$\Psi$ denotes the class of all continuous and non-decreasing functions $\psi :[0,\infty )\to [0,\infty )$, such that:

• $\psi \left(t\right)=0$ if, and only if $t=0$.
  $\Phi$ denotes the class of all lower semi-continuous functions $\varphi :[0,\infty )\to [0,\infty )$, such that:
• $\varphi \left(t\right)=0$ if, and only if $t=0$.

For more results on contraction principles involving the above said control functions, we refer the reader to [16,17,18].

In [19], the family of $𝒞$-class functions were introduced as follows: $F:{[0,\infty )}^2\to \mathbb{R}$ belongs to the $𝒞$-class functions if:

• F is continuous,
• $F(s,t)\le s$,
• $F(s,t)=s$ implies that either $s=0$ or $t=0$; for all $s,t\in [0,\infty )$.

In [20], Samet et al. introduced the concept of $\alpha$-admissible mappings, and proved fixed point theorems for $\alpha$-$\psi$ contractive-type mappings, which paved a way to prove new results and generalise existing results in the fixed point theory. For some recent results on fixed point theorems of $\alpha$-admissible mappings, the reader may refer to [21,22,23,24].

In this work, we utilised the C-class functions to give modified versions of contraction principles involving $\Psi$ class functions and $\Phi$ class functions in the sense that we have relaxed the condition $\alpha \left(t\right)=0$ if, and only if $t=0$ in the $\Psi$ and $\Phi$ class functions to $\alpha \left(t\right)=0\Rightarrow t=0$. As an application, we have also deduced some common fixed point theorems for a pair of $\alpha$-admissible mappings.

Throughout this work, $(X_G,d_G)$ will denote the metric space endowed with a directed graph G with $V\left(G\right)=X_G$ and $\Delta \subseteq E\left(G\right)$, where $V\left(G\right)$ denotes the set of vertices, $E\left(G\right)$ denotes the set of edges of the graph G and $\Delta =\{(x_G,x_G):x_G\in X\}$.

Definition 1.

[3] $(X_G,d_G)$ is said to have property A if $x_n\to x_G$ and $(x_n,x_{n+1})\in E\left(G\right)$ implies $(x_n,x_G)\in E\left(G\right)$, for all sequences ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ in $X_G$.

Definition 2.

[16] The pair $(f,g)$ of self mappings of $X_G$ is g-edge preserving in G, if

$$\begin{array}{cc}\hfill (gx,gy)\in E\left(G\right)& \Rightarrow (fx,fy)\in E\left(G\right).\hfill \end{array}$$

(2)

Definition 3.

[11] $E\left(G\right)$ satisfies transitivity property if, and only if for all $x,y,z\in X_G$, $(x,z)\in E\left(G\right)$ and $(z,y)\in E\left(G\right)$ implies $(x,y)\in E\left(G\right)$.

Let mappings $S,T:X_G\to CL\left(X_G\right)$ be given. We will make use of the following notations:

• $COFIX\{S,T\}=\{u\in X_G:u\in Su\bigcap Tu\}$ is the set of all common fixed points of S and T
• $FIX\left\{T\right\}=\{u\in X_G:u\in Tu\}$ is the set of all fixed points of T.

2. Main Results

Let ${\Theta }^{*}$ be the set of all continuous functions $\theta :[0,\infty )\to [0,1)$.

${\Psi }^{*}$ be the set of all continuous and non decreasing functions $\psi :[0,\infty )\to [0,\infty )$, such that:

• $\psi \left(t\right)=0\Rightarrow t=0$.
  Let ${\Phi }^{*}$ be the set of all lower semi-continuous functions $\varphi :[0,\infty )\to [0,\infty )$, such that:
• $\varphi \left(t\right)=0\Rightarrow t=0$.

Definition 4.

Let $S,T:X_G\to CB\left(X_G\right)$ be two given mappings. We say that the pair $(S,T)$ belongs to the class of $(𝒞,{\Psi }^{*},G)$ contractions if, and only if for all $x_G,y_G\in X_G$ with $(x_G,y_G)\in E\left(G\right)$, the following conditions are satisfied:

(4.1) 

For $u_G\in S{x}_G$, there exists $v_G\in T{y}_G$ such that $(u_G,v_G)\in E\left(G\right)$

(4.2) 

For $u_G\in T{x}_G$, there exists $v_G\in S{y}_G$ such that $(u_G,v_G)\in E\left(G\right)$

(4.3) 

there exists $F\in 𝒞$, $\psi \in {\Psi }^{*}$, such that

• $\psi \left(H(S{x}_G,T{y}_G)\right)\le F(\psi \left(M(x_G,y_G)\right),M(x_G,y_G))$ and
• $\psi \left(H(T{x}_G,S{y}_G)\right)\le F(\psi \left(M(x_G,y_G)\right),M(x_G,y_G))$

where

$$M(x_G,y_G)=max\left\{d_G(x_G,y_G),d_G(S{x}_G,x_G),d_G(T{y}_G,y_G),{\displaystyle \frac{d_G(y_G,S{x}_G)+d_G(x_G,T{y}_G)}{2}}\right\}$$

Theorem 1.

Let $(X_G,d_G)$ be complete and $S,T:X_G\to CB\left(X_G\right)$ satisfy the following:

(1.1) 

There exists ${x_G}_0,{x_G}_1\in X_G$ such that ${x_G}_1\in T{x_G}_0\bigcup S{x_G}_0$ and $({x_G}_0,{x_G}_1)\in E\left(G\right)$,

(1.2) 

$E\left(G\right)$ satisfy transitivity property,

(1.3) 

$(S,T)\in (𝒞,{\Psi }^{*},G)$ for some $F\in {𝒞}^{*}$ and $\psi \in {\Psi }^{*}$.

Then $COFIX\{S,T\}\ne \varphi$.

Proof. 

By condition (1.1), suppose ${x_G}_0\in X_G$, and ${x_G}_1\in S\left({x_G}_0\right)$. By condition (4.1), we can find ${x_G}_2\in T\left({x_G}_1\right)$ with $({x_G}_1,{x_G}_2)\in E\left(G\right)$ and

$$\begin{array}{ccc}\hfill \psi \left(d_G({x_G}_1,{x_G}_2)\right)& \le \psi \left(H(S\left({x_G}_0\right),T\left({x_G}_1\right))\right)\hfill & \hfill \le F(\psi \left(M({x_G}_0,{x_G}_1)\right),\varphi \left(M({x_G}_0,{x_G}_1)\right))\end{array}$$

Now again by condition (4.2), for ${x_G}_2\in T\left({x_G}_1\right)$, there exists ${x_G}_3\in S\left({x_G}_2\right)$ with $({x_G}_2,{x_G}_3)\in E\left(G\right)$ and

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_2,{x_G}_3)\right)& \le \psi \left(H(T{x_G}_1,S{x_G}_2)\right)\hfill \\ & \le F(\psi \left(M({x_G}_1,{x_G}_2)\right),M({x_G}_1,{x_G}_2)).\hfill \end{array}$$

Continuing inductively, we construct the sequence $\left\{{x_G}_n\right\}$ recursively as for $n\ge 0$, as

$$\begin{array}{c}\hfill {x_G}_{2n+1}\in S\left({x_G}_{2n}\right),{x_G}_{2n}\in T\left({x_G}_{2n-1}\right)\end{array}$$

(3)

as well as $({x_G}_n,{x_G}_{n+1})\in E\left(G\right)$. Our first task is to establish that $COFIX\{S,T\}\ne \varphi$. Note that if $M({x_G}_m,{x_G}_n)=0$ for any n,$m\in N$ then

$$\begin{array}{cc}\hfill M({x_G}_m,{x_G}_n)& =max\{d_G({x_G}_m,{x_G}_n),d_G(S{x_G}_m,{x_G}_m),d_G(T{x_G}_n,{x_G}_n),\hfill \\ & {\displaystyle \frac{d_G({x_G}_n,S{x_G}_m)+d_G({x_G}_m,T{x_G}_n)}{2}}\}=0\hfill \end{array}$$

which shows that ${x_G}_m={x_G}_n\in COFIX\{S,T\}$, and our first task will be complete. So let $M({x_G}_m,{x_G}_n)\ne 0$ for any $n,m\in N$. Then, by definition of $\psi$, $\psi \left(M({x_G}_{n-1},{x_G}_n)\right)\ne 0$.

If n is odd, we have

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_n,{x_G}_{n+1})\right)& \le \psi \left(H(S{x_G}_{n-1},T{x_G}_n)\right)\hfill \\ & \le F(\psi \left(M({x_G}_{n-1},{x_G}_n)\right),M({x_G}_{n-1},{x_G}_n))\hfill \end{array}$$

(4)

Since $\psi \left(M({x_G}_{n-1},{x_G}_n)\right)\ne 0$, we have

$$\begin{array}{cc}\hfill F(\psi \left(M({x_G}_{n-1},{x_G}_n)\right),M({x_G}_{n-1},{x_G}_n))& <\psi \left(M({x_G}_{n-1},{x_G}_n)\right)\hfill \end{array}$$

Then, by (4), we get

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_n,{x_G}_{n+1})\right)& <\psi \left(M({x_G}_{n-1},{x_G}_n)\right)\hfill \end{array}$$

(5)

where

$$\begin{array}{cc}\hfill M({x_G}_{n-1},{x_G}_n)& =max\{d_G({x_G}_{n-1},{x_G}_n),d_G(S{x_G}_{n-1},{x_G}_{n-1}),d_G(T{x_G}_n,{x_G}_n),\hfill \\ & {\displaystyle \frac{d_G({x_G}_n,S{x_G}_{n-1})+d_G({x_G}_{n-1},T{x_G}_n)}{2}}\}\hfill \\ & \le max\{d_G({x_G}_{n-1},{x_G}_n),d_G({x_G}_n,{x_G}_{n-1}),d_G({x_G}_{n+1},{x_G}_n),\hfill \\ & {\displaystyle \frac{d_G({x_G}_n,{x_G}_n)+d_G({x_G}_{n-1},{x_G}_{n+1})}{2}}\}\hfill \\ & \le max\left\{d_G({x_G}_{n-1},{x_G}_n),d_G({x_G}_{n+1},{x_G}_n),{\displaystyle \frac{d_G({x_G}_{n-1},{x_G}_{n+1})}{2}}\right\}\hfill \\ & \le max\{d_G({x_G}_{n-1},{x_G}_n),d_G({x_G}_{n+1},{x_G}_n),\hfill \\ & {\displaystyle \frac{d_G({x_G}_{n-1},{x_G}_n)+d_G({x_G}_n,{x_G}_{n+1})}{2}}\}\hfill \\ & \le max\{d_G({x_G}_{n-1},{x_G}_n),d_G({x_G}_n,{x_G}_{n+1})\}\hfill \end{array}$$

If $d_G({x_G}_{n+1},{x_G}_n)>d_G({x_G}_{n-1},{x_G}_n)$, then $M({x_G}_{n-1},{x_G}_n)\le d_G({x_G}_{n+1},{x_G}_n)$. Then (5) gives

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_n,{x_G}_{n+1})\right)& <\psi \left(d_G({x_G}_n,{x_G}_{n+1})\right)\hfill \end{array}$$

a contradiction. So, we have

$$\begin{array}{c}\hfill d_G({x_G}_n,{x_G}_{n+1})\le d_G({x_G}_{n-1},{x_G}_n)\end{array}$$

(6)

For an even number n, a similar argument leads to inequality (6). Thus, $\left\{d_G(x_{n+1},{x_G}_n)\right\}$ is a monotonically non-increasing sequence which is bounded below, and thereby,

$$\underset{n\to \infty }{lim}{d}_G({x_G}_n,{x_G}_{n+1})=\underset{n\to \infty }{lim}M({x_G}_{n-1},{x_G}_n)=r\ge 0.$$

Assume that $r>0$, so that $\psi \left(r\right)>0$. Taking lim inf on both sides of the inequality (5), we obtain

$$\begin{array}{c}\hfill \psi \left(r\right)<\psi \left(r\right)\end{array}$$

a contradiction. Hence $r=0$. Consequently, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{d}_G({x_G}_n,{x_G}_{n+1})& =0.\hfill \end{array}$$

(7)

Next, we prove that $\left\{{x_G}_n\right\}$ is a Cauchy sequence. By (7), it is enough if we show that the subsequence $\left\{{x_G}_{2n}\right\}$ is a Cauchy sequence. Suppose, if possible, $\left\{{x_G}_{2n}\right\}$ is not a Cauchy sequence. Then, there exists $ϵ>0$ and subsequences $\left\{{x_G}_{2m\left(k\right)}\right\}$ and $\left\{{x_G}_{2n\left(k\right)}\right\}$, such that $n\left(k\right)$ is the smallest index for which $n\left(k\right)>m\left(k\right)>k,d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)})\ge ϵ$. That is,

$$\begin{array}{cc}\hfill d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)-2})& <ϵ\hfill \end{array}$$

(8)

Now, we have

$$\begin{array}{cc}\hfill ϵ& \le d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)})\hfill \\ & \le d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)-2})+d_G({x_G}_{2n\left(k\right)-2},{x_G}_{2n\left(k\right)-1})+d_G({x_G}_{2n\left(k\right)-1},{x_G}_{2n\left(k\right)})\hfill \\ & <ϵ+d_G({x_G}_{2n\left(k\right)-2},{x_G}_{2n\left(k\right)-1})+d_G({x_G}_{2n\left(k\right)-1},{x_G}_{2n\left(k\right)})\hfill \end{array}$$

AS $k\to \infty$, we get

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{d}_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)})& =ϵ\hfill \end{array}$$

(9)

Also, we have

$$\begin{array}{cc}\hfill |d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)+1})-d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)})|& \le d_G({x_G}_{2n\left(k\right)},{x_G}_{2n\left(k\right)+1})\hfill \\ \hfill \mathrm{and}|d_G({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})-d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)})|& \le d_G({x_G}_{2m\left(k\right)},x_{2m\left(k\right)-1}).\hfill \end{array}$$

Letting $k\to \infty$ and using (7) and (9), we get

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{d}_G({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})=\underset{k\to \infty }{lim}{d}_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)+1})=ϵ.\end{array}$$

(10)

From

$$|d_G({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)+1})-d_G({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})|\le d_G({x_G}_{2n\left(k\right)},{x_G}_{2n\left(k\right)+1})$$

and making use of (7) and (10), we get

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{d}_G({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)+1})& =ϵ\hfill \end{array}$$

(11)

Also, from the definition of M and from (7) and (9)–(11), we have

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}M({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})& =ϵ\hfill \end{array}$$

(12)

Also by the transitivity property of G, we have $({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})\in E\left(G\right)$. Thus, we have

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_{2m\left(k\right)},{x_G}_{2n\left(k\right)+1})\right)& =\psi \left(H(T{x_G}_{2m\left(k\right)-1},S{x_G}_{2n\left(k\right)})\right)\hfill \\ & \le F(\psi \left(M({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})\right),\left(M({x_G}_{2m\left(k\right)-1},{x_G}_{2n\left(k\right)})\right)\hfill \end{array}$$

Letting $k\to \infty$ and making use of (10) and (11), the above inequality yields

$$\begin{array}{cc}\hfill \psi \left(ϵ\right)& <\psi \left(ϵ\right)\hfill \end{array}$$

a contradiction. Thus, $\left\{{x_G}_n\right\}$ is a Cauchy sequence. By completeness of $X_G$, we can find $u_G\in X_G$, such that ${x_G}_n\to u_G$ as $n\to \infty$.

We will now prove that $u_G\in COFIX\{S,T\}$. Note that $({x_G}_{2n+1},u_G)\in E\left(G\right)$, and so

$$\begin{array}{cc}\hfill \psi \left(d_G({x_G}_{2n+1},T{u}_G)\right)& \le \psi \left(H(S{x_G}_{2n},T{u}_G)\right)\hfill \\ & \le F(\psi \left(M({x_G}_{2n},u_G)\right),M({x_G}_{2n},u_G))\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill M({x_G}_{2n},u_G)& =max\{d_G({x_G}_{2n},u_G),d_G(S{x_G}_{2n},{x_G}_{2n}),d_G(T{u}_G,u_G),\hfill \\ & {\displaystyle \frac{d_G({x_G}_{2n},T{u}_G)+d_G(u_G,S{x_G}_{2n})}{2}}\}\hfill \end{array}$$

Note that as $n\to \infty$, $d_G(S{x_G}_{2n},{x_G}_{2n})\to 0$, $d_G(u_G,S{x_G}_{2n})\to 0$, and so $M(u_G,{x_G}_{2n})\to d_G(T{u}_G,u_G)$ Now, if $d_G(T{u}_G,u_G)\ne 0$, then from (13) as $n\to \infty$, we have

$$\psi (d_G(T{u}_G,u_G)<\psi \left(d_G(T{u}_G,u_G)\right),$$

again, a contradiction. Thus, $d_G(T{u}_G,u_G)=0$, which implies that $u_G\in \overline{T{u}_G}$, and since $T{u}_G$ is closed, we have $u_G\in T{u}_G$.

Now again, we have $M(u_G,u_G)=d_G(u_G,S{u}_G)$, and if $d_G(u_G,S{u}_G)\ne 0$, since $(u_G,u_G)\in \Delta \subset E\left(G\right)$, we get

$$\begin{array}{cc}\hfill \psi \left(d_G(S{u}_G,u_G)\right)& \le \psi \left(H(S{u}_G,T{u}_G)\right)\hfill \\ & \le F\left(\psi \left(M(u_G,u_G)\right)M(u_G,u_G)\right)\hfill \\ & <\psi \left(d_G(u_G,S{u}_G)\right)\hfill \end{array}$$

a contradiction, and thereby, $d_G(u_G,S{u}_G)=0$ or $u_G\in S{u}_G$. Hence, $COFIX\{S,T\}\ne \varphi$. ☐

We will deduce the following important results from Theorem 1:

Corollary 1.

Let $(X_G,d_G)$ be complete and $S,T:X_G\to CB\left(X_G\right)$ satisfy conditions (4.1), condition (4.2), condition (1.1), condition (1.2), and the following:

(1.1) For all $x_G,y_G\in X_G$ with $(x_G,y_G)\in E\left(G\right)$

• $\psi \left(H(S{x}_G,T{y}_G)\right)\le \psi \left(M(x_G,y_G)\right)-\varphi \left(M(x_G,y_G)\right)$ and
• $\psi \left(H(T{x}_G,S{y}_G)\right)\le \psi \left(M(x_G,y_G)\right)-\varphi \left(M(x_G,y_G)\right)$

where $\psi \in {\Psi }^{*}$, $\varphi \in {\Phi }^{*}$ and $M(x_G,y_G)$ is as in Definition 4. Then, $COFIX\{S,T\}\ne \varphi$.

Proof. 

Take $F(r,t)=r-\varphi \left(t\right)$ in Theorem 1. ☐

Corollary 2.

Let $(X_G,d_G)$ be complete and $S,T:X_G\to CB\left(X_G\right)$ satisfy the conditions (4.1), condition (4.2), condition (1.1), condition (1.2), and the following:

(2.1) For all $x_G,y_G\in X_G$ with $(x_G,y_G)\in E\left(G\right)$

• $\psi \left(H(S{x}_G,T{y}_G)\right)\le \theta \left(M(x_G,y_G)\right)\psi \left(M(x_G,y_G)\right)$ and
• $\psi \left(H(T{x}_G,S{y}_G)\right)\le \theta \left(M(x_G,y_G)\right)\psi \left(M(x_G,y_G)\right)$

where $\psi \in {\Psi }^{*}$, $\theta \in {\Theta }^{*}$ and $M(x_G,y_G)$ is as in Definition 4. $COFIX\{S,T\}\ne \varphi$.

Proof. 

Take $F(r,t)=\theta \left(t\right).r$ in Theorem 1. ☐

Corollary 3.

Let $(X_G,d_G)$ be complete and $S,T:X_G\to CB\left(X_G\right)$ satisfy the conditions (4.1), (4.2), (1.1) and (1.2), and the following:

(3.1) For all $x_G,y_G\in X_G$ with $(x_G,y_G)\in E\left(G\right)$, there exist $0<\lambda <1$, such that

• $H(S{x}_G,T{y}_G)\le \lambda \left(M(x_G,y_G)\right)$ and
• $H(T{x}_G,S{y}_G)\le \lambda \left(M(x_G,y_G)\right)$

where $M(x_G,y_G)$ is as in Definition 4. Then $COFIX\{S,T\}\ne \varphi$.

Proof. 

For some $k>0$, set $k^{*}=k(1-\lambda )$. Then,

• $H(S{x}_G,T{y}_G))\le \lambda \left(M(x_G,y_G)\right)$ and
• $H(T{x}_G,S{y}_G)\le \lambda \left(M(x_G,y_G)\right)$

implies

• $H(S{x}_G,T{y}_G)\le {\displaystyle \frac{k-k^{*}}{k}}\left(M(x_G,y_G)\right)$ and
• $H(T{x}_G,S{y}_G)\le {\displaystyle \frac{k-k^{*}}{k}}\left(M(x_G,y_G)\right)$

or

• $kH(S{x}_G,T{y}_G)+1\le kM(x_G,y_G)+1-k^{*}M(x_G,y_G)$ and
• $kH(T{x}_G,S{y}_G)\le kM(x_G,y_G)-k^{*}M(x_G,y_G)$
• Now, let $\psi \left(t\right)=kt+1$ and $\varphi \left(t\right)=k^{*}\left(t\right)$. Then, the above inequality leads to
• $\psi \left(H(S{x}_G,T{y}_G)\right)\le \psi \left(M(x_G,y_G)\right)-\varphi \left(M(x_G,y_G)\right)$ and
• $\psi \left(H(T{x}_G,S{y}_G)\right)\le \psi \left(M(x_G,y_G)\right)-\varphi \left(M(x_G,y_G)\right)$

Thus, all conditions of Corollary 1 are satisfied, and hence, $COFIX\{S,T\}\ne \varphi$. ☐

Corollary 4.

Let $(X_G,d_G)$ be complete and $T:X_G\to CB\left(X_G\right)$ satisfy the following:

(4.1) 

There exists ${x_G}_0,{x_G}_1\in X_G$, such that ${x_G}_1\in T{x_G}_0$ and $({x_G}_0,{x_G}_1)\in E\left(G\right)$;

(4.2) 

For any $u\in T{x}_G$, there exists $w\in T{y}_G$, such that $(u,w)\in E\left(G\right)$;

(4.3) 

$E\left(G\right)$ satisfies the transitivity property;

(4.4) 

$\psi \left(H(T{x}_G,T{y}_G)\right)\le \psi \left(d_G(x_G,y_G)\right)-\varphi \left(d_G(x_G,y_G)\right)$

where $\psi \in {\Psi }^{*}$, $\varphi \in {\Phi }^{*}$. Then, $FIX\left\{T\right\}\ne \varphi$.

Proof. 

Take $S=T$ in Corollary 1. ☐

Corollary 5.

Let $(X_G,d_G)$ be complete and $T:X_G\to CB\left(X_G\right)$ satisfy conditions (4.1)–(4.3), and the following:

(5.1) $\psi \left(H(T{x}_G,T{y}_G)\right)\le \theta \left(d_G(x_G,y_G)\right)\psi \left(d_G(x_G,y_G)\right)$

where $\psi \in {\Psi }^{*}$, $\theta \in {\Theta }^{*}$. Then, $FIX\left\{T\right\}\ne \varphi$.

Proof. 

Take $F(r,t)=\theta \left(t\right).r$ in Corollary 2. ☐

Example 1.

Let $X_G=\{0,{\displaystyle \frac{1}{2^n}}:n\in \mathbb{N}\}$, $d_G(x_G,y_G)=|x_G-y_G|$, $G=(V,E)$, with $V(G)=X_G$ and $E(G)=\{(0,0),({\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^n}}),({\displaystyle \frac{1}{2^n}},0)\}$ and $S,T:X_G\to CB(X_G)$ be defined by

$$\begin{array}{c}\hfill S{x}_G=\left\{\begin{array}{cc}\{0\},\hfill & if{x}_G=0\hfill \\ \{{\displaystyle \frac{1}{2^{n+1}}},0\},\hfill & if{x}_G={\displaystyle \frac{1}{2^n}}.\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill T{x}_G=\left\{\begin{array}{cc}\{0\},\hfill & if{x}_G=0\hfill \\ \{{\displaystyle \frac{1}{2^{n+2}}},0\},\hfill & if{x}_G={\displaystyle \frac{1}{2^n}}.\hfill \end{array}\right.\end{array}$$

Define $\psi ,\varphi :[0,\infty )\to [0,\infty )$ by $\psi (t)=2t+1$ and $\varphi (t)={\displaystyle \frac{t}{4}}$ for all $t\in [0,\infty )$. Clearly, $\psi (t)\in {\Psi }^{*}$ (note that $\psi \notin \Psi$) and $\varphi (t)\in {\Phi }^{*}$.

• If $x_G={\displaystyle \frac{1}{2^n}}$ and $y_G=0$ with $({\displaystyle \frac{1}{2^n}},0)\in E(G)$, then $S{x}_G=\{{\displaystyle \frac{1}{2^{n+1}}},0\},Sy=\{0\},T{x}_G=\{{\displaystyle \frac{1}{2^{n+2}}},0\},T{y}_G=\{0\},d_G(x_G,y_G)={\displaystyle \frac{1}{2^n}},H(S{x}_G,T{y}_G)={\displaystyle \frac{1}{2^{n+1}}},\psi (H(S{x}_G,T{y}_G))={\displaystyle \frac{1}{2^n}}+1,M(x_G,y_G)={\displaystyle \frac{1}{2^n}},\psi (M(x_G,y_G))={\displaystyle \frac{1}{2^{n-1}}}+1,\varphi (M(x_G,y_G))={\displaystyle \frac{1}{2^{n+2}}}$ and

  $$\begin{array}{c}\hfill \psi (H(S{x}_G,T{y}_G))={\displaystyle \frac{1}{2^n}}+1<{\displaystyle \frac{1}{2^{n-1}}}+1-{\displaystyle \frac{1}{2^{n+2}}}=\psi (M(x_G,y_G))-\varphi (M(x_G,y_G))\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$
  Also, for ${\displaystyle \frac{1}{2^{n+1}}}\in S{x}_G$, there exists $0\in T{y}_G$, such that $({\displaystyle \frac{1}{2^{n+1}}},0)\in E(G)$ and for $0\in S{x}_G$, there exists $0\in T{y}_G$ such that $(0,0)\in E(G)$.
  For ${\displaystyle \frac{1}{2^{n+2}}}\in T{x}_G$, there exists $0\in S{y}_G$ such that $({\displaystyle \frac{1}{2^{n+2}}},0)\in E(G)$ and for $0\in T{x}_G$, there exists $0\in S{y}_G$ such that $(0,0)\in E(G)$.
• If $x_G=0,y_G=0$ with $(0,0)\in E(G)$, then $S{x}_G=\{0\}=S{y}_G=T{x}_G=T{y}_G,0=d_G(x_G,y_G)=H(S{x}_G,T{y}_G)=M(x_G,y_G),\psi (H(S{x}_G,T{y}_G))=\psi (M(x_G,y_G))=1,\varphi (M(x_G,y_G))=0$ and

  $$\begin{array}{c}\hfill \psi (H(S{x}_G,T{y}_G))=1=\psi (M(x_G,y_G))-\varphi (M(x_G,y_G)).\end{array}$$
  Also, for $0\in S{x}_G$, there exists $0\in T{y}_G$ such that $(0,0)\in E(G)$ and for $0\in S{x}_G$, there exists $0\in T{y}_G$, such that $(0,0)\in E(G)$.
  For $0\in T{x}_G$, there exists $0\in S{y}_G$, such that $(0,0)\in E(G)$ and for $0\in T{x}_G$, there exists $0\in S{y}_G$, such that $(0,0)\in E(G)$
• Also, if $x_G={\displaystyle \frac{1}{2^n}},y_G={\displaystyle \frac{1}{2^n}}$ with $({\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^n}})\in E(G)$, then $S{x}_G=\{{\displaystyle \frac{1}{2^{n+1}}},0\}=S{y}_G,T{x}_G=\{{\displaystyle \frac{1}{2^{n+2}}},0\}=T{y}_G,d_G(x_G,y_G)=0,H(S{x}_G,T{y}_G)={\displaystyle \frac{1}{2^{n+2}}},\psi (H(S{x}_G,T{y}_G))={\displaystyle \frac{1}{2^{n+1}}}+1,M(x_G,y_G)={\displaystyle \frac{1}{2^{n+1}}},\psi (M(x_G,y_G))={\displaystyle \frac{1}{2^n}}+1,\varphi (M(x_G,y_G))={\displaystyle \frac{1}{2^{n+3}}}$ and

  $$\begin{array}{c}\hfill \psi (H(S{x}_G,T{y}_G))={\displaystyle \frac{1}{2^{n+1}}}+1<{\displaystyle \frac{1}{2^n}}+1-{\displaystyle \frac{1}{2^{n+3}}}=\psi (M(x_G,y_G))-\varphi (M(x_G,y_G))\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$
  Also, for ${\displaystyle \frac{1}{2^{n+1}}}\in S{x}_G$, there exists $0\in T{y}_G$, such that $({\displaystyle \frac{1}{2^{n+1}}},0)\in E(G)$ and for $0\in S{x}_G$, there exists $0\in T{y}_G$, such that $(0,0)\in E(G)$.
  For ${\displaystyle \frac{1}{2^{n+2}}}\in T{x}_G$, there exists $0\in S{y}_G$, such that $({\displaystyle \frac{1}{2^{n+2}}},0)\in E(G)$ and for $0\in T{x}_G$, there exists $0\in S{y}_G$, such that $(0,0)\in E(G)$.

Thus, we see that for all $x_G,y_G\in X_G$ with $(x_G,y_G)\in E(G)$

$$\psi (H(S{x}_G,T{y}_G))\le \psi (M(x_G,y_G))-\varphi (M(x_G,y_G))\mathrm{for}\mathrm{all}{x}_G,y_G\in X_G.$$

Also, for $u_G\in S{x}_G$, there exists $v_G\in Ty$ such that $(u_G,v_G)\in E(G)$ and for $u_G\in T{x}_G$, there exists $v_G\in Sy$, such that $(u_G,v_G)\in E(G)$. Hence, the pair $(S,T)\in (𝒞,{\Psi }^{*},G)$ with $F(r,t)=r-\varphi (t)$. Thus, all conditions of Theorem 1 are satisfied, and $COFIX\{S,T\}=\{0\}$.

Remark 1.

Corollary 3 (and hence, Corollary 1 and Theorem 1) are proper extensions and generalisations of Theorem 3.1 of [4] and Theorem 4.2 of [8].

Remark 2.

Note that in Example 1, the graph G is a directed graph and not connected, and so Theorem 3.1 of [4] cannot be applied to neither of the mappings S or T. Also note that $({\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^n}})\in E(G)$, $H(S{\displaystyle \frac{1}{2^n}},T{\displaystyle \frac{1}{2^n}})={\displaystyle \frac{1}{2^{n+2}}}>0=d_G({\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^n}})$ and hence a simple extension of Theorem 3.1 of [4] and Theorem 4.2 of [8] to two mappings cannot be applied. However, we see that the mappings S and T satisfy the conditions of Corollary 3, and so Corollary 3 also ensures the existence of a common fixed point of S and T.

Remark 3.

In Theorem 1, if the directed graph G is replaced with an undirected graph $G^{\prime }$ with $E\left(G^{\prime }\right)=E\left(G\right)\bigcup E\left(G^{-1}\right)$, then Condition (4.3) in Definition 4 can be replaced with only one inequality:

• $\psi \left(H(S{x}_G,T{y}_G)\right)\le F(\psi \left(M(x_G,y_G)\right),M(x_G,y_G))$
• Similar arguments follow in Corollaries 1–3 also.

Definition 5.

Let $f,g:X_G\to X_G$. We say that the pair $(f,g)$ belongs to the class of Jungck type $(𝒞,{\Psi }^{*},G)$ contractions if

(5.1) 

$fisg-edgepreservinginG$.

(5.2) 

$Forall{x}_G,y_G\in X_{G}with(g{x}_G,g{y}_G)\in E\left(G\right)$

$$\begin{array}{cc}\hfill \psi \left(d_G(f{x}_G,f{y}_G)\right)& \le F(\psi \left(M(g{x}_G,g{y}_G)\right),M(g{x}_G,g{y}_G)),forsome\psi \in {\Psi }^{*},F\in 𝒞\hfill \end{array}$$

(14)

$M(g{x}_G,g{y}_G)=max\{d_G(g{x}_G,g{y}_G),d_G(g{x}_G,f{x}_G),d_G(g{y}_G,f{y}_G),{\displaystyle \frac{d_G(g{x}_G,f{y}_G)+d_G(g{y}_G,f{x}_G)}{2}}\}$.

Let mappings $f,g:X_G\to X_G$ be given. We will make use of the following notations:

• $X_G(f,g):\{u\in X_G:(gu,fu)\in E\left(G\right)\}$,
• $C(f,g):\{u\in X_G:fu=gu\}$ is the set of all coincidence points of mappings f and g,
• $C_m(f,g):\{u\in X_G:fu=gu=u\}$ is the set of all common fixed points of mappings f and (g).
• $CS(X,d)$: Collection of all Cauchy sequences in the metric space $(X,d)$.

Lemma 1.

Let f and g satisfy the following:

(1.1) 

$x_G,y_G\in C(f,g)impliesg{x}_G=g{y}_G$

(2.1) 

$(f,g)iscompatible$

Then, $C_m(f,g)\ne \varphi$.

Proof. 

Let $x_G\in C(f,g)$ and $g{x}_G=w$. Then, since (f, g) is compatible, $gw=gg{x}_G=gf{x}_G=fg{x}_G=fw$, or in other words, $w\in C(f,g)$. By Lemma (1), $gw=g{x}_G=w$, which, in turn, shows that $w\in C_m(f,g)$. ☐

Theorem 2.

Let $d_G^{\prime }$ and $d_G$ be any two metrics defined on $X_G$, and $(X_G,d_G^{\prime })$ is complete. Suppose $f,g:X_G\to X_G$ satisfy the following:

(2.1) 

$(f,g)\in Jungcktype(𝒞,{\Psi }^{*},G)withrespectto{d}_G$

(2.2) 

$giscontinuousandg\left(X_G\right)isclosedwithrespectto{d}_G^{\prime }$

(2.3) 

$f\left(X_G\right)\subseteq g\left(X_G\right)$

(2.4) 

$E\left(G\right)satisfiesthetransitivityproperty$

(2.5) 

$if{d}_G\ge d_G^{\prime },thenf:(X_G,d_G)\to (X_G,d_G^{\prime })isg-Cauchy$

(2.6) 

$fisG-continuouswithrespectto{d}_G^{\prime },fandgare{d}_G^{\prime }-compatible.$

Then,

$$\begin{array}{c}\hfill X_G(f,g)\ne \varphi iffC(f,g)\ne \varphi \end{array}$$

Proof. 

Suppose that $C(f,g)\ne \varphi$. Let $u\in C(f,g)$. Then, $(gu,fu)=(gu,gu)\in \Delta \subset E\left(G\right)$ and so $u\in X_G(f,g)$; that is, $X_G(f,g)\ne \varphi$.

Suppose now, $X_G(f,g)\ne \varphi$. Let ${x_G}_0,\in X_G$, such that $(g{x_G}_0,f{x_G}_0)\in E\left(G\right)$. Now, since $F\left(X_G\right)\subseteq g\left(X_G\right)$, using condition (5.1) we can construct sequence $\left\{{x_G}_n\right\}$ in $X_G$, such that

$$\begin{array}{cc}\hfill g{x_G}_n& =f{x_G}_{n-1},(g{x_G}_{n-1},g{x_G}_n)\in E\left(G\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$. It is easy to see that if $M({x_G}_m,{x_G}_n)=0$ for any $m,n\in N,$, then ${x_G}_m,{x_G}_n\in C(f,g)$ and the proof is done. So we assume that for all $m,n\in \mathbb{N}$, $M({x_G}_m,{x_G}_n)\ne 0$. Then,

$$\begin{array}{cc}\hfill \psi \left(d_G(g{x_G}_{n+1},g{x}_{n+2})\right)& =\psi \left(d_G(f{x_G}_n,f{x_G}_{n+1})\right)\hfill \\ & \le F(\psi \left(M(g{x_G}_n,g{x_G}_{n+1})\right),M(g{x_G}_n,g{x_G}_{n+1}))\hfill \\ & <\psi \left(M(g{x_G}_n,g{x_G}_{n+1})\right)\hfill \end{array}$$

(15)

We also have

$$\begin{array}{cc}\hfill M(g{x_G}_n,g{x_G}_{n+1})& =max\{d_G(g{x_G}_n,g{x_G}_{n+1}),d_G(g{x_G}_n,f{x_G}_n),d_G(g{x_G}_{n+1},f{x_G}_{n+1}),\hfill \\ & {\displaystyle \frac{d_G(g{x_G}_n,f{x_G}_{n+1})+d_G(g{x_G}_{n+1},f{x_G}_n)}{2}}\}\hfill \\ & =max\left\{d_G(g{x_G}_n,g{x_G}_{n+1}),d_G(g{x_G}_{n+1},g{x}_{n+2}),{\displaystyle \frac{d_G(g{x_G}_n,g{x}_{n+2})}{2}}\right\}\hfill \\ & \le max\{d_G(g{x_G}_n,g{x_G}_{n+1}),d_G(g{x_G}_n,g{x}_{n+2})\}\hfill \end{array}$$

If $M(g{x_G}_n,g{x_G}_{n+1})=d_G(g{x_G}_{n+1},g{x}_{n+2})$, then by (15), we obtain that

$$\begin{array}{cc}\hfill \psi \left(d_G(g{x_G}_{n+1},g{x}_{n+2})\right)& <\psi \left(d_G(g{x_G}_{n+1},g{x}_{n+2})\right)\hfill \end{array}$$

a contradiction. Hence,

$$\begin{array}{cc}\hfill M(g{x_G}_n,g{x_G}_{n+1})& =d_G(g{x_G}_n,g{x_G}_{n+1})\hfill \end{array}$$

Substituting in (15), we get t $\psi \left(d_G(g{x_G}_{n+1},g{x}_{n+2})\right)<\psi \left(d_G(g{x_G}_n,g{x_G}_{n+1})\right)$. So by the definition of $\psi$, we have

$$\begin{array}{cc}\hfill d_G(g{x_G}_{n+1},g{x}_{n+2})& \le d_G(g{x_G}_n,g{x_G}_{n+1}),\forall n\in \mathbb{N}\hfill \end{array}$$

Hence, the sequence $\left\{d_G(g{x_G}_n,g{x_G}_{n+1})\right\}$ is non-negative and non-increasing, and thereby we can find $r\ge 0$, such that ${lim}_{n\to \infty }{d}_G(g{x_G}_n,g{x_G}_{n+1})=r$. We claim that $r=0$. Suppose, on the contrary, that $r>0$. Letting $n\to \infty$ in (15), we obtain

$$\begin{array}{cc}\hfill \psi \left(r\right)& \le F\left(\psi \right(r),r)<\psi \left(r\right)\hfill \end{array}$$

a contradiction. Thus,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{d}_G(g{x_G}_n,g{x_G}_{n+1})& =0.\hfill \end{array}$$

(16)

We will show that $\left\{g{x_G}_n\right\}\in CS(X_G,d_G)$. Suppose $\left\{g{x_G}_n\right\}\notin CS(X_G,d_G)$ and for $ϵ>0$, $k\in \mathbb{N}$, let $n\left(k\right)\in \mathbb{N}$ be the smallest integer with $n\left(k\right)>m\left(k\right)\ge k$ and

$$\begin{array}{cc}\hfill d_G(g{x_G}_{n\left(k\right)},g{x_G}_{m\left(k\right)})& \ge ϵ\hfill \\ \hfill d_G(g{x_G}_{n\left(k\right)-1},g{x_G}_{m\left(k\right)})& <ϵ.\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill ϵ& \le d_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})\hfill \\ & \le d_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)-1})+d_G(g{x_G}_{n\left(k\right)-1},g{x_G}_{n\left(k\right)})\hfill \\ & <ϵ+d_G(g{x_G}_{n\left(k\right)-1},g{x_G}_{n\left(k\right)})\hfill \end{array}$$

Using (16) in the above inequality, we get

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{d}_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})& =ϵ>0.\hfill \end{array}$$

By condition (2.4) we get $(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})\in E\left(G\right)$. Thus, we have

$$\begin{array}{cc}\hfill \psi \left(d_G(g{x_G}_{m\left(k\right)+1},g{x_G}_{n\left(k\right)+1})\right)& =\psi \left(d_G(f{x_G}_{m\left(k\right)},f{x_G}_{n\left(k\right)})\right)\hfill \\ & \le F(\psi \left(M(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})\right),M(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)}))\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill M(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})& =max\{d_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)}),d_G(g{x_G}_{m\left(k\right)},f{x_G}_{m\left(k\right)}),d_G(g{x_G}_{n\left(k\right)},f{x_G}_{n\left(k\right)}),\hfill \\ & {\displaystyle \frac{d_G(g{x_G}_{m\left(k\right)},f{x_G}_{n\left(k\right)})+d_G(g{x_G}_{n\left(k\right)},f{x_G}_{m\left(k\right)})}{2}}\}\hfill \\ & =max\{d_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)}),d_G(g{x_G}_{m\left(k\right)},g{x}_{m\left(k\right)+1}),d_G(g{x_G}_{n\left(k\right)},g{x}_{n\left(k\right)+1}),\hfill \\ & {\displaystyle \frac{d_G(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)+1})+d_G(g{x_G}_{n\left(k\right)},g{x}_{m\left(k\right)+1})}{2}}\}\hfill \end{array}$$

Letting $k\to \infty$, we obtain

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}M(g{x_G}_{m\left(k\right)},g{x_G}_{n\left(k\right)})& =ϵ\hfill \end{array}$$

By inequality (17), we get

$$\psi \left(ϵ\right)<\psi \left(ϵ\right)$$

a contradiction. So $\left\{g{x_G}_n\right\}\in CS(X_G,d_G)$.

We will show that $\left\{g{x_G}_n\right\}\in CS(X_G,d_G^{\prime })$. If $d_G\ge d_G^{\prime }$, it is trivial. Thus, suppose $d_G\ge d_G^{\prime }$. Let $ϵ>0$. Since $\{g{x_G}_n\in CS(X_G,d_G)\}$, by condition (2.5) we see that $\left\{f{x_G}_n\right\}\in CS(X_G,d_G^{\prime })$. Then, there exists $N_0\in \mathbb{N}$ with

$$\begin{array}{cc}\hfill d_G^{\prime }(g{x_G}_{n+1},g{x}_{m+1})& =d_G^{\prime }(f{x_G}_n,f{x}_m)<ϵ\hfill \end{array}$$

whenever $n,m\ge N_0$. So $\left\{g{x_G}_n\right\}\in CS(X_G,d_G^{\prime })$.

Since $g\left(X_G\right)$ is $d_G^{\prime }$ - closed and $(X_G,d_G^{\prime })$ is complete, there exists $u_G=g{x}_G\in g\left(X_G\right)$, such that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}g{x_G}_n& =\underset{n\to \infty }{lim}f{x_G}_n=u_G.\hfill \end{array}$$

By $d_G^{\prime }$ - compatibility of f and g, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{d}_G^{\prime }(gf{x_G}_n,fg{x_G}_n)& =0\hfill \end{array}$$

(18)

Then,

$$\begin{array}{cc}\hfill d_G^{\prime }(g{u}_G,f{u}_G)& \le d_G^{\prime }(g{u}_G,gf{x_G}_n)+d_G^{\prime }(gf{x_G}_n,fg{x_G}_n)+d_G^{\prime }(fg{x_G}_n,f{u}_G)\hfill \end{array}$$

Letting $n\to \infty$ and using (18), the continuity of g, and the G- continuity of f, it follows that $d_G^{\prime }(g{u}_G,f{u}_G)=0$, which implies that $g{u}_G=f{u}_G$. So $u_G\in C(f,g)$ and the proof is complete. ☐

If $d_G=d_G^{\prime }$, we have the following

Theorem 3.

Let $(X_G,d_G)$ be complete and $f,g:X_G\to X_G$ satisfy the following:

(3.1) 

($f,g)\in Jungcktype(𝒞,{\Psi }^{*},G)$

(3.2) 

g is continuous and $g\left(X_G\right)$ is closed

(3.3) 

$F\left(X_G\right)\subset g\left(X_G\right)$

(3.4) 

$E\left(G\right)$ satisfies the transitivity property

(3.5) 

(a)  f is G-continuous and f and g are $d_G$-compatible or

 (b)  $(X_G,d_G,G)$ has property A.

Then,

$$\begin{array}{c}\hfill X_G(f,g)\ne \varphi \mathit{iff}C(f,g)\ne \varphi .\end{array}$$

Proof. 

Proceeding as in the proof of Theorem 2, we see that if $C(f,g)\ne \varphi$ then $X_G(f,g)\ne \varphi$ and if $X_G(f,g)\ne \varphi$ then $\left\{g{x_G}_n\right\}\in CS(X_G,d_G)$ Now since $g\left(X_G\right)$ is closed in $X_G$, there exists $u_G\in X_G$, such that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}g{x_G}_n& =g{u}_G=\underset{n\to \infty }{lim}f{x_G}_n.\hfill \end{array}$$

(19)

We will show that $u_G\in C(f,g)$. Suppose $u_G\notin C(f,g)$. Then $d_G(f{u}_G,g{u}_G)>0$. Note that if $M({x_G}_m,u_G)=0$ for any $m\in N,$, then ${x_G}_m,u_G\in C(f,g)$ and the proof is done. So we assume that for all $m\in \mathbb{N}$, $M({x_G}_m,u_G)\ne 0$. If condition (3.5a) is satisfied, then proof follows from a similar argument as in Theorem 2. If condition (3.5b) is satisfied, then $(g{x_G}_n,gu)\in E\left(G\right)$ for each $n\in \mathbb{N}$. Thus, we have

$$\begin{array}{cc}\hfill d_G\left(g{u}_{G}f{u}_G\right)& \le d_G(g{u}_G,f{x_G}_{n\left(k\right)})+d_G(f{x_G}_{n\left(k\right)},f{u}_G)\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill d_G(g{u}_G,f{u}_G)-d_G(g{u}_G,f{x_G}_{n\left(k\right)})& \le d_G(f{x_G}_{n\left(k\right)},f{u}_G)\hfill \end{array}$$

Since $\psi$ is non-decreasing, we get

$$\begin{array}{cc}\hfill \psi (d_G(g{u}_G,f{u}_G)-d_G(g{u}_G,f{x_G}_{n\left(k\right)}))& \le \psi \left(d_G(f{x_G}_{n\left(k\right)},f{u}_G)\right)\hfill \\ & \le F(\psi \left(M(g{x_G}_{n\left(k\right)},g{u}_G)\right),M(g{x_G}_{n\left(k\right)},g{u}_G))\hfill \end{array}$$

(20)

where

$$\begin{array}{cc}\hfill M(g{x_G}_{n\left(k\right)},g{u}_G)& =max\{d_G(g{x_G}_{n\left(k\right)},g{u}_G),d_G(g{x_G}_{n\left(k\right)},f{x_G}_{n\left(k\right)}),d_G(g{u}_G,f{u}_G),\hfill \\ & {\displaystyle \frac{d_G(g{x_G}_{n\left(k\right)},f{u}_G)+d_G(g{u}_G,f{x_G}_{n\left(k\right)}))}{2}}\}\hfill \end{array}$$

Using (19), we obtain

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}M(g{x_G}_{n\left(k\right)},g{u}_G)& =d_G(g{u}_G,f{u}_G)>0.\hfill \end{array}$$

Thus, taking $k\to \infty$ in (22), we get $\psi \left(d_G(g{u}_G,f{u}_G)\right)<\psi \left(d_G(g{u}_G,f{u}_G)\right)$, a contradiction. Therefore, $f{u}_G=g{u}_G$ and so $C(f,g)\ne \varphi$. ☐

Theorem 4.

Suppose f and g satisfy condition (2.1)–(2.6), condition (2.6) and the following:

(4.1) If $x_G,y_G\in C(f,g)$ and $g{x}_G\ne g{y}_G$, then $(g{x}_G,g{y}_G)\in E\left(G\right)$.

  If $X_G(f,g)\ne \varphi$, then $C_m(f,g)\ne \varphi$.

Proof. 

By Theorem 2 $C(f,g)\ne \varphi$. Let $x_G,y_G\in C(f,g)$ and suppose $g{x}_G\ne g{y}_G$ so that $M(g{x}_G,g{y}_G)\ne 0$. By assumption $\left(K\right),(g{x}_G,g{y}_G)\in E\left(G\right)$, and we have

$$\begin{array}{cc}\hfill \psi \left(d_G(f{x}_G,f{y}_G)\right)& \le F(\psi \left(M(g{x}_G,g{y}_G)\right),M(g{x}_G,g{y}_G))\hfill \\ & <\psi \left(M(g{x}_G,g{y}_G)\right)\hfill \\ & =\psi \left(d_G(f{x}_G,f{y}_G)\right)\hfill \end{array}$$

(21)

which is a contradiction. Therefore, $g{x}_G=g{y}_G$. Now by Lemma 1, $C_m(f,g)\ne \varphi$. ☐

Corollary 6.

Let $d_G^{\prime }$ and $d_G$ be any two metrics defined on $X_G$, and $(X_G,d_G^{\prime })$ is complete. Suppose $f,g:X_G\to X_G$ satisfy conditions (5.1) and Theorem (2.1) to Theorem (2.5), and the following: for some $\psi \in {\Psi }^{*}$, $\varphi \in {\Phi }^{*}$ and all $x_G,y_G\in X_G$ with $(g{x}_G,g{y}_G)\in E\left(G\right)$

$$\begin{array}{cc}\hfill \psi \left(d_G(f{x}_G,f{y}_G)\right)& \le \psi \left(M(g{x}_G,g{y}_G)\right)-\varphi \left(M(g{x}_G,g{y}_G)\right)\hfill \end{array}$$

(22)

Then,

$$\begin{array}{c}\hfill X_G(f,g)\ne \varphi iffC(f,g)\ne \varphi \end{array}$$

Corollary 7.

Let $d_G^{\prime }$ and $d_G$ be any two metrics defined on $X_G$ and $(X_G,d_G^{\prime })$ is complete. Suppose $f,g:X_G\to X_G$ satisfy condition (5.1) and condition (2.1) to condition (2.5) and the following: for some $\psi \in {\Psi }^{*}$, $\theta \in {\Theta }^{*}$ and all $x_G,y_G\in X_G$ with $(g{x}_G,g{y}_G)\in E\left(G\right)$

$$\begin{array}{cc}\hfill \psi \left(d_G(f{x}_G,f{y}_G)\right)& \le \theta \left(M(g{x}_G,g{y}_G)\right)\psi \left(M(g{x}_G,g{y}_G)\right)\hfill \end{array}$$

(23)

Then,

$$\begin{array}{c}\hfill X_G(f,g)\ne \varphi iffC(f,g)\ne \varphi \end{array}$$

Let $X_G=[0,\infty )$ and $d_G,d_G^{\prime }:X_G\times X_G\to [0,\infty )$ be defined by

$$\begin{array}{cc}\hfill d_G(x_G,y_G)& =\left\{\begin{array}{cc}0\hfill & \mathrm{if}{x}_G=y_G\hfill \\ max\left\{x_G,y_G\right\}\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}\begin{array}{c}\hfill d_G^{\prime }(x_G,y_G)=|x_G-y_G|\end{array}$$

(24)

Then clearly, $d_G\ge d_G^{\prime }$. We define

$$\begin{array}{cc}\hfill E\left(G\right)& =\{(x_G,y_G):x_G=y_{G}or{x}_G,y_G\in [0,{\displaystyle \frac{1}{2}}]with{x}_G\le y_G\}\hfill \end{array}$$

Consider the mappings $f:X_G\to X_G$ and $g:X_G\to X_G$, defined by

$$\begin{array}{ccc}\hfill f{x}_G& =\left\{\begin{array}{cc}4x_G^4,\hfill & \mathrm{if}0\le x_G\le {\displaystyle \frac{1}{2}}\hfill \\ 2x_G^2,\hfill & \mathrm{if}{x}_G>{\displaystyle \frac{1}{2}}\hfill \end{array}\right.g{x}_G\hfill & \hfill =\left\{\begin{array}{cc}2x_G^2,\hfill & \mathrm{if}0\le x_G\le {\displaystyle \frac{1}{2}}\hfill \\ 8x_G^4,\hfill & \mathrm{if}{x}_G>{\displaystyle \frac{1}{2}}\hfill \end{array}\right.\end{array}$$

(25)

for all $x_G\in X_G$.

Let $\psi :[0,\infty )\to [0,\infty )$ be defined by

$$\begin{array}{cc}\hfill \psi \left(t\right)& =2t\hfill \end{array}$$

and $\theta :[0,\infty )\to [0,1)$ be defined by

$$\begin{array}{cc}\hfill \theta \left(t\right)& =\left\{\begin{array}{cc}t,\hfill & \mathrm{if}0\le t\le {\displaystyle \frac{1}{2}}\hfill \\ {\displaystyle \frac{1}{2t+1}},\hfill & t>{\displaystyle \frac{1}{2}}\hfill \end{array}\right.\hfill \end{array}$$

and $F\in 𝒞$ be given by

$$\begin{array}{c}\hfill F(r,t)=\theta (t).r\end{array}$$

We will show that the pair $(f,g)$ is a $(F,\psi ,G)$ contraction.

Let $(g{x}_G,g{y}_G)\in E(G)$. If $g{x}_G=g{y}_G$ (which is possible only if $x_G=y_G=0$), then $f{x}_G=g{y}_G=0$ and so $(f{x}_G,f{y}_G)\in E(G)$. If $(g{x}_G,g{y}_G)\in E(G)$ with $g{x}_G,g{y}_G\in [0,{\displaystyle \frac{1}{2}}]$ and $g{x}_G\le g{y}_G$, then we obtain $x_G,y_G\in [0,{\displaystyle \frac{1}{2}}]$, $x_G\le y_G$ and then $x_G^4,y_G^4\in [0,{\displaystyle \frac{1}{2}}]$, $f{x}_G=4x_G^4\le 4y_G^4=f{y}_G$, and thus $(f{x}_G,fy)\in E(G)$. Thus, the pair $(f,g)$ is g-edge preserving in G.

Now, let $x_G,y_G\in X_G$ and $(g{x}_G,g{y}_G)\in E(G)$. From the argument given above, if $g{x}_G=g{y}_G$, then $\psi (d_G(f{x}_G,f{y}_G))=0$ and $(f,g)$ satisfy the condition condition (5.2). If $(g{x}_G,g{y}_G)\in E(G)$ with $g{x}_G,g{y}_G\in [0,{\displaystyle \frac{1}{2}}]$ and $g{x}_G\le g{y}_G$, then

$$\begin{array}{cc}\hfill \psi (d_G(f{x}_G,f{y}_G)& =8{y_G}^4\hfill \\ & \le 2{y_G}^2\hfill \\ & \le y_G.2y_G\hfill \\ & \le max\{x_G,y_G\}.2max\{x_G,y_G\}\hfill \\ & =\theta (d_G(x_G,y_G))\psi (d_G(x_G,y_G))\hfill \\ & \le \theta (M(g{x}_G,g{y}_G))\psi (M(g{x}_G,g{y}_G))\hfill \end{array}$$

and so $(f,g)$ satisfy the condition condition (5.1).

We will show that g and f are $d^{\prime }$ - compatible. Let $\left\{{x_G}_n\right\}$ be a sequence in $X_G$, such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{x_G}_n=\underset{n\to \infty }{lim}f{x_G}_n=u\end{array}$$

Note that

$$\begin{array}{c}\hfill d^{\prime }(gf{x_G}_n,fg{x_G}_n)=32{x_G}_N^8\to 0asn\to \infty .\end{array}$$

Clearly, $f:(X_G,d)\to (X_G,d^{\prime })$ is G-continuous, and consequently, all the conditions of Theorem 2 are satisfied. Also note that $0,{\displaystyle \frac{1}{2}}\in C(f,g)$, $g0\ne g({\displaystyle \frac{1}{2}})$, and $(g0,g({\displaystyle \frac{1}{2}})\in E(G)$, and thus condition Theorem (4.1) of Theorem 4 is satisfied. Consequently, $\{0,{\displaystyle \frac{1}{2}}\}\subset X_G(f,g)\bigcap C(f,g)\bigcap C_m(f,g)$.

3. Applications

In this section, as an application of our results, we will give some fixed point results for a pair of set valued $\alpha$-admissible contraction mappings in a metric space.

Throughout this section, $(X,d)$ is any metric space, $S,T:X\to CB\left(X\right)$ two given mappings, and $\alpha :X\times X\to [0,\infty )$.

Definition 6.

We say that the pair $(S,T)$ is α-admissible if, and only if for all $x,y\in X$ with $\alpha (x,y)\ge 1$, the following conditions hold:

(6.1) 

For $u\in Sx$, there exists $v\in Ty$, such that $\alpha (u,v)\ge 1$.

(6.2) 

For $u\in Tx$, there exists $v\in Sy$, such that $\alpha (u,v)\ge 1$.

Theorem 5.

Suppose the following conditions hold:

(5.1) 

There exists $x_0,x_1\in X$ such that $x_1\in T{x}_0\bigcup S{x}_0$ and $\alpha (x_0,x_1)\ge 1$,

(5.2) 

α is a triangular function,

(5.3) 

The pair $(S,T)$ is α-admissible,

(5.4) 

there exists $F\in 𝒞$, $\psi \in {\Psi }^{*}$, such that for all $x,y\in X$ with $\alpha (x,y)\ge 1$

• $\psi \left(H\right(Sx,Ty\left)\right)\le F\left(\psi \right(M(x,y)),M(x,y\left)\right)$ and
• $\psi \left(H\right(Tx,Sy\left)\right)\le F\left(\psi \right(M(x,y)),M(x,y\left)\right)$

where

$$M(x,y)=max\left\{d(x,y),d(Sx,x),d_G(Ty,y),{\displaystyle \frac{d(y,Sx)+d(x,Ty)}{2}}\right\}$$

Then $COFIX\{S,T\}$ is a singleton set.

Proof. 

Consider the graph G on $(X,d)$, defined by $V\left(G\right)=X$ and $E\left(G\right)=(x,y)\in X\times X:\alpha (x,y)\ge 1)$. It is easy to see that the functions S and T satisfy all conditions of Theorem 1, and hence, $COFIX\{S,T\}$ is a singleton set. ☐

Similarly, we have the following results:

Theorem 6.

Suppose conditions Theorem (5.1), Theorem (6.1), Theorem (6.3), and the following hold:

(6.1) for all $x,y\in X$ with $\alpha (x,y)\ge 1$

   $\psi \left(H\right(Sx,Ty\left)\right)\le \psi \left(M\right(x,y\left)\right)-\varphi \left(M\right(x,y\left)\right)$ and

   $\psi \left(H\right(Tx,Sy\left)\right)\le \psi \left(M\right(x,y\left)\right)-\varphi \left(M\right(x,y\left)\right)$

where $\psi \in {\Psi }^{*}$, $\varphi \in {\Phi }^{*}$ and $M(x,y)$ is as in Theorem 5. Then $COFIX\{S,T\}\ne \varphi$.

Theorem 7.

Suppose conditions Theorem (5.1), Theorem (6.1), Theorem (5.3), and the following hold:

(7.1) for all $x,y\in X$ with $\alpha (x,y)\ge 1$

   $\psi \left(H\right(Sx,Ty\left)\right)\le \theta \left(M\right(x,y\left)\right)\psi \left(M\right(x,y\left)\right)$ and

   $\psi \left(H\right(Tx,Sy\left)\right)\le \theta \left(M\right(x,y\left)\right)\psi \left(M\right(x,y\left)\right)$

where $\psi \in {\Psi }^{*}$, $\varphi \in {\Phi }^{*}$ and $M(x,y)$ is as in Theorem 5. Then, $COFIX\{S,T\}\ne \varphi$.

Example 2.

Let $X_G=[0,\infty )\subseteq \mathbb{R}$, and let the metrics $d_G,d_G^{\prime }:X_G\times X_G\to [0,\infty )$ be defined by

$$\begin{array}{c}\hfill d_G(x_G,y_G)=max\left\{x_G,y_G\right\}\\ \hfill d^{\prime }(x_G,y_G)=L|x_G-y_G|\end{array}$$

(26)

for all $x_G,y_G\in X_G$, respectively, where L is a constant real number, such that $L\in (1,\infty )$. It is easy to see that $d<d^{\prime }$. Now, $E\left(G\right)$ is given by

$$\begin{array}{cc}\hfill E\left(G\right)& =\{(x_G,y_G):x_G=y_{G}or{x}_G,y_G\in [0,1]with{x}_G\le y_G\}\hfill \end{array}$$

(27)

Consider the mappings $f:X_G\to X_G$ and $g:X_G\to X_G$ defined by

$$\begin{array}{c}\hfill fx=ln\left(1+{\displaystyle \frac{x_G^2}{2}}\right),gx=x_G^2\end{array}$$

(28)

for all $x_G\in X_G$, respectively.

Let $\psi ,\varphi :[0,\infty )\to [0,\infty )$ be defined by

$$\begin{array}{cc}\hfill \psi (t)& =\left\{\begin{array}{cc}t,\hfill & if0\le t\le 1\hfill \\ t^2,\hfill & ift>1\hfill \end{array}\right.\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \varphi (t)& =\left\{\begin{array}{cc}{\displaystyle \frac{t^2}{2}},\hfill & if0\le t\le 2\hfill \\ {\displaystyle \frac{1}{2}},\hfill & otherwiset>1\hfill \end{array}\right.\hfill \end{array}$$

(30)

Next, let $(gx,gy)\in E(G)$ if $x_G=y_G$. Then, $(fx,fy)\in E(G)$, and if $(gx,gy)\in E(G)$ with $gx\le gy$, then we obtain $gx=x_G^2,gy=y_G^2\in [0,1]$ and $x_G^2=gx\le gy=y_G^2$, and we have $fx=ln(1+{\displaystyle \frac{x_G^2}{2}})\le ln(1+{\displaystyle \frac{y_G^2}{2}})=fy$ and $f{x}_G,fy\in [0,1]$. This implies that $(f{x}_G,fy)\in E(G)$.

Since $d<d^{\prime }$, we need to prove that the function $f:(X_G,d)\to (X_G,d^{\prime })$ is a g-Cauchy sequence in $X_G$. Let $ϵ>0$, and let $\left\{{x_G}_n\right\}$ be a sequence in $X_G$ such that $\left\{g{x_G}_n\right\}$ is a Cauchy sequence in $(X_G,d)$. Then, there exists $k\in \mathbb{N}$, such that for all $n,m\ge k,d_G(g{x_G}_n,g{x}_m)<{\displaystyle \frac{ϵ}{L}}$. Then, we have

$$\begin{array}{cc}\hfill d^{\prime }(f{x_G}_n,f{x}_m)& =L|f{x_G}_n-f{x}_m|\hfill \\ & =L\left|ln\left(1+{\displaystyle \frac{{({x_G}_n)}^2}{2}}\right)-ln\left(1+{\displaystyle \frac{{(x_m)}^2}{2}}\right)\right|\hfill \\ & =L\left|ln{\displaystyle \frac{1+{\displaystyle \frac{{(x_m)}^2}{2}}}{1+{\displaystyle \frac{{({x_G}_n)}^2}{2}}}}\right|\hfill \\ & =L\left|ln\left(1+{\displaystyle \frac{{\displaystyle \frac{{(x_m)}^2}{2}}-{\displaystyle \frac{{({x_G}_n)}^2}{2}})}{1+{\displaystyle \frac{{({x_G}_n)}^2}{2}}}}\right)\right|\hfill \\ & \le L\left[ln\left(1+|{\displaystyle \frac{{({x_G}_n)}^2}{2}}-{\displaystyle \frac{{(x_M)}^2}{2}}|\right)\right]\hfill \\ & \le L\left[{\displaystyle \frac{2ln(1+{\displaystyle \frac{1}{2}}|{({x_G}_n)}^2-{(x_m)}^2|)}{|{({x_G}_n)}^2-{(x_m)}^2|}}|{({x_G}_n)}^2-{(x_m)}^2|\right]\hfill \\ & <L|{({x_G}_n)}^2-{(x_m)}^2|\hfill \\ & =L{d}_G(g{x_G}_n,g{x}_m)\hfill \\ & <L·{\displaystyle \frac{ϵ}{L}}=ϵ.\hfill \end{array}$$

This implies that $f:(X_G,d)\to (X_G,d^{\prime })$ is a g- Cauchy on $X_G$.

It can easily be shown that $f:(X_G,d)\to (X_G,d^{\prime })$ is G-continuous. As a result, we will only need to prove that g and f are $d^{\prime }$-compatible. Let $\left\{{x_G}_n\right\}$ be a sequence in $X_G$, such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{x_G}_n=\underset{n\to \infty }{lim}f{x_G}_n=u\end{array}$$

(31)

Then, we have $ln(1+u/2)=a$ and so it follows that $u=0$. Now, we have

$$\begin{array}{c}\hfill d^{\prime }(gf{x_G}_n,fg{x_G}_n)=L\left|ln{\left(1+{\displaystyle \frac{{\left({x_G}_n\right)}^2}{2}}\right)}^2-ln{\left(1+{\displaystyle \frac{{\left({x_G}_n\right)}^4}{2}}\right)}^2\right|\to 0\end{array}$$

(32)

as $n\to \infty$. It is easy to see that there exists a point $u\in X_G$, such that $(gu,fu)\in E\left(G\right)$, and thus $X_G(f,g)\ne \varphi$.

Consequently, all the conditions of Theorem 2 are satisfied.

Example 3.

Let $X_G=[0,\infty )\subseteq \mathbb{R}$ and let the metrics $d_G,d_G^{\prime }:X_G\times X_G\to [0,\infty )$ be defined by

$$\begin{array}{c}\hfill d_G(x_G,y_G)=max\left\{x_G,y_G\right\}\\ \hfill d_G^{\prime }(x_G,y_G)=L.max\left\{x_G,y_G\right\}\end{array}$$

(33)

for all $x_G,y_G\in X_G$ and some $L\in (1,\infty )$. Then clearly, $d_G<d_G^{\prime }$. We define

$$\begin{array}{cc}\hfill E\left(G\right)& =\{(x_G,y_G):x_G=y_{G}or{x}_G,y_G\in [0,1]with{x}_G\le y_G\}\hfill \end{array}$$

Consider the mappings $f:X_G\to X_G$ and $g:X_G\to X_G$ defined by

$$\begin{array}{ccc}\hfill f{x}_G& =\left\{\begin{array}{cc}{x}_G^4,\hfill & if0\le x_G\le 1\hfill \\ x_G^2,\hfill & if{x}_G>1\hfill \end{array}\right.g{x}_G\hfill & \hfill =\left\{\begin{array}{cc}2x_G^2,\hfill & if0\le x_G\le 1\hfill \\ 2x_G^4,\hfill & if{x}_G>1\hfill \end{array}\right.\end{array}$$

(34)

for all $x_G\in X_G$.

Let $\psi :[0,\infty )\to [0,\infty )$ be defined by

$$\begin{array}{cc}\hfill \psi \left(t\right)& =2t\hfill \end{array}$$

and, $\theta :[0,\infty )\to [0,1)$ be defined by

$$\begin{array}{cc}\hfill \theta \left(t\right)& =\left\{\begin{array}{cc}t,\hfill & if0\le t<1\hfill \\ {\displaystyle \frac{1}{t+1}},\hfill & t\ge 1\hfill \end{array}\right.\hfill \end{array}$$

Next, if $(g{x}_G,g{y}_G)\in E\left(G\right)$ and $x_G=y_G$ then $(f{x}_G,f{y}_G)\in E\left(G\right)$ and if $(g{x}_G,g{y}_G)\in E\left(G\right)$ with $g{x}_G\le g{y}_G$, then we obtain $x_G,y_G\in [0,{\displaystyle \frac{1}{\sqrt{2}}}$, $x_G\le y_G$ and then $x_G^4,y_G^4\in [0,{\displaystyle \frac{1}{4}}$, $f{x}_G=x_G^4\le y_G^4=f{y}_G$, and thus $(f{x}_G,fy)\in E\left(G\right)$.

Since $d_G<d_G^{\prime }$, we need to prove that the function $f:(X_G,d)\to (X_G,d^{\prime })$ is a g-Cauchy sequence in $X_G$. Let $ϵ>0$, and let $\left\{{x_G}_n\right\}$ be a sequence in $X_G$ such that $\left\{g{x_G}_n\right\}$ is a Cauchy sequence in $(X_G,d)$. Then, there exists $k\in \mathbb{N}$ such that, for all $n,m\ge k,d_G(g{x_G}_n,g{x}_m)<{\displaystyle \frac{ϵ}{L}}$. Then, we have

$$\begin{array}{cc}\hfill d_G^{\prime }(f{x_G}_n,f{x_G}_m)& =Lmax\left\{f{x_G}_n,f{x_G}_m\right\}\hfill \end{array}$$

We will consider the following cases:

Case 1: ${x_G}_n,{x_G}_m\in [0,1]$. Then, we have

$$d_G^{\prime }(f{x_G}_n,f{x}_m)=Lmax\left\{{x_G}_n^4,{x_G}_m^4\right\}\le Lmax\left\{{2x_G}_n^2,2{x_G}_m^2\right\}$$

$$=Lmax\left\{g{x_G}_n,g{x_G}_m\right\}=L.d_G(g{x_G}_n,g{x_G}_m)<L.{\displaystyle \frac{ϵ}{L}}=ϵ$$

Case 2: ${x_G}_n,{x_G}_m\in (0,\infty$. Then, we have

$$d_G^{\prime }(f{x_G}_n,f{x}_m)=Lmax\left\{{x_G}_n^2,{x_G}_m^2\right\}\le Lmax\left\{{2x_G}_n^4,2{x_G}_m^4\right\}$$

$$=Lmax\left\{g{x_G}_n,g{x_G}_m\right\}=L.d_G(g{x_G}_n,g{x_G}_m)<L.{\displaystyle \frac{ϵ}{L}}=ϵ$$

Case 3: ${x_G}_n\in [0,1],{x_G}_m\in (1,\infty )$. Then, we have

$$d_G^{\prime }(f{x_G}_n,f{x}_m)=Lmax\left\{{x_G}_n^4,{x_G}_m^2\right\}\le Lmax\left\{{2x_G}_n^2,2{x_G}_m^4\right\}$$

$$=Lmax\left\{g{x_G}_n,g{x_G}_m\right\}=L.d_G(g{x_G}_n,g{x_G}_m)<L.{\displaystyle \frac{ϵ}{L}}=ϵ$$

Case 1: ${x_G}_n\in (0,\infty ),{x_G}_n\in [0,1]$. Then, we have

$$d_G^{\prime }(f{x_G}_n,f{x}_m)=Lmax\left\{{x_G}_n^2,{x_G}_m^4\right\}\le Lmax\left\{{2x_G}_n^4,2{x_G}_m^2\right\}$$

$$=Lmax\left\{g{x_G}_n,g{x_G}_m\right\}=L.d_G(g{x_G}_n,g{x_G}_m)<L.{\displaystyle \frac{ϵ}{L}}=ϵ$$

Thus, $f:(X_G,d)\to (X_G,d^{\prime })$ is a g-Cauchy on $X_G$.

It can be easily shown that $f:(X_G,d)\to (X_G,d^{\prime })$ is G-continuous. We will show that g and f are $d^{\prime }$ compatible. Let $\left\{{x_G}_n\right\}$ be a sequence in $X_G$, such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}g{x_G}_n=\underset{n\to \infty }{lim}f{x_G}_n=u\end{array}$$

Then, we have $ln(1+u/2)=a$, and so it follows that $u=0$. Now, we have

$$\begin{array}{c}\hfill d^{\prime }(gf{x_G}_n,fg{x_G}_n)=L\left|ln{\left(1+{\displaystyle \frac{{\left({x_G}_n\right)}^2}{2}}\right)}^2-ln{\left(1+{\displaystyle \frac{{\left({x_G}_n\right)}^4}{2}}\right)}^2\right|\to 0\end{array}$$

(35)

as $n\to \infty$. It is easy to see that there exists a point $u\in X_G$ such that $(gu,fu)\in E\left(G\right)$, and thus $X_G(f,g)\ne \varphi$.

Consequently, all the conditions of Theorem 2 are satisfied.