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.

1 Introduction and preliminaries

Ran and Reurings [1] gave a generalization of Banach contraction principle to partially ordered metric spaces. Since then, many authors obtained generalization and extension of the results of [2–7].

In particular, Ćirić et al. [3] extended the results of [1, 5, 6] to partially ordered Menger probabilistic metric spaces.

Samet et al. [8] introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems for such mappings in complete metric spaces.

Cho [9] obtained a generalization of the results of [3] by introducing the concept of α-contractive type mappings in Menger probabilistic metric spaces.

Recently, Wu [10] obtained a generalization of the results of [3], and improved and extended the fixed point results of [4, 11, 12]. Also, Kamran et al. [13] 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) f is nondecreasing and left-continuous;

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

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

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

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

• (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 [14] 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

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

is of Hadžić-type.

It is easy to see that the following are equivalent (see [14]):

• (1) for a t-norm Δ,

  $$ \mbox{it is of Had\v{z}i\'{c}-type}; $$

  (1.1)

• (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 [15]). 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:

• (PM1) $F_{x,y}(t)=\epsilon_{0}(t)$ for all $t\in \mathbb {R}$ if and only if $x=y$;

• (PM2) $F_{x,y}=F_{y,x}$ for all $x,y\in X$;

• (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 [18] 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) condition (1.3) is the weakest condition which ensure the existence of the $(\epsilon,\lambda)$-topology (see [19]);

• (2) condition (1.1) ⟹ condition (1.3) (see [22]).

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) $\{x_{n}\}$ 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) $\{x_{n}\}$ 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) $(X,F,\Delta)$ 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 [18]).

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 [23] gave the corrected version of Theorem 12 of [11] 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

[23]

The following are satisfied:

• (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) 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) 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

[23]

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

Lemma 1.3

[23]

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

[18]

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

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

which is a contradiction.

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

[24]

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

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

Lemma 1.7

[25]

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

• (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 G̃ 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 ℜ 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 ℜ 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 [26].

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 [13].

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) 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) 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) 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) 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 $k\in \mathbb {N}$,

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

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

• (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 G̃. 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 G̃. 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 $x_{*}=0\in [0]_{\widetilde{G}}$. 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 [23] 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) f preserves edges of G;

• (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) Corollary 2.2, in part, is a generalization of Theorem 3.9 and Theorem 3.15 of [13].

• (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 [9].

• (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 [10].

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 [10].

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 [23] 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) $f(X) \subset h(X)$;

• (2) $h(X)$ is closed;

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

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

• (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) 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 [3]. 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 [3].

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

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

• (4a) f is continuous, or

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

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

• (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 [5]. 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 [5].