# Generalized probabilistic G-contractions

## Abstract

In this paper, the notion of generalized probabilistic G-contractions in Menger probabilistic metric spaces endowed with a directed graph G is introduced and some new fixed point theorems for such mappings are established.

## Introduction and preliminaries

Ran and Reurings  gave a generalization of Banach contraction principle to partially ordered metric spaces. Since then, many authors obtained generalization and extension of the results of .

In particular, Ćirić et al.  extended the results of [1, 5, 6] to partially ordered Menger probabilistic metric spaces.

Samet et al.  introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems for such mappings in complete metric spaces.

Cho  obtained a generalization of the results of  by introducing the concept of α-contractive type mappings in Menger probabilistic metric spaces.

Recently, Wu  obtained a generalization of the results of , and improved and extended the fixed point results of [4, 11, 12]. Also, Kamran et al.  introduced the notion of probabilistic G-contractions in Menger PM-spaces endowed with a graph G and obtained some fixed point results. Especially, they obtained the following result.

### Theorem 1.1

Let $$(X,F,\Delta)$$ be a complete Menger PM-space, where Δ is of Hadžić-type. Let $$G=(V(G),E(G))$$ be a directed graph such that $$V(G)=X$$ and $$\Omega\subset E(G)$$. Suppose that a map $$f:X\to X$$ satisfies f preserves edges and there exists $$k\in(0,1)$$ such that, for all $$x,y \in X$$ with $$(x,y)\in E(G)$$,

$$F_{fx,fy}(kt)\geq F_{x,y}(t).$$

Assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$. If either f is orbitally G-continuous or G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Further if $$(x,y)\in E(G)$$ for any $$x,y\in M$$, where $$M=\{x\in X:(x,fx)\in E(G)\}$$, then f has a unique fixed point.

In this paper, we give some new fixed point theorems which are generalizations of the results of [3, 9, 10, 13], by introducing a concept of generalized probabilistic G-contractions in Menger PM-spaces with a directed graph $$G=(V(G),E(G))$$ such that $$V(G)=X$$ and $$\Omega\subset E(G)$$.

We recall some definitions and results which will be needed in the sequel.

A mapping $$f:\mathbb {R}\to[0,\infty)$$ is called a distribution if the following conditions hold:

1. (1)

f is nondecreasing and left-continuous;

2. (2)

$$\sup\{f(t):t\in \mathbb {R}\}=1$$;

3. (3)

$$\inf\{f(t):t\in \mathbb {R}\}=0$$.

We denote by D the set of all distribution functions.

Let $$\epsilon_{0}:\mathbb {R}\to[0,\infty)$$ be a function defined by

$$\epsilon_{0}(t)= \textstyle\begin{cases} 0&(t\leq0),\cr 1&(t>0). \end{cases}$$

Then $$\epsilon_{0} \in D$$.

Let $$\Delta:[0,1]\times[0,1]\to[0,1]$$ be a mapping such that

1. (1)

$$\Delta(a,b)=\Delta(b,a)$$ for all $$a,b\in[0,1]$$;

2. (2)

$$\Delta(\Delta(a,b),c)=\Delta(a, \Delta(b,c))$$ for all $$a,b,c\in[0,1]$$;

3. (3)

$$\Delta(a,1)=a$$ for all $$a\in[0,1]$$;

4. (4)

$$\Delta(a,b)\geq\Delta(c,d)$$, whenever $$a\geq c$$ and $$b\geq d$$ for all $$a,b,c,d\in[0,1]$$.

Then Δ is called a triangular norm (for short t-norm).

We denote $$\mathbb {N}$$ by the set of all natural numbers.

For a t-norm Δ, we consider the following notation:

$$\Delta^{1}(t)=\Delta(t,t),\qquad \Delta^{n}(t)=\Delta\bigl(t, \Delta^{n-1}(t)\bigr) \quad\mbox{for all } n\in \mathbb {N} \mbox{ and } t \in[0,1].$$

A t-norm Δ is said to be of Hadžić-type  whenever the family of $$\{\Delta^{n}(t)\}_{n=1}^{\infty}$$ is equicontinuous at $$t=1$$.

For example, the minimum t-norm $$\Delta_{m}$$ defined by

$$\Delta_{m}(a,b)=\min\{a,b\},\quad \forall a,b\in[0,1],$$

It is easy to see that the following are equivalent (see ):

1. (1)

for a t-norm Δ,

$$\mbox{it is of Had\v{z}i\'{c}-type};$$
(1.1)
2. (2)

given $$\epsilon\in(0,1)$$, there is a $$\delta\in(0,1)$$ such that $$\Delta^{n}(x)>1-\epsilon$$ for all $$n\in \mathbb {N}$$, whenever $$x>1-\delta$$.

Also, it is well known that if Δ satisfies condition $$\Delta(a,a)\geq a$$ for all $$a\in [0,1]$$, then $$\Delta=\Delta_{m}$$ (see ). Hence we have

$$\forall a\in[0,1],\quad\Delta(a,a)\geq a \quad\Longleftrightarrow\quad\Delta= \Delta_{m}.$$

Let X be a nonempty set, and let Δ be a t-norm. Suppose that a mapping $$F:X\times X\to D$$ (for $$x,y\in X$$, we denote $$F(x,y)$$ by $$F_{x,y}$$) satisfies the following conditions:

1. (PM1)

$$F_{x,y}(t)=\epsilon_{0}(t)$$ for all $$t\in \mathbb {R}$$ if and only if $$x=y$$;

2. (PM2)

$$F_{x,y}=F_{y,x}$$ for all $$x,y\in X$$;

3. (PM3)

$$F_{x,y}(t+s)\geq\Delta(F_{x,z}(t),F_{z,y}(s))$$ for all $$x, y, z \in X$$ and all $$t,s\geq0$$.

Then a 3-tuple $$(X,F,\Delta)$$ is called a Menger probabilistic metric space (briefly, Menger PM-space) [16, 17].

Let $$(X,F,\Delta)$$ be a Menger PM-space and X, and let $$\epsilon >0$$ and $$\lambda\in(0,1]$$.

Schweizer and Sklar  brought in the notion of neighborhood $$U_{x}(\epsilon,\lambda)$$ of x, where $$U_{x}(\epsilon,\lambda)$$ is defined as follows:

$$U_{x}(\epsilon,\lambda)=\bigl\{ y\in X:F_{x,y}(\epsilon)>1- \lambda\bigr\} .$$

The family

$$\bigl\{ U_{x}(\epsilon,\lambda):x\in X, \epsilon>0, \lambda\in(0,1]\bigr\}$$
(1.2)

does not necessarily determine a topology on X (see [19, 20]).

It is well known that if Δ satisfies condition

$$\sup\bigl\{ \Delta(t,t):0< t< 1\bigr\} =1$$
(1.3)

then (1.2) determines a Hausdorff topology on X, and it is called $$(\epsilon,\lambda)$$-topology.

So if (1.3) holds, then Menger space $$(X,F,\Delta)$$ is a Hausdorff topological space in the $$(\epsilon,\lambda)$$-topology (see [18, 21]).

### Remark 1.1

The following are satisfied:

1. (1)

condition (1.3) is the weakest condition which ensure the existence of the $$(\epsilon,\lambda)$$-topology (see );

2. (2)

condition (1.1) condition (1.3) (see ).

Let $$(X,F,\Delta)$$ be a Menger PM-space, and let $$\{x_{n}\}$$ be a sequence in X and $$x\in X$$. Then we say that

1. (1)

$$\{x_{n}\}$$ is convergent to x (we write $$\lim_{n\to\infty}x_{n}=x$$) if and only if, given $$\epsilon>0$$ and $$\lambda \in(0,1)$$, there exists $$n_{0}\in \mathbb {N}$$ such that $$F_{x_{n},x}(\epsilon)>1-\lambda$$, for all $$n\geq n_{0}$$.

2. (2)

$$\{x_{n}\}$$ is a Cauchy sequence if and only if, given $$\epsilon>0$$ and $$\lambda\in(0,1)$$, there exists $$n_{0}\in \mathbb {N}$$ such that $$F_{x_{n},x_{m}}(\epsilon)>1-\lambda$$, for all $$m>n\geq n_{0}$$.

3. (3)

$$(X,F,\Delta)$$ is complete if and only if each Cauchy sequence in X is convergent to some point in X.

### Example 1.1

Let D be a distribution function defined by

$$D(t)= \textstyle\begin{cases}0 &(t\leq0), \cr 1-e^{-t} &(t>0). \end{cases}$$

Let

$$F_{x,y}(t)= \textstyle\begin{cases} \epsilon_{0}(t) &(x=y), \cr D({t\over d(x,y)}) &(x\neq y), \end{cases}$$

for all $$x,y\in X$$ and $$t>0$$, where d is a metric on a nonempty set X.

Then $$(X,F,\Delta_{m})$$ is a Menger PM-space (see ).

### Remark 1.2

If $$(X,d)$$ is complete, then $$(X,F,\Delta_{m})$$ is complete. In fact, let $$\{x_{n}\}$$ be any Cauchy sequence in $$(X,F,\Delta_{m})$$.

Then

$$\lim_{n,m\to\infty}D\biggl({t\over d(x_{n},x_{m})}\biggr)=\lim _{n,m\to\infty }F_{x_{n},x_{m}}(t)=1$$

for all $$t>0$$, which implies $$\lim_{n,m\to\infty} d(x_{n},x_{m})=0$$.

Hence, $$\{x_{n}\}$$ is a Cauchy sequence in $$(X,d)$$. Since $$(X,d)$$ is complete, there exists $$x_{*}\in X$$ such that $$\lim_{n\to\infty }d(x_{n},x_{*})=0$$.

Thus, we have

$$\lim_{n\to\infty}F_{x_{n},x_{*}}(t)=\lim_{n\to\infty}D \biggl({t\over d(x_{n},x_{*})}\biggr)=1$$

for all $$t>0$$. Hence, $$(X,F,\Delta_{m})$$ is complete.

From now on, let

$$\Phi=\Bigl\{ \phi:[0,\infty)\to[0,\infty) \mid\lim_{n\to\infty}\phi ^{n}(t)=0, \forall t>0\Bigr\}$$

and let

$$\Phi_{w}=\Bigl\{ \phi:[0,\infty)\to[0,\infty) \mid\forall t>0, \exists r\geq t \mbox{ s.t. }\lim_{n\to\infty}\phi^{n}(r)=0\Bigr\} .$$

Note that $$\Phi\subset\Phi_{w}$$.

Fang  gave the corrected version of Theorem 12 of  by introducing the notion of right-locally monotone functions as follows: $$\phi:[0,\infty) \to[0,\infty)$$ is right-locally monotone if and only if $$\forall t\geq0$$, $$\exists\delta>0$$ s.t. it is monotone on $$[t,t+\delta)$$.

### Lemma 1.1



The following are satisfied:

1. (1)

If a right-locally monotone function $$\phi:[0,\infty) \to [0,\infty)$$ satisfies

$$\phi(0)=0,\qquad \phi(t)< t \quad\textit{and}\quad \lim_{r\to t^{+}}\inf \phi(r)< t\quad \textit{for all } t>0,$$

then $$\phi\in\Phi$$.

2. (2)

If a function $$\phi:[0,\infty) \to[0,\infty)$$ satisfies

$$\phi(t)< t \quad\textit{and}\quad \lim_{r\to t^{+}}\sup\phi(r)< t \quad \textit{for all } t>0,$$

then $$\phi\in\Phi_{w}$$.

3. (3)

If a function $$\alpha:[0,\infty) \to[0,1)$$ is piecewise monotone and

$$\phi(t)=\alpha(t)t\quad \textit{for all } t\geq0,$$

then $$\phi\in\Phi$$.

### Lemma 1.2



If $$\phi\in\Phi_{w}$$, then $$\forall t>0$$, $$\exists r\geq t$$ s.t. $$\phi(r)< t$$.

### Lemma 1.3



Let $$(X,F,\Delta)$$ be a Menger PM-space, and let $$x,y\in X$$. If

$$F_{x,y}\bigl(\phi(t)\bigr)\geq F_{x,y}(t)$$

for all $$t>0$$, where $$\phi\in\Phi_{w}$$, then $$x=y$$.

### Lemma 1.4



Let $$(X,F,\Delta)$$ be a Menger PM-space and $$x,y\in X$$, where Δ is continuous. Suppose that $$\{x_{n}\}$$ is a sequence of points in X. If $$\lim_{n\to\infty}x_{n}=x$$, then $$\lim_{n\to\infty}\inf F_{x_{n},y}(t)=F_{x,y}(t)$$ for all $$t>0$$.

### Lemma 1.5

Let $$(X,F,\Delta)$$ be a Menger PM-space, where Δ is of Hadžić-type. Let $$\{x_{n}\}$$ be a sequence of points in X such that $$x_{n-1}\neq x_{n}$$ for all $$n\in \mathbb {N}$$. If there exists $$\phi\in\Phi_{w}$$ such that

$$F_{x_{n},x_{m}}\bigl(\phi(s)\bigr)\geq\min\bigl\{ F_{x_{n-1},x_{m-1}}(s),F_{x_{n-1},x_{n}}(s),F_{x_{m-1},x_{m}}(s)\bigr\}$$
(1.4)

for all $$s>0$$ and all $$n,m\in \mathbb {N}$$, then for each $$t>0$$ there exists $$r\geq t$$ such that

$$F_{x_{n},x_{m}}(t)\geq\Delta^{m-n}\bigl(F_{x_{n},x_{n+1}} \bigl(t-\phi(r)\bigr)\bigr) \quad\textit{for all }m\geq n+1.$$
(1.5)

### Proof

It is easy to see that (1.4) implies that $$\phi(t)>0$$ for all $$t>0$$. In fact, if there exists $$t_{0}>0$$ such that $$\phi(t_{0})=0$$, then we obtain

$$0=F_{x_{n},x_{n}}\bigl(\phi(t_{0})\bigr)\geq F_{x_{n-1},x_{n}}(t_{0})>0$$

We claim that

$$F_{x_{n},x_{n+1}}(u)\geq F_{x_{n-1},x_{n}}(u) \quad\mbox{for all }u>0 \mbox{ and } n\in \mathbb {N}.$$

From (1.4) we have

$$F_{x_{n},x_{n+1}}\bigl(\phi(s)\bigr)\geq\min\bigl\{ F_{x_{n-1},x_{n}}(s),F_{x_{n},x_{n+1}}(s) \bigr\}$$

for all $$s>0$$ and all $$n\in \mathbb {N}$$.

If there exists $$n\in \mathbb {N}$$ such that $$F_{x_{n-1},x_{n}}(s)\geq F_{x_{n},x_{n+1}}(s)$$ for all $$s>0$$, then $$F_{x_{n},x_{n+1}}(\phi (s))\geq F_{x_{n},x_{n+1}}(s)$$ for all $$s>0$$. Thus, $$x_{n}=x_{n+1}$$, which is a contradiction. Hence we have $$F_{x_{n-1},x_{n}}(s)< F_{x_{n},x_{n+1}}(s)$$ for all $$s>0$$ and $$n \in \mathbb {N}$$, and so

$$F_{x_{n},x_{n+1}}\bigl(\phi(s)\bigr)\geq F_{x_{n-1},x_{n}}(s)$$

for all $$s>0$$ and $$n \in \mathbb {N}$$.

Since $$\phi\in\Phi_{w}$$, for each $$u>0$$, there exists $$v\geq u$$ such that

$$\phi(v)< u.$$

Hence,

$$F_{x_{n},x_{n+1}}(u)\geq F_{x_{n},x_{n+1}}\bigl(\phi(v)\bigr)\geq F_{x_{n-1},x_{n}}(v)\geq F_{x_{n-1},x_{n}}(u)$$

for all $$u>0$$ and $$n\in \mathbb {N}$$. So the claim is proved.

Let $$t>0$$ be given. By Lemma 1.2, there exists $$r\geq t$$ such that

$$\phi(r)< t.$$
(1.6)

By induction, we show that (1.5) holds.

Let $$m=n+1$$.

Then

\begin{aligned} &F_{x_{n},x_{n+1}}(t) \\ &\quad\geq F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr) \\ &\quad=\Delta\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r),1\bigr)\bigr) \\ &\quad\geq\Delta^{1}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)\bigr). \end{aligned}

Thus, (1.5) holds for $$m=n+1$$.

Assume that (1.5) holds for some fixed $$m>n+1$$. That is,

$$F_{x_{n},x_{m}}(t)\geq\Delta^{m-n}\bigl(F_{x_{n},x_{n+1}} \bigl(t-\phi(r)\bigr)\bigr) \quad\mbox{holds for some }m>n+1.$$
(1.7)

Then

\begin{aligned} &F_{x_{n},x_{m+1}}(t) \\ &\quad= F_{x_{n},x_{m+1}}\bigl(t-\phi(r)+\phi(r)\bigr) \\ &\quad\geq \Delta\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr),F_{x_{n+1},x_{m+1}} \bigl(\phi (r)\bigr)\bigr). \end{aligned}
(1.8)

From (1.4) we obtain

\begin{aligned} &F_{x_{n+1},x_{m+1}}\bigl(\phi(r)\bigr) \\ &\quad\geq\min\bigl\{ F_{x_{n},x_{m}}(r),F_{x_{n},x_{n+1}}(r),F_{x_{m},x_{m+1}}(r) \bigr\} . \end{aligned}

By the above claim, since $$F_{x_{m}, x_{m+1}}(t)\geq F_{x_{n}, x_{n+1}}(t)$$, from (1.4) and (1.7) we obtain

\begin{aligned}[b] &F_{x_{n+1},x_{m+1}}\bigl(\phi(r)\bigr) \\ &\quad\geq\min\bigl\{ F_{x_{n},x_{m}}(t),F_{x_{n},x_{n+1}}(t)\bigr\} \\ &\quad\geq\min\bigl\{ \Delta^{m-n}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi (r) \bigr)\bigr),F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)\bigr\} \\ &\quad=\Delta^{m-n}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)\bigr). \end{aligned}
(1.9)

Thus, from (1.8) and (1.9) we have

\begin{aligned} &F_{x_{n},x_{m+1}}(t) \\ &\quad\geq\Delta\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr),\Delta ^{m-n}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)\bigr)\bigr) \\ &\quad=\Delta^{m-n+1}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)\bigr). \end{aligned}

Hence, (1.5) holds for all $$m\geq n+1$$. □

### Lemma 1.6



Let $$(X,d)$$ be a metric space. Suppose that $$F:X\times X \to D$$ is a mapping defined by

$$F(x,y) (t)=F_{x,y}(t)=\epsilon_{0}\bigl(t-d(x,y)\bigr)$$

for all $$x,y\in X$$ and all $$t>0$$.

Then $$(X,F,\Delta_{m})$$ is a Menger PM-space, which is called a Menger PM-space induced by the metric d.

### Remark 1.3

Let $$(X,d)$$ be a metric space. Suppose that $$(X,F,\Delta_{m})$$ is a Menger PM-space induced by d.

Then we have the following.

1. (1)

If $$f:X\to X$$ is continuous in $$(X,d)$$, then it is continuous in $$(X,F,\Delta_{m})$$.

2. (2)

If a sequence $$\{x_{n}\}$$ is convergent to a point x in $$(X,d)$$, then it is convergent to x in $$(X,F,\Delta_{m})$$.

3. (3)

If $$(X,d)$$ is complete, then $$(X,F,\Delta_{m})$$ is complete.

### Lemma 1.7



If X is a nonempty set and $$h:X\to X$$ is a function, then there exists $$Y \subset X$$ such that $$h(Y)=h(X)$$ and $$h:Y\to X$$ is one-to-one.

Let X be a nonempty set, and let $$\Omega=\{(x,x):x\in X\}$$ the diagonal of the Cartesian product $$X\times X$$.

Let G be a directed graph such that the following conditions are satisfied:

1. (1)

the set $$V(G)$$ of its vertices coincides with X, i.e. $$V(G)=X$$;

2. (2)

the set $$E(G)$$ of its edges contains all loops, i.e. $$\Omega\subset E(G)$$.

If G has no parallel edges, then we can identify G with the pair $$(V(G), E(G))$$.

Let $$G=(V(G), E(G))$$ be a directed graph.

Then the conversion of the graph G (denoted by $$G^{-1}$$) is an ordered pair $$(V(G^{-1}), E(G^{-1}))$$ consisting of a set $$V(G^{-1})$$ of vertices and a set $$E(G^{-1})$$ of edges, where

$$V\bigl(G^{-1}\bigr)=V(G) \quad\mbox{and}\quad E\bigl(G^{-1} \bigr)=\bigl\{ (x,y)\in X\times X:(y,x)\in E(G)\bigr\} .$$

Note that $$G^{-1}=(V(G), E(G^{-1}))$$.

Given a directed graph $$G=(V(G), E(G))$$, let $$\widetilde {G}=(V(\widetilde{G}), E(\widetilde{G}))$$ be a directed graph such that

$$V(\widetilde{G})=V(G) \quad\mbox{and}\quad E(\widetilde{G})=E(G)\cup E \bigl(G^{-1}\bigr).$$

For $$x,y\in V(G)$$, let $$p=(x=x_{0}, x_{1}, x_{2}, \ldots, x_{N}=y)$$ be a finite sequence such that

$$(x_{n-1},x_{n})\in E(G) \quad\mbox{for } n=1,2,\ldots, N.$$

Then p is called a path in G from x to y of length N.

Denote $$\Xi(G)$$ by the family of all path in G.

If, for any $$x,y\in V(G)$$, there is a path $$p\in\Xi(G)$$ from x to y, then the graph G called connected. A graph G is called weakly connected, whenever is connected.

Let G be a graph such that $$E(G)$$ is symmetric and $$x\in V(G)$$.

Then the subgraph $$G_{x}=(V(G_{x}),E(G_{x}))$$ is called component of G containing x if and only if there is a path $$p\in \Xi(G)$$ beginning at x such that

$$v\in p \quad\mbox{for all }v\in V(G_{x}) \quad\mbox{and}\quad e \subset p \quad\mbox{for all }e\in E(G_{x}).$$

Define a relation on $$V(G)$$ as follows:

$$(y,z)\in\Re\quad\Longleftrightarrow\quad\mbox{there is a } p\in\Xi(G) \mbox{ from }y \mbox{ to } z.$$

Then the relation is an equivalence relation on $$V(G)$$, and $$[x]_{G}=V(G_{x})$$, where $$[x]_{G}$$ is the equivalence class of $$x\in V(G)$$.

Note that the component $$G_{x}$$ of G containing x is connected.

For the details of the graph theory, we refer to .

Let $$(X,F,\Delta)$$ be a Menger PM-space, and let $$G=(V(G),E(G))$$ be a directed graph such that $$V(G)=X$$ and $$\Omega\subset E(G)$$.

Then the graph G is said to be a C-graph if and only if, for any sequence $$\{x_{n}\}\subset X$$ with $$\lim_{n\to\infty }x_{n}=x_{*}\in X$$, there exist a subsequence $$\{x_{n_{k}}\}$$ of $$\{ x_{n}\}$$ and an $$N\in \mathbb {N}$$ such that $$(x_{n_{k}},x_{*})\in E(G)$$ (resp. $$(x_{*},x_{n_{k}})\in E(G)$$) for all $$k \geq N$$ whenever $$(x_{n},x_{n+1})\in E(G)$$ (resp. $$(x_{n+1},x_{n})\in E(G)$$) for all $$n\in \mathbb {N}$$.

The following definitions are in .

Let $$(X,F,\Delta)$$ be a Menger PM-space, and let $$G=(V(G),E(G))$$ be a directed graph such that $$V(G)=X$$ and $$\Omega\subset E(G)$$. Let $$f:X\to X$$ be a map. Then we say that:

1. (1)

f is continuous if and only if, for any $$x\in X$$ and a sequence $$\{x_{n}\}\subset X$$ with $$\lim_{n\to\infty}x_{n}=x$$,

$$\lim_{n\to\infty}fx_{n}=fx.$$
2. (2)

f is G-continuous if and only if, for any $$x\in X$$ and a sequence $$\{x_{n}\}\subset X$$ with $$\lim_{n\to\infty}x_{n}=x$$ and $$(x_{n},x_{n+1})\in E(G)$$ for all $$n\in \mathbb {N}$$,

$$\lim_{n\to\infty}fx_{n}=fx.$$
3. (3)

f is orbitally continuous if and only if, for all $$x,y\in X$$ and any sequence $$\{k_{n}\}\subset \mathbb {N}$$ with $$\lim_{n\to \infty}f^{k_{n}}x=y$$,

$$\lim_{n\to\infty}ff^{k_{n}}x=fy.$$
4. (4)

f is orbitally G-continuous if and only if, for all $$x,y\in X$$ and any sequence $$\{k_{n}\}\subset \mathbb {N}$$ with $$\lim_{n\to\infty}f^{k_{n}}x=y$$ and $$(f^{k_{n}}x,f^{k_{n}+1}x)\in E(G)$$ for all $$k\in \mathbb {N}$$,

$$\lim_{n\to\infty}ff^{k_{n}}x=fy.$$

## Main results

From now on, let $$(X,F,\Delta)$$ be a Menger PM-space, where Δ is a t-norm of Hadžić-type. Let $$G=(V(G),E(G))$$ be a directed graph satisfying conditions

$$V(G)=X \quad\mbox{and}\quad \Omega\subset E(G).$$

A map $$f:X \to X$$ is said to be a generalized probabilistic G-contraction if and only if the following conditions are satisfied:

1. (1)

f preserves edges of G, i.e. $$(x,y)\in E(G) \Longrightarrow(fx,fy)\in E(G)$$;

2. (2)

there exists $$\phi\in\Phi_{w}$$ such that

$$F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\}$$
(2.1)

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$.

### Theorem 2.1

Let $$(X,F,\Delta)$$ be complete. Suppose that a map $$f:X\to X$$ is a generalized probabilistic G-contraction. Assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$. If either f is orbitally G-continuous or Δ is a continuous t-norm and G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Further if $$(x,y)\in E(G)$$ for any $$x,y\in M$$, where $$M=\{x\in X:(x,fx)\in E(G)\}$$, then f has a unique fixed point.

### Proof

Let $$x_{0}\in X$$ be such that $$(x_{0},fx_{0})\in E(G)$$. Let $$x_{n}=f^{n}x_{0}$$ for all $$n\in \mathbb {N}\cup\{0\}$$.

If there exists $$n_{0}\in \mathbb {N}$$ such that $$x_{n_{0}}=x_{n_{0}+1}$$, then $$x_{n_{0}}=x_{n_{0}+1}=fx_{n_{0}}$$, and so $$x_{n_{0}}$$ is a fixed point of f.

Consider the path p in G from $$x_{0}$$ to $$x_{n_{0}+1}$$:

$$p=(x_{0},x_{1},x_{2}, \ldots, x_{n_{0}}=x_{n_{0}+1})\in\Xi(G).$$

Then the above path is in . Hence, $$x_{n_{0}}=x_{n_{0}+1}\in[x_{0}]_{\widetilde{G}}$$.

Hence, the proof is finished.

Assume that $$x_{n-1}\neq x_{n}$$ for all $$n\in \mathbb {N}$$.

As in the proof of Lemma 1.4, we have $$\phi(t)>0$$ for all $$t>0$$.

Since f is a generalized probabilistic G-contraction, $$(x_{n},x_{n+1})\in E(G)$$ for all $$n=0,1,2,\ldots$$ , and from (2.1) with $$x=x_{n-1}$$, $$y=x_{n}$$ we have

\begin{aligned} F_{x_{n},x_{n+1}}\bigl(\phi(t)\bigr) &=F_{fx_{n-1},fx_{n}}\bigl(\phi(t)\bigr) \\ & \geq\min\bigl\{ F_{x_{n-1},x_{n}}(t),F_{x_{n-1}, fx_{n-1}}(t), F_{x_{n},fx_{n}}(t)\bigr\} \\ & =\min\bigl\{ F_{x_{n-1},x_{n}}(t),F_{x_{n},x_{n+1}}(t)\bigr\} \end{aligned}

for all $$t>0$$ and $$n\in \mathbb {N}$$.

If there exists $$n\in \mathbb {N}$$ such that $$F_{x_{n-1},x_{n}}(t)\geq F_{x_{n},x_{n+1}}(t)$$ for all $$t>0$$, then

$$F_{x_{n},x_{n+1}}\bigl(\phi(t)\bigr)\geq F_{x_{n},x_{n+1}}(t)$$

for all $$t>0$$.

By Lemma 1.3, $$x_{n}=x_{n+1}$$, which is a contradiction. Thus, we have $$F_{x_{n-1},x_{n}}(t)< F_{x_{n},x_{n+1}}(t)$$ for all $$t>0$$ and $$n\in \mathbb {N}$$, and so

$$F_{x_{n},x_{n+1}}\bigl(\phi(t)\bigr)\geq F_{x_{n-1},x_{n}}(t)$$

for all $$t>0$$ and $$n\in \mathbb {N}$$. Thus, we have

$$F_{x_{n},x_{n+1}}\bigl(\phi^{n}(t)\bigr)\geq F_{x_{0},x_{1}}(t)$$

for all $$t>0$$ and $$n\in \mathbb {N}$$.

We now show that

$$\lim_{n\to\infty}F_{x_{n},x_{n+1}}(t)=1$$
(2.2)

for all $$t>0$$. Since $$\lim_{t\to\infty}F_{x_{0},x_{1}}(t)=1$$, for any $$\epsilon\in (0,1)$$ there exists $$t_{0}>0$$ such that

$$F_{x_{0},x_{1}}(t_{0})>1-\epsilon.$$

Because $$\phi\in\Phi_{w}$$, there exists $$t_{1}\geq t_{0}$$ such that

$$\lim_{t\to\infty}\phi^{n}(t_{1})=0.$$

Thus, for each $$t>0$$, there exists N such that $$\phi^{n}(t_{1})< t$$ for all $$n>N$$. Hence, we have

$$F_{x_{n},x_{n+1}}(t)\geq F_{x_{n},x_{n+1}}\bigl(\phi^{n}(t_{1}) \bigr)\geq F_{x_{0},x_{1}}(t_{1})\geq F_{x_{0},x_{1}}(t_{0})>1- \epsilon$$

for all $$n>N$$. Thus, $$\lim_{n\to\infty}F_{x_{n},x_{n+1}}(t)=1$$ for all $$t>0$$.

Next, we show that $$\{x_{n}\}$$ is a Cauchy sequence.

Let $$\epsilon\in(0,1)$$ be given.

Since Δ is of Hadžić-type, there exists $$\lambda\in (0,1)$$ such that

$$\Delta^{n}(s)>1-\epsilon\quad\mbox{for all } n=1,2, \ldots, \mbox{whenever } s>1-\lambda.$$
(2.3)

Since $$\phi\in\Phi_{w}$$, for each $$t>0$$, there exists $$r\geq t$$ such that $$\phi(r)< t$$. From (2.2) we have

$$\lim_{n\to\infty}F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)=1.$$

Thus, there exists $$N_{1}$$ such that

$$F_{x_{n},x_{n+1}}\bigl(t-\phi(r)\bigr)>1-\lambda$$
(2.4)

for all $$n>N_{1}$$.

Since (1.4) is satisfied,

$$F_{x_{n},x_{m}}(t) \geq\Delta^{m-n}\bigl(F_{x_{n},x_{n+1}} \bigl(t-\phi(r)\bigr)\bigr)$$
(2.5)

holds for all $$m\geq n+1$$ by Lemma 1.5.

By applying (2.3) with (2.4) and (2.5),

$$F_{x_{n},x_{m}}(t)>1-\epsilon$$

for all $$m>n>N_{1}$$.

Thus, $$\{x_{n}\}$$ is a Cauchy sequence in X. It follows from the completeness of X that there exists $$x_{*}\in X$$ such that

$$\lim_{n\to\infty}x_{n}=x_{*}.$$

If f is orbitally G-continuous, then $$\lim_{n\to\infty }x_{n}=fx_{*}$$. Hence, $$x_{*}=fx_{*}$$.

Suppose that Δ is continuous and G is C-graph.

Then there exist a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ and an $$N\in \mathbb {N}$$ such that

$$(x_{n_{k}},x_{*})\in E(G)$$

for all $$k\geq N$$. Since f is a generalized probabilistic G-contraction and $$(x_{n_{k}},x_{*})\in E(G)$$ for all $$k\geq N$$, from (2.1) with $$x=x_{n_{k}}$$ and $$y=x_{*}$$ we have

\begin{aligned} &F_{x_{n_{k}+1},fx_{*}}\bigl(\phi(t)\bigr) \\ &\quad=F_{fx_{n_{k}},fx_{*}}\bigl(\phi(t)\bigr) \\ &\quad\geq \min\bigl\{ F_{x_{n_{k}},x_{*}}(t),F_{x_{n_{k}},fx_{n_{k}}}(t),F_{x_{*},fx_{*}}(t) \bigr\} \\ &\quad= \min\bigl\{ F_{x_{n_{k}},x_{*}}(t),F_{x_{n_{k}},x_{n_{k}+1}}(t),F_{x_{*},fx_{*}}(t) \bigr\} \end{aligned}

for all $$t>0$$.

By Lemma 1.4, we obtain

\begin{aligned}& F_{x_{*},fx_{*}}\bigl(\phi(t)\bigr) \\& \quad= \lim_{k\to\infty} \inf F_{x_{n_{k}+1},fx_{*}}\bigl(\phi(t)\bigr) \\& \quad\geq \lim_{k\to\infty} \inf\min\bigl\{ F_{x_{n_{k}},x_{*}}(t), F_{x_{n_{k}},fx_{n_{k}}}(t),F_{x_{*},fx_{*}}(t)\bigr\} \\& \quad= \min\bigl\{ 1,1,F_{x_{*},fx_{*}}(t)\bigr\} \\& \quad= F_{x_{*},fx_{*}}(t) \end{aligned}

for all $$t>0$$. By Lemma 1.3, $$x_{*}=fx_{*}$$.

Consider the path q in G from $$x_{0}$$ to $$x_{*}$$:

$$q=(x_{0},x_{1},x_{2}, \ldots,x_{n_{N}}, x_{*})\in\Xi(G).$$

Then the above path is in . Hence, $$x_{*}\in [x_{0}]_{\widetilde{G}}$$.

Suppose that $$(x,y)\in E(G)$$ for any $$x,y\in M$$.

Let $$x_{*}$$ and $$y_{*}$$ be two fixed point of f.

Then $$x_{*},y_{*}\in M$$. By assumption, $$(x_{*},y_{*})\in E(G)$$.

From (2.1) with $$x=x_{*}$$, $$y=y_{*}$$ we have

\begin{aligned} F_{x_{*},y_{*}}\bigl(\phi(t)\bigr)&=F_{fx_{*},fy_{*}}\bigl(\phi(t)\bigr) \\ &\geq \min\bigl\{ F_{x_{*},y_{*}}(t),F_{x_{*},fx_{*}}(t),F_{y_{*},fy_{*}}(t) \bigr\} \\ &= \min\bigl\{ F_{x_{*},y_{*}}(t),1,1\bigr\} \\ &= F_{y_{*},x_{*}}(t) \end{aligned}

for all $$t>0$$. By Lemma 1.3, $$x_{*}=y_{*}$$. Thus, f has a unique fixed point. □

### Example 2.1

Let $$X=[0,\infty)$$, and let $$d(x,y)=| x-y |$$ for all $$x,y\in X$$.

Let

$$F_{x,y}(t)= \textstyle\begin{cases} \epsilon_{0}(t) &(x=y), \cr D({t\over d(x,y)}) &(x\neq y), \end{cases}$$

for all $$x,y\in X$$ and $$t>0$$, where D is a distribution function defined by

$$D(t)= \textstyle\begin{cases} 0 &(t\leq0), \cr 1-e^{-t} &(t>0). \end{cases}$$

Then $$(X,F,\Delta_{m})$$ is a complete Menger PM-space.

Let $$fx={1\over 2}x$$ for all $$x\in X$$, and let

$$\phi(t)= \textstyle\begin{cases} {1\over 2}t&(0\leq t< 1), \cr -{1\over 3}t+{4\over 3}&(1\leq t\leq{3 \over 2}), \cr t-{2\over 3}&( {3\over 2}< t< \infty). \end{cases}$$

Then $$\phi\in\Phi_{w}$$ and $$\phi(t)\geq{1\over 2}t$$ for all $$t\geq0$$.

Further assume that X is endowed with a graph G consisting of $$V(G)=X$$ and $$E(G)=\{(x,y)\in X\times X:y \preceq x\}$$.

Obviously, f preserves edges, and it is orbitally G-continuous. If $$x_{0}=0$$, then $$(x_{0}, fx_{0})=(0,0)\in E(G)$$.

We have

\begin{aligned} F_{fx,fy}\bigl(\phi(t)\bigr)&=D\biggl({\phi(t) \over {| fx-fy |}}\biggr) \\ &\geq D\biggl({{1\over 2}t \over {1\over 2}t{| x-y |}}\biggr) =D\biggl({t \over t{| x-y |}} \biggr) \\ &= F_{x,y}(t) \\ &\geq \min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} \end{aligned}

for all $$(x,y)\in E(G)$$ and $$t>0$$.

Thus, (2.1) is satisfied. Hence, all the conditions of Theorem 2.1 are satisfied and f has a fixed point $$x_{*}=0\in _{\widetilde{G}}$$. Furthermore, $$M=\{0\}$$ and the fixed point is unique.

### Remark 2.1

Note that in Theorem 2.1 the assumption of orbitally G-continuity can be replaced by orbitally continuity, G-continuity or continuity.

### Remark 2.2

Theorem 2.1 is a generalization of Theorem 3.1 in  to the case of a Menger PM-space endowed with a graph.

### Corollary 2.2

Let $$(X,F,\Delta)$$ be complete, and let $$f:X\to X$$ be a map. Suppose that the following are satisfied:

1. (1)

f preserves edges of G;

2. (2)

there exists $$\phi\in\Phi$$ such that

$$F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\}$$

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$.

Assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$. If either f is orbitally G-continuous or Δ is a continuous t-norm and G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

### Remark 2.3

1. (1)

Corollary 2.2, in part, is a generalization of Theorem 3.9 and Theorem 3.15 of .

2. (2)

In Corollary 2.2, let $$\phi(s)=ks$$ for all $$s\geq 0$$, where $$k\in(0,1)$$. If G is a graph such that $$V(G)=X$$ and $$E(G)=\{(x,y)\in X\times X:\alpha(x,y)\geq1\}$$, where $$\alpha:X\times X \to[0,\infty)$$ is a function, then Corollary 2.2 reduces to Theorem 2.1 of .

3. (3)

If G is a graph such that $$V(G)=X$$ and $$E(G)=\{(x,y)\in X\times X:x\preceq y\}$$, where is a partial order on X, then Corollary 2.2 become to Theorem 2.1 of .

### Corollary 2.3

Let $$(X,F,\Delta)$$ be complete. Suppose that a map $$f:X\to X$$ is generalized probabilistic G-contraction. Assume that either f is continuous or Δ is a continuous t-norm and G is a C-graph.

Then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$ for some $$x_{0}\in Q$$ if and only if $$Q\neq\emptyset$$, where $$Q=\{x\in X:(x,fx)\in E(\widetilde{G})\}$$. Further if, for any $$x,y\in Q$$, $$(x,y)\in E(\widetilde{G})$$ then f has a unique fixed point.

### Proof

If f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$, say $$x_{*}$$, then $$(x_{*},fx_{*})=(x_{*},x_{*})\in\Omega\subset E(\widetilde{G})$$. Thus, $$Q\neq\emptyset$$.

Suppose that $$Q\neq\emptyset$$.

Then there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(\widetilde{G})$$.

We have two cases: $$(x_{0},fx_{0})\in E(G)$$ or $$(x_{0},fx_{0})\in E(G^{-1})$$.

If $$(x_{0},fx_{0})\in E(G)$$, then following Theorem 2.1 f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Assume that $$(x_{0},fx_{0})\in E(G^{-1})$$.

Then $$(fx_{0},x_{0})\in E(G)$$. Since f is preserves edges of G, $$(f^{n+1}x_{0},f^{n}x_{0})\in E(G)$$ for all $$n\in \mathbb {N}\cup\{0\}$$.

In the same way as the proof of Theorem 2.1 with condition (PM2), we deduce that f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Suppose that, for any $$x,y\in Q$$, $$(x,y)\in E(\widetilde{G})$$.

Let $$x_{*}$$ and $$y_{*}$$ be two fixed points of f.

Then $$x_{*},y_{*}\in Q$$. By assumption, $$(x_{*},y_{*})\in E(\widetilde{G})$$.

If $$(x_{*},y_{*})\in E(G)$$, then

$$F_{x_{*},y_{*}}\bigl(\phi(t)\bigr)\geq\min\bigl\{ F_{x_{*},y_{*}}(t),F_{x_{*},x_{*}}(t),F_{y_{*},y_{*}}(t) \bigr\} =F_{x_{*},y_{*}}(t)$$

for all $$t>0$$. By Lemma 1.1, $$x_{*}=y_{*}$$.

Let $$(x_{*},y_{*})\in E(G^{-1})$$, then $$(y_{*},x_{*})\in E(G)$$.

Then

$$F_{y_{*},x_{*}}\bigl(\phi(t)\bigr)\geq F_{y_{*},x_{*}}(t)$$

for all $$t>0$$. Hence, $$y_{*}=x_{*}$$. Thus, f has a unique fixed point. □

### Remark 2.4

If $$\phi\in\Phi$$ and G is a graph such that $$V(G)=X$$ and $$E(G)=\{ (x,y)\in X\times X:{x\preceq y}\}$$, where is a partial order on X, then Corollary 2.3 reduces to Theorem 2.2 of .

In the following result, we can drop continuity of the t-norm Δ.

### Corollary 2.4

Let $$(X,F,\Delta)$$ be complete. Suppose that a map $$f:X\to X$$ satisfies

$$F_{fx,fy}\bigl(\phi(t)\bigr)\geq F_{x,y}(t)$$
(2.6)

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$, where $$\phi\in \Phi_{w}$$.

Assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$. If either f is orbitally G-continuous or G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Further if $$(x,y)\in E(G)$$ for any $$x,y\in M$$, where $$M=\{x\in X:(x,fx)\in E(G)\}$$, then f has a unique fixed point.

### Proof

Let $$x_{0}\in X$$ be such that $$(x_{0},fx_{0})\in E(G)$$, and let $$x_{n}=f^{n}x_{0}$$ for all $$n\in \mathbb {N}\cup\{0\}$$.

Note that (2.6) to be satisfied implies that (2.1) is satisfied.

As in the proof of Theorem 2.1, $$x_{n-1}\neq x_{n}$$ and $$(x_{n-1},x_{n})\in E(G)$$ for all $$n\in \mathbb {N}$$ and there exists

$$\lim_{n\to\infty}x_{n}=x_{*}\in X.$$

If f is orbitally G-continuous, then $$\lim_{n\to\infty }x_{n}=fx_{*}$$, and so $$x_{*}=fx_{*}$$.

Assume that G is a C-graph.

Then there exist a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ and an $$N\in \mathbb {N}$$ such that

$$(x_{n_{k}},x_{*})\in E(G)$$

for all $$k\geq N$$.

Since $$\phi\in\Phi_{w}$$, for each $$t>0$$, there exists $$r\geq t$$ such that $$\phi(r)< t$$.

We have

\begin{aligned} &F_{x_{*},fx_{*}}(t) \\ &\quad\geq\Delta\bigl(F_{x_{*},x_{n_{k}+1}}\bigl(t-\phi(r)\bigr),F_{fx_{n_{k}},fx_{*}} \bigl(\phi (r)\bigr)\bigr) \\ &\quad\geq\Delta\bigl(F_{x_{*},x_{n_{k}+1}}\bigl(t-\phi(r)\bigr),F_{x_{n_{k}},x_{*}}(r) \bigr) \\ &\quad\geq\Delta\bigl(F_{x_{*},x_{n_{k}+1}}\bigl(t-\phi(r)\bigr),F_{x_{n_{k}},x_{*}}(t) \bigr) \\ &\quad\geq \Delta(a_{n},a_{n}) \end{aligned}
(2.7)

for all $$t>0$$, where $$a_{n}=\min\{F_{x_{*},x_{n_{k}+1}}(t-\phi (r)),F_{x_{n_{k}},x_{*}}(t)\}$$.

Since $$\lim_{n\to\infty}a_{n}=1$$ and $$\Delta(t,t)$$ is continuous at $$t=1$$, $$\lim_{n\to\infty}\Delta(a_{n},a_{n})=\Delta(1,1)=1$$. Hence, from (2.7) we have $$F_{x_{*},fx_{*}}(t)=1$$ for all $$t>0$$, and so $$x_{*}=fx_{*}$$. □

### Remark 2.5

Corollary 2.4 is a generalization of Theorem 3.1 in  to the case of a Menger PM-space endowed with a graph.

### Theorem 2.5

Let $$(X,F,\Delta)$$ be complete such that Δ is continuous. Let $$f,h:X\to X$$ be maps, and let G be a directed graph satisfying $$V(G)=h(X)$$ and $$\{(hx,hx):x\in X\}\subset E(G)$$. Suppose that the following are satisfied:

1. (1)

$$f(X) \subset h(X)$$;

2. (2)

$$h(X)$$ is closed;

3. (3)

$$(hx,hy)\in E(G)$$ implies $$(fx,fy)\in E(G)$$;

4. (4)

there exists $$x_{0}\in X$$ such that $$(hx_{0},fx_{0})\in E(G)$$;

5. (5)

there exists $$\phi\in\Phi_{w}$$ such that

$$F_{fx,fy}\bigl(\phi(t)\bigr) \geq\min\bigl\{ F_{hx,hy}(t),F_{hx,fx}(t),F_{hy,fy}(t)\bigr\}$$
(2.8)

for all $$x,y\in X$$ with $$(hx,hy)\in E(G)$$ and all $$t>0$$;

6. (6)

if $$\{x_{n}\}$$ is a sequence in X such that $$(hx_{n},hx_{n+1})\in E(G)$$ for all $$n\in \mathbb {N}\cup\{0\}$$ and $$\lim_{n\to\infty}hx_{n}=hu$$ for some $$u\in X$$, then $$(hx_{n},hu)\in E(G)$$ for all $$n\in \mathbb {N}\cup\{0\}$$.

Then f and h have a coincidence point in X. Further if f and h commute at their coincidence points and $$(hu,hhu)\in E(G)$$, then f and h have a common fixed point in X.

### Proof

By Lemma 1.7, there exists $$Y\subset X$$ such that $$h(Y)=h(X)$$ and $$h:Y\to X$$ is one-to-one. Define a mapping $$U:h(Y) \to h(Y)$$ by $$U(hx)=fx$$. Since $$h:Y\to X$$ is one-to-one, U is well defined.

By (3), $$(hx,hy)\in E(G)$$ implies $$(U(hx),U(hy))\in E(G)$$.

By (4), $$(hx_{0},U(hx_{0}))\in E(G)$$ for some $$x_{0}\in X$$. We have

\begin{aligned}& F_{U(hx),U(hy)}\bigl(\phi(t)\bigr) \\& \quad= F_{fx,fy}\bigl(\phi(t)\bigr) \\& \quad\geq \min\bigl\{ F_{hx,hy}(t),F_{hx,fx}(t),F_{hy,fy}(t) \bigr\} \\& \quad= \min\bigl\{ F_{hx,hy}(t),F_{hx,U(hx)}(t),F_{hy,U(hy)}(t) \bigr\} \end{aligned}

for all $$hx,hy\in h(Y)$$ with $$(hx,hy)\in E(G)$$. Since $$h(Y)=h(X)$$ is complete, by applying Theorem 2.1, there exists $$u\in X$$ such that $$U(hu)=hu$$, and so $$hu=fu$$. Hence, u is a coincidence point of f and h.

Suppose that f and h commute at their coincidence points and $$(hu,hhu)\in E(G)$$. Let $$w=hu=fu$$. Then $$fw=fhu=hfu=hw$$, and $$(hu,hw)=(hu,hhu)\in E(G)$$.

Applying inequality (2.8) with $$x=u$$, $$y=w$$, we have

\begin{aligned}& F_{w,fw}\bigl(\phi(t)\bigr) \\& \quad= F_{fu,fw}\bigl(\phi(t)\bigr) \\& \quad\geq \min\bigl\{ F_{hu,hw}(t),F_{hu,fu}(t),F_{hw,fw}(t) \bigr\} \\& \quad= \min\bigl\{ F_{w,fw}(t),F_{w,w}(t),F_{fw,fw}(t) \bigr\} \\& \quad= \min\bigl\{ F_{w,fw}(t),1,1\bigr\} \\& \quad= F_{fw,w}(t) \end{aligned}

for all $$t>0$$.

By Lemma 1.2, $$w=fw$$. Hence $$w=fw=hw$$. Thus, w is a common fixed point of f and h. □

### Remark 2.6

Theorem 2.5 is a generalization of Theorem 3.4 of . If we have $$\phi(s)=ks$$ for all $$s\geq0$$, where $$k\in(0,1)$$, and $$V(G)=X$$ and $$E(G)=\{(x,y):x\leq y\}$$, where ≤ is a partial order on X, then Theorem 2.5 reduces to Theorem 3.4 of .

### Theorem 2.6

Let $$(X,F,\Delta)$$ be complete. Suppose that maps $$f_{0},f_{1}:X\to X$$ satisfy the following:

$$F_{f_{0}x,f_{0}y}\bigl(\phi(t)\bigr)\geq F_{x,y}(t) ,$$
(2.9)

where $$\phi\in\Phi_{w}$$ and

$$F_{f_{1}x,f_{1}y}(t)\geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\}$$
(2.10)

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$.

Suppose that f preserves edges, and assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$, where $$f=f_{0}f_{1}$$. If either f is orbitally G-continuous or Δ is a continuous t-norm and G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Further if $$(x,y)\in E(G)$$ for any $$x,y\in M$$, where $$M=\{x\in X:(x,fx)\in E(G)\}$$, then $$f_{0}$$ and $$f_{1}$$ have a common fixed point whenever $$f_{0}$$ is commutative with $$f_{1}$$.

### Proof

From (2.9) and (2.10) we have

$$F_{fx,fy}\bigl(\phi(t)\bigr)\geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\}$$

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$. By Theorem 2.1, f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$, say $$x_{*}$$.

Suppose that $$(x,y)\in E(G)$$ for any $$x,y\in M$$.

Then from Theorem 2.1 f has a unique fixed point.

Since $$f_{0}$$ is commutative with $$f_{1}$$ and $$fx_{*}=x_{*}$$, $$ff_{0}x_{*}=f_{0}(f_{1}f_{0}x_{*}) =f_{0}(f_{0}f_{1}x_{*})=f_{0}fx_{*}=f_{0}x_{*}$$. Similarly, we obtain $$ff_{1}x_{*}=f_{1}x_{*}$$. From the uniqueness of fixed point of f, we have $$x_{*}=f_{0}x_{*}=f_{1}x_{*}$$. □

### Example 2.2

Let $$X=[0,\infty)$$, and let $$F_{x,y}(t)= {t\over {t+d(x,y)}}$$ for all $$x,y \in X$$ and all $$t>0$$, where

$$d(x,y)= \textstyle\begin{cases} \max\{x,y\} &(x\neq y), \cr 0 &(\mbox{otherwise}). \end{cases}$$

Then $$(X,F,\Delta_{m})$$ is a complete Menger PM-space.

Let

$$\phi(t)= \textstyle\begin{cases}{1\over 2}t&(0\leq t< 1), \cr -{1\over 3}t+{4\over 3}&(1\leq t\leq{3 \over 2}), \cr t-{2\over 3}&( {3\over 2}< t< \infty). \end{cases}$$

Then $$\phi\in\Phi_{w}$$ and $$\phi(t)\geq{1\over 2}t$$ for all $$t\geq0$$.

Further assume that X is endowed with a graph G consisting of $$V(G)=X$$ and $$E(G)=\{(x,y)\in X\times X:y \preceq x\}$$.

Obviously, G is a C-graph.

Let $$f_{0}:X\to X$$ be a map defined by $$f_{0}x={1\over 2}x$$ for all $$x\geq0$$, and define a map $$f_{1}:X\to X$$ by

$$f_{1}x= \textstyle\begin{cases} {x\over 4(1+x)} &(0\leq x\leq2), \cr {1\over 12}x &(x>2). \end{cases}$$

Then

$$fx=f_{0}f_{1}x= \textstyle\begin{cases} {x\over 8(1+x)} &(0\leq x\leq2), \cr {1\over 24}x &(x>2). \end{cases}$$

Obviously, f preserves edges.

Let $$(x,y)\in E(G)$$.

Then $$y\preceq x$$, and we obtain

\begin{aligned} F_{f_{0}x,f_{0}y}\bigl(\phi(t)\bigr)&={\phi(t) \over {\phi(t)+d({1\over 2}x,{1\over 2}y)}} \\ &\geq {{1\over 2}t \over {{1\over 2}t+{1\over 2}x}}={t \over {t+x}} \\ &= {t \over {t+\max\{x,y\}}}=F_{x,y}(t) \end{aligned}

for all $$t>0$$. Hence, (2.9) is satisfied.

We consider the following three cases:

Case 1. $$0\leq y< x\leq2$$:

\begin{aligned} F_{f_{1}x,f_{1}y}(t)&={t \over {t+d({x\over 4(1+x)}, {y\over 4(1+y)})}} \\ &= {t \over {t+{x\over 4(1+x)}}} \geq{t \over {t+x}} \\ &= {t \over {t+\max\{x,y\}}} ={t \over {t+d(x,y)}} =F_{x,y}(t) \\ &\geq \min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} \end{aligned}

for all $$t>0$$.

Case 2. $$2< y< x$$:

\begin{aligned} F_{f_{1}x,f_{1}y}(t)&={t \over {t+d({x\over 12}, {y\over 12})}} \\ &={t \over {t+{x\over 12}}} \geq{t \over {t+x}} ={t \over {t+\max\{x,y\}}} \\ &={t \over {t+d(x,y)}} =F_{x,y}(t) \\ &\geq \min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} \end{aligned}

for all $$t>0$$.

Case 3. $$0\leq y\leq2$$ and $$2< x$$:

\begin{aligned} F_{f_{1}x,f_{1}y}(t)&={t \over {t+d({x\over 12}, {y\over 4(1+y)})}} \\ &={t \over {t+{x\over 12}}} \geq{t \over {t+x}} ={t \over {t+\max\{x,y\}}} \\ &={t \over {t+d(x,y)}} =F_{x,y}(t) \\ &\geq \min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\} \end{aligned}

for all $$t>0$$.

Thus, (2.10) is satisfied.

For $$x_{0}=4$$, $$(x_{0},fx_{0})=(4,{1\over 6})\in E(G)$$. Hence, all the conditions of Theorem 2.6 are satisfied and f has a fixed point $$x_{*}=0\in[x_{0}]_{\widetilde{G}}$$.

### Corollary 2.7

Let $$(X,F,\Delta)$$ be complete. Suppose that maps $$f_{0},f_{1}:X\to X$$ satisfy the following:

$$F_{f_{0}x,f_{0}y}\bigl(\phi(t)\bigr)\geq F_{x,y}(t) ,$$
(2.11)

where $$\phi\in\Phi_{w}$$ and

$$F_{f_{1}x,f_{1}y}(t)\geq F_{x,y}(t)$$
(2.12)

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$.

Suppose that f preserves edges, and assume that there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$, where $$f=f_{0}f_{1}$$. If f is orbitally G-continuous or G is a C-graph, then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

Further if $$(x,y)\in E(G)$$ for any $$x,y\in M$$, where $$M=\{x\in X:(x,fx)\in E(G)\}$$, then $$f_{0}$$ and $$f_{1}$$ have a common fixed point whenever $$f_{0}$$ is commutative with $$f_{1}$$.

### Proof

From (2.11) and (2.12) we have

$$F_{fx,fy}\bigl(\phi(t)\bigr)\geq F_{x,y}(t)$$

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$ and all $$t>0$$. By Corollary 2.4, f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$, say $$x_{*}$$.

Suppose that $$(x,y)\in E(G)$$ for any $$x,y\in M$$.

Then from Corollary 2.4 f has a unique fixed point.

Since $$f_{0}$$ is commutative with $$f_{1}$$, as in the proof of Theorem 2.6 we have $$x_{*}= f_{0}x_{*}= f_{1}x_{*}$$. □

### Remark 2.7

Corollary 2.7 is a generalization of Corollary 2.1 of  to the case of Menger PM-space endowed with a graph.

### Corollary 2.8

Let $$(X,d)$$ be a complete metric space, and let $$G=(V(G),E(G))$$ be a directed graph satisfying $$V(G)=X$$ and $$\Omega\subset E(G)$$. Let $$f:X\to X$$ be a map. Suppose that the following are satisfied:

1. (1)

$$(x,y)\in E(G)$$ implies $$(fx,fy)\in E(G)$$;

2. (2)

there exists $$\phi\in\Phi_{w}$$ such that

\begin{aligned} &d(fx,fy) \\ &\quad\leq\phi\bigl(\max\bigl\{ d(x,y),d(x,fx),d(y,fy)\bigr\} \bigr) \end{aligned}
(2.13)

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$, where ϕ is nondecreasing;

3. (3)

there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$;

4. (4a)

f is continuous, or

5. (4b)

if $$\{x_{n}\}$$ is a sequence in X such that $$\lim_{n\to \infty}x_{n}=x_{*}\in X$$ and $$(x_{n},x_{n+1})\in E(G)$$ for all $$n\in \mathbb {N}$$, then there exists a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ such that $$(x_{n_{k}},x_{*})\in E(G)$$ for all $$k\in \mathbb {N}$$.

Then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

### Proof

Suppose that equality holds in (2.13) and $$x\neq fx$$ for all $$x\in X$$.

Let $$x_{0}\in X$$ be fixed. Then $$(x_{0},x_{0})\in E(G)$$, and from (2.13) we have

\begin{aligned}& 0=d(fx_{0},fx_{0}) \\& \quad= \phi\bigl(\max\bigl\{ d(x_{0},x_{0}),d(x_{0},fx_{0}),d(x_{0},fx_{0}) \bigr\} \bigr) \\& \quad= \phi\bigl(d(x_{0},fx_{0})\bigr), \end{aligned}

which implies $$d(x_{0},fx_{0})=0$$ and so $$x_{0}=fx_{0}$$, which is a contradiction.

Thus, if equality holds in (2.13), then f has a fixed point.

Assume that equality is not satisfied in (2.13).

Let $$(X,F, \Delta_{m})$$ be the induced Menger PM-space by $$(X,d)$$.

By Lemma 1.6, $$(X,F, \Delta_{m})$$ is complete. By Remark 1.3, (4a) implies f is continuous in $$(X,F, \Delta_{m})$$, and (4b) implies G is C-graph.

We show that (2.1) is satisfied.

We know that the values of each distribution function $$F_{u,v}(\cdot)$$, $$u,v\in X$$, in the induced Menger PM-space only can equal 0 or 1. Hence, without loss of generality, we may assume that

$$\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t)\bigr\} =1$$

for all $$x,y\in E(G)$$ and $$t>0$$. Then

$$t>d(x,y),\qquad t>d(x,fx) \quad\mbox{and}\quad t>d(y,fy).$$

Thus,

$$t>\max\bigl\{ d(x,y),d(x,fx),d(y,fy)\bigr\} .$$

Since ϕ is nondecreasing,

$$\phi\bigl(\max\bigl\{ d(x,y),d(x,fx),d(y,fy)\bigr\} \bigr)\leq\phi(t).$$

By assumption, we have

$$d(fx,fy)< \phi(t).$$

Hence, $$\phi(t)-d(fx,fy)>0$$. So $$F_{fx,fy}(\phi(t))=1$$. Thus we have

$$F_{fx,fy}\bigl(\phi(t)\bigr)\geq\min\bigl\{ F_{x,y}(t),F_{x,fx}(t),F_{y,fy}(t) \bigr\}$$

for all $$x,y\in X$$ with $$(x,y) \in E(G)$$ and all $$t>0$$.

Hence, (2.1) is satisfied. By Theorem 2.1 and Remark 2.1, f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$. □

### Corollary 2.9

Let $$(X,d)$$ be a complete metric space, and let $$G=(V(G),E(G))$$ be a directed graph satisfying $$V(G)=X$$ and $$\Omega\subset E(G)$$. Let $$f:X\to X$$ be a map.

Suppose that the following are satisfied:

1. (1)

$$(x,y)\in E(G)$$ implies $$(fx,fy)\in E(G)$$;

2. (2)

there exists $$\phi\in\Phi_{w}$$ such that

$$d(fx,fy)\leq\phi\bigl(d(x,y)\bigr)$$

for all $$x,y\in X$$ with $$(x,y)\in E(G)$$, where ϕ is nondecreasing;

3. (3)

there exists $$x_{0}\in X$$ such that $$(x_{0},fx_{0})\in E(G)$$;

4. (4)

either f is continuous or if $$\{x_{n}\}$$ is a sequence in X such that $$\lim_{n\to\infty}x_{n}=x_{*}\in X$$ and $$(x_{n},x_{n+1})\in E(G)$$ for all $$n\in \mathbb {N}$$, then there exists a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ such that $$(x_{n_{k}},x_{*})\in E(G)$$ for all $$k\in \mathbb {N}$$.

Then f has a fixed point in $$[x_{0}]_{\widetilde{G}}$$.

### Remark 2.8

Corollary 2.9 is a generalization of the results of . If we have a graph G such that $$V(G)=X$$ and $$E(G)=\{(x,y)\in X\times X:x\preceq y\}$$, where is a partial order on X, and $$\phi (s)=ks$$ for all $$s\geq0$$, where $$k\in[0,1)$$, then Corollary 2.9 reduces to Theorem 2.1 and Theorem 2.2 of .

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