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Generalized probabilistic Gcontractions
Fixed Point Theory and Applications volume 2016, Article number: 50 (2016)
Abstract
In this paper, the notion of generalized probabilistic Gcontractions 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 [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 αψcontractive 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 αcontractive 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 Gcontractions in Menger PMspaces 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 PMspace, where Δ is of Hadžićtype. 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 \((x,y)\in E(G)\),
Assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\). If either f is orbitally Gcontinuous or G is a Cgraph, 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 Gcontractions in Menger PMspaces 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 leftcontinuous;

(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
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 tnorm).
We denote \(\mathbb {N}\) by the set of all natural numbers.
For a tnorm Δ, we consider the following notation:
A tnorm Δ is said to be of Hadžićtype [14] whenever the family of \(\{\Delta^{n}(t)\}_{n=1}^{\infty}\) is equicontinuous at \(t=1\).
For example, the minimum tnorm \(\Delta_{m}\) defined by
is of Hadžićtype.
It is easy to see that the following are equivalent (see [14]):

(1)
for a tnorm 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
Let X be a nonempty set, and let Δ be a tnorm. 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 3tuple \((X,F,\Delta)\) is called a Menger probabilistic metric space (briefly, Menger PMspace) [16, 17].
Let \((X,F,\Delta)\) be a Menger PMspace 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:
The family
does not necessarily determine a topology on X (see [19, 20]).
It is well known that if Δ satisfies condition
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)
Let \((X,F,\Delta)\) be a Menger PMspace, 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
Let
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 PMspace (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
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
for all \(t>0\). Hence, \((X,F,\Delta_{m})\) is complete.
From now on, let
and let
Note that \(\Phi\subset\Phi_{w}\).
Fang [23] gave the corrected version of Theorem 12 of [11] by introducing the notion of rightlocally monotone functions as follows: \(\phi:[0,\infty) \to[0,\infty)\) is rightlocally 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 rightlocally 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 PMspace, and let \(x,y\in X\). If
for all \(t>0\), where \(\phi\in\Phi_{w}\), then \(x=y\).
Lemma 1.4
[18]
Let \((X,F,\Delta)\) be a Menger PMspace 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 PMspace, where Δ is of Hadžićtype. Let \(\{x_{n}\}\) be a sequence of points in X such that \(x_{n1}\neq x_{n}\) for all \(n\in \mathbb {N}\). If there exists \(\phi\in\Phi_{w}\) such that
for all \(s>0\) and all \(n,m\in \mathbb {N}\), then for each \(t>0\) there exists \(r\geq t\) such that
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
which is a contradiction.
We claim that
From (1.4) we have
for all \(s>0\) and all \(n\in \mathbb {N}\).
If there exists \(n\in \mathbb {N}\) such that \(F_{x_{n1},x_{n}}(s)\geq F_{x_{n},x_{n+1}}(s)\) 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 \(F_{x_{n1},x_{n}}(s)< F_{x_{n},x_{n+1}}(s)\) for all \(s>0\) and \(n \in \mathbb {N}\), and so
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
Hence,
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
By induction, we show that (1.5) holds.
Let \(m=n+1\).
Then
Thus, (1.5) holds for \(m=n+1\).
Assume that (1.5) holds for some fixed \(m>n+1\). That is,
Then
From (1.4) we obtain
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
Thus, from (1.8) and (1.9) we have
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
for all \(x,y\in X\) and all \(t>0\).
Then \((X,F,\Delta_{m})\) is a Menger PMspace, which is called a Menger PMspace induced by the metric d.
Remark 1.3
Let \((X,d)\) be a metric space. Suppose that \((X,F,\Delta_{m})\) is a Menger PMspace 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 onetoone.
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
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
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
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
Define a relation ℜ on \(V(G)\) as follows:
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 PMspace, 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 Cgraph 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 PMspace, 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 Gcontinuous 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 Gcontinuous 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 PMspace, where Δ is a tnorm of Hadžićtype. Let \(G=(V(G),E(G))\) be a directed graph satisfying conditions
A map \(f:X \to X\) is said to be a generalized probabilistic Gcontraction 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 Gcontraction. Assume that there exists \(x_{0}\in X\) such that \((x_{0},fx_{0})\in E(G)\). If either f is orbitally Gcontinuous or Δ is a continuous tnorm and G is a Cgraph, 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 \(x_{n_{0}+1}\):
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 \(x_{n1}\neq x_{n}\) 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 Gcontraction, \((x_{n},x_{n+1})\in E(G)\) for all \(n=0,1,2,\ldots\) , and from (2.1) with \(x=x_{n1}\), \(y=x_{n}\) we have
for all \(t>0\) and \(n\in \mathbb {N}\).
If there exists \(n\in \mathbb {N}\) such that \(F_{x_{n1},x_{n}}(t)\geq F_{x_{n},x_{n+1}}(t)\) for all \(t>0\), then
for all \(t>0\).
By Lemma 1.3, \(x_{n}=x_{n+1}\), which is a contradiction. Thus, we have \(F_{x_{n1},x_{n}}(t)< F_{x_{n},x_{n+1}}(t)\) for all \(t>0\) and \(n\in \mathbb {N}\), and so
for all \(t>0\) and \(n\in \mathbb {N}\). Thus, we have
for all \(t>0\) and \(n\in \mathbb {N}\).
We now show that
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
Because \(\phi\in\Phi_{w}\), there exists \(t_{1}\geq t_{0}\) such that
Thus, for each \(t>0\), there exists N such that \(\phi^{n}(t_{1})< t\) for all \(n>N\). Hence, we have
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 Hadžićtype, there exists \(\lambda\in (0,1)\) such that
Since \(\phi\in\Phi_{w}\), for each \(t>0\), there exists \(r\geq t\) such that \(\phi(r)< t\). From (2.2) we have
Thus, there exists \(N_{1}\) such that
for all \(n>N_{1}\).
Since (1.4) is satisfied,
holds for all \(m\geq n+1\) by Lemma 1.5.
By applying (2.3) with (2.4) and (2.5),
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
If f is orbitally Gcontinuous, then \(\lim_{n\to\infty }x_{n}=fx_{*}\). Hence, \(x_{*}=fx_{*}\).
Suppose that Δ is continuous and G is Cgraph.
Then there exist a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) and an \(N\in \mathbb {N}\) such that
for all \(k\geq N\). Since f is a generalized probabilistic Gcontraction 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
for all \(t>0\).
By Lemma 1.4, we obtain
for all \(t>0\). By Lemma 1.3, \(x_{*}=fx_{*}\).
Consider the path q in G from \(x_{0}\) to \(x_{*}\):
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
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 \(d(x,y)= xy \) for all \(x,y\in X\).
Let
for all \(x,y\in X\) and \(t>0\), where D is a distribution function defined by
Then \((X,F,\Delta_{m})\) is a complete Menger PMspace.
Let \(fx={1\over 2}x\) for all \(x\in X\), and let
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 Gcontinuous. If \(x_{0}=0\), then \((x_{0}, fx_{0})=(0,0)\in E(G)\).
We have
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 Gcontinuity can be replaced by orbitally continuity, Gcontinuity or continuity.
Remark 2.2
Theorem 2.1 is a generalization of Theorem 3.1 in [23] to the case of a Menger PMspace 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 Gcontinuous or Δ is a continuous tnorm and G is a Cgraph, 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 Gcontraction. Assume that either f is continuous or Δ is a continuous tnorm and G is a Cgraph.
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
for all \(t>0\). By Lemma 1.1, \(x_{*}=y_{*}\).
Let \((x_{*},y_{*})\in E(G^{1})\), then \((y_{*},x_{*})\in E(G)\).
Then
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 tnorm Δ.
Corollary 2.4
Let \((X,F,\Delta)\) be complete. Suppose that a map \(f:X\to X\) satisfies
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 Gcontinuous or G is a Cgraph, 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, \(x_{n1}\neq x_{n}\) and \((x_{n1},x_{n})\in E(G)\) for all \(n\in \mathbb {N}\) and there exists
If f is orbitally Gcontinuous, then \(\lim_{n\to\infty }x_{n}=fx_{*}\), and so \(x_{*}=fx_{*}\).
Assume that G is a Cgraph.
Then there exist a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) and an \(N\in \mathbb {N}\) such that
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
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 PMspace 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 onetoone. Define a mapping \(U:h(Y) \to h(Y)\) by \(U(hx)=fx\). Since \(h:Y\to X\) is onetoone, 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
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
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:
where \(\phi\in\Phi_{w}\) and
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 Gcontinuous or Δ is a continuous tnorm and G is a Cgraph, 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
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
Then \((X,F,\Delta_{m})\) is a complete Menger PMspace.
Let
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 Cgraph.
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
Then
Obviously, f preserves edges.
Let \((x,y)\in E(G)\).
Then \(y\preceq x\), and we obtain
for all \(t>0\). Hence, (2.9) is satisfied.
We consider the following three cases:
Case 1. \(0\leq y< x\leq2\):
for all \(t>0\).
Case 2. \(2< y< x\):
for all \(t>0\).
Case 3. \(0\leq y\leq2\) and \(2< x\):
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:
where \(\phi\in\Phi_{w}\) and
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 Gcontinuous or G is a Cgraph, 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
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 PMspace 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
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 PMspace 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 Cgraph.
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 PMspace only can equal 0 or 1. Hence, without loss of generality, we may assume that
for all \(x,y\in E(G)\) and \(t>0\). Then
Thus,
Since ϕ is nondecreasing,
By assumption, we have
Hence, \(\phi(t)d(fx,fy)>0\). So \(F_{fx,fy}(\phi(t))=1\). Thus we have
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].
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MSC
 47H10
 54H25
Keywords
 fixed point
 coincidence point
 directed graph
 Menger probabilistic metric space