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. (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 [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. (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 [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:

  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 [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. (1)

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

  2. (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. (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 [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. (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

[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. (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

[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. (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 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. (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 \(k\in \mathbb {N}\),

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

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. (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 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. (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 [13].

  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 [9].

  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 [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. (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 [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. (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 [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].