Abstract
In this paper, we introduce the concepts of generalized probabilistically bounded set \(\Omega^{*}\) and Menger-Hausdorff metric \(\widetilde{G}^{*}\) in Menger probabilistic G-metric spaces, and prove that \((\Omega^{*},\widetilde{G}^{*},\Delta)\) is also a Menger probabilistic G-metric space. Utilizing these concepts, we establish some common fixed point theorems for three hybrid pairs of mappings satisfying the common property \((E.A)\) in Menger probabilistic G-metric spaces. Finally, an example is given to exemplify the theorems.
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1 Introduction and preliminaries
As a generalization of a metric space, the concept of a probabilistic metric space has been introduced by Menger [1, 2]. Fixed point theory in a probabilistic metric space is an important branch of probabilistic analysis, and many results on the existence of fixed points or solutions of nonlinear equations in Menger PM-spaces have been studied by many scholars (see e.g. [3, 4]). Egbert [5] defined the notion of the distance between two sets in a Menger PM-space, i.e., the so-called Menger-Hausdorff metric. In 2006, Mustafa and Sims [6] introduced the concept of a generalized metric space, and many fixed point results have been obtained by many authors (see e.g. [7–12]). On the other hand, Kaewcharoen and Kaewkhao [13] introduced the concept of a Hausdorff G-distance in a G-metric space. Moreover, Zhou et al. [14] defined the notion of a generalized probabilistic metric space or a PGM-space as a generalization of a PM-space and a G-metric space. After that, Zhu et al. [15] obtained some fixed point theorems in generalized probabilistic metric spaces. However, the concept of a Menger-Hausdorff \(G^{*}\)-metric in a PGM-space has not been introduced and studied yet.
To fill this gap, we introduce the concept of a generalized probabilistically bounded set and a Menger-Hausdorff \(G^{*}\)-metric in Menger probabilistic G-metric spaces, and we prove that \((\Omega ^{*},\widetilde{G}^{*},\Delta)\) is also a Menger probabilistic G-metric space. Based on these, we obtain some useful results. As an application, we establish some common fixed point theorems for three hybrid pairs of mappings satisfying the common property \((E.A)\) in Menger probabilistic G-metric spaces. Finally, an example is given to illustrate the theorems.
Throughout this paper, let \(\mathbb{R}=(-\infty,+\infty)\), \(\mathbb {R}^{+}=[0,+\infty)\), and \(\mathbb{Z}^{+}\) be the set of all positive integers.
A mapping \(F:\mathbb{R}\rightarrow\mathbb{R}^{+}\) is called a distribution function if it is nondecreasing left-continuous with \(\sup_{t\in\mathbb{R}}F(t)=1\) and \(\inf_{t\in\mathbb{R}}F(t)=0\).
We shall denote by \(\mathscr{D}\) the set of all distribution functions while H will always denote the specific distribution function defined by
A mapping \(\Delta:[0,1]\times[0,1]\rightarrow[0,1]\) is called a triangular norm (for short, a t-norm) if the following conditions are satisfied:
-
(1)
\(\Delta(a,1)=a\);
-
(2)
\(\Delta(a,b)= \Delta(b,a)\);
-
(3)
\(a\geq b,c\geq d\Rightarrow\Delta(a,c)\geq\Delta(b,d)\);
-
(4)
\(\Delta(a,\Delta(b,c))= \Delta(\Delta(a,b),c)\).
A typical example of t-norm is \(\Delta_{m}\), where \(\Delta _{m}(a,b)=\min\{a,b\}\), for each \(a,b\in[0,1]\).
Remark 1.1
From (4), it is not difficult to find that
Definition 1.1
[16]
A triplet \((X, \mathscr{F}, \Delta)\) is called a Menger probabilistic metric space (for short, a Menger PM-space) if X is a nonempty set, Δ is a t-norm and \(\mathscr{F}\) is a mapping from \(X\times X\) into \(\mathscr{D}\) satisfying the following conditions (we denote \(\mathscr{F}(x,y)\) by \(F_{x,y}\)):
-
(MS-1)
\(F_{x,y}(t)=H(t)\) for all \(t\in R\) if and only if \(x=y\);
-
(MS-2)
\(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in R\);
-
(MS-3)
\(F_{x,y}(t+s)\geq\Delta(F_{x,z}(t),F_{z,y}(s))\) for all \(x,y,z\in X\) and \(t,s\geq0\).
Let \((X,\mathscr{F},\Delta)\) be a PM-space and A be a nonempty subset of X. Then the function
is called the probabilistic diameter of A. If \(\sup_{t>0}D_{A}(t)=1\), then A is said to be probabilistically bounded.
Let \((X, \mathscr{F}, \Delta)\) be a Menger PM-space and Ω be the family of all nonempty probabilistically bounded \(\mathscr {T}\)-closed subsets of X. For any \(A,B\in\Omega\), define the distribution functions as follows:
where \(\tilde{\mathscr{F}}\) is called the Menger-Hausdorff metric induced by \(\mathscr{F}\).
Lemma 1.1
[16]
Let \((X,\mathscr{F},\Delta)\) be a Menger PM-space. Then for any \(A, B, C\in\Omega\) and any \(x,y\in X\), we have the following:
-
(i)
\(\tilde{F}_{A,B}(t)=1\) if and only if \(A=B\);
-
(ii)
\(F_{x,A}(t)=1\) if and only if \(x\in A\);
-
(iii)
for any \(x\in A\), \(F_{x,B}(t)\geq\tilde{F}_{A,B}(t)\), for all \(t \geq0\);
-
(iv)
\(F_{x,A}(t_{1}+t_{2})\geq\Delta(F_{x,y}(t_{1}),F_{y,A}(t_{2}))\), for all \(t_{1},t_{2}\geq0\);
-
(v)
\(F_{x,A}(t_{1}+t_{2})\geq\Delta(F_{x,B}(t_{1}),F_{A,B}(t_{2}))\), for all \(t_{1},t_{2}\geq0\);
-
(vi)
\(\tilde{F}_{A,C}(t_{1}+t_{2})\geq\Delta(\tilde{F}_{A,B}(t_{1}), \tilde{F}_{B,C}(t_{2}))\), for all \(t_{1},t_{2}\geq0\).
Definition 1.2
[14]
A Menger probabilistic G-metric space (for brevity, a PGM-space) is a triple \((X,G^{*},\Delta)\), where X is a nonempty set, Δ is a continuous t-norm and \(G^{*}\) is a mapping from \(X\times X\times X\) into \(\mathscr{D}\) (\(G^{*}_{x,y,z}\) denote the value of \(G^{*}\) at the point \((x,y,z)\)) satisfying the following conditions:
-
(PGM-1)
\(G^{*}_{x,y,z}(t)=1\) for all \(x,y,z\in X\) and \(t>0\) if and only if \(x=y=z\);
-
(PGM-2)
\(G^{*}_{x,x,y}(t)\geq G^{*}_{x,y,z}(t)\) for all \(x,y,z\in X\) with \(z\neq y\) and \(t>0\);
-
(PGM-3)
\(G^{*}_{x,y,z}(t)=G^{*}_{x,z,y}(t)=G^{*}_{y,x,z}(t)=\cdots\) (symmetry in all three variables);
-
(PGM-4)
\(G^{*}_{x,y,z}(t+s)\geq\Delta(G^{*}_{x,a,a}(s), G^{*}_{a,y,z}(t))\) for all \(x,y,z,a\in X\) and \(s,t\geq0\).
Definition 1.3
[14]
Let \((X,G^{*},\Delta)\) be a Menger PGM-space and \(x_{0}\) be any point in X. For any \(\epsilon>0\) and δ with \(0<\delta <1\), and \((\epsilon,\delta)\)-neighborhood of \(x_{0}\) is the set of all points y in X for which \(G^{*}_{x_{0},y,y}(\epsilon)>1-\delta \) and \(G^{*}_{y,x_{0},x_{0}}(\epsilon)>1-\delta\). We write
which means that \(N_{x_{0}}(\epsilon,\delta)\) is the set of all points y in X for which the probability of the distance from \(x_{0}\) to y being less than ϵ is greater than \(1-\delta\).
Lemma 1.2
[14]
Let \((X,G^{*},\Delta)\) be a Menger PGM-space. Then \((X,G^{*},\Delta)\) is a Hausdorff space in the topology introduced by the family \(\{N_{x_{0}}(\epsilon,\delta)\}\) of \((\epsilon,\delta )\)-neighborhoods.
Definition 1.4
[14]
Let \((X, G^{*}, \Delta)\) be a PGM-space, and \(\{x_{n}\}\) is a sequence in X.
-
(1)
\(\{x_{n}\}\) is said to be convergent to a point \(x\in X\) (write \(x_{n}\rightarrow x\)), if for any \(\epsilon>0\) and \(0<\delta<1\), there exists a positive integer \(M_{\epsilon,\delta}\) such that \(x_{n}\in N_{x_{0}}(\epsilon,\delta )\) whenever \(n>M_{\epsilon,\delta}\);
-
(2)
\(\{x_{n}\}\) is called a Cauchy sequence, if for any \(\epsilon>0\) and \(0<\delta<1\), there exists a positive integer \(M_{\epsilon,\delta}\) such that \(G^{*}_{x_{n},x_{m},x_{l}}(\epsilon )>1-\delta\) whenever \(n,m,l>M_{\epsilon,\delta}\);
-
(3)
\((X, G^{*}, \Delta)\) is said to be complete, if every Cauchy sequence in X converges to a point in X.
We can analogously prove the following lemma as in Menger PM-spaces.
Lemma 1.3
Let \((X,G^{*},\Delta)\) be a Menger PGM-space with Δ a continuous t-norm, \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{z_{n}\}\) be sequences in X and \(x, y, z\in X\), if \(\{x_{n}\} \rightarrow x\), \(\{ y_{n}\} \rightarrow y\) and \(\{z_{n}\} \rightarrow z\) as \(n \rightarrow \infty\). Then
-
(1)
\(\liminf_{n\rightarrow\infty }G^{*}_{x_{n},y_{n},z_{n}}(t)\geq G^{*}_{x,y,z}(t)\) for all \(t>0\);
-
(2)
\(G^{*}_{x,y,z}(t+0)\geq\limsup_{n\rightarrow\infty }G^{*}_{x_{n},y_{n},z_{n}}(t)\) for all \(t>0\).
Particularly, if \(t_{0}\) is a continuous point of \(G_{x,y,z}(\cdot)\), then \(\lim_{n\rightarrow\infty }G_{x_{n},y_{n},z_{n}}(t_{0})=G_{x,y,z}(t_{0})\).
Definition 1.5
[17]
A pair of self-mappings S and T on X are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence point, i.e., if \(Tu=Su\) for some \(u\in X\) implies that \(TSu=STu\).
Definition 1.6
[18]
Let \(F_{1},F_{2}\in\mathscr{D}\). The algebraic sum \(F_{1}\oplus F_{2}\) of \(F_{1}\) and \(F_{2}\) is defined by
for all \(t\in\mathbb{R}\).
As a generalization, we give the following definition.
Definition 1.7
Let \(F_{1},F_{2},F_{3}\in\mathscr{D}\). The algebraic sum \(F_{1}\oplus F_{2}\oplus F_{3}\) of \(F_{1}\), \(F_{2}\), and \(F_{3}\) is defined by
for all \(t\in\mathbb{R}\).
Remark 1.2
Let \(F_{3}(t)=H(t)\). Then Definition 1.6 and Definition 1.7 are equivalent.
For two functions f and g, \(f>g\) means that \(f(t)\geq g(t)\) and there exists some \(t_{0}\) such that \(f(t_{0})>g(t_{0})\).
Definition 1.8
[19]
Let f and g be self-mappings of a set X. If \(w=fx=gx\) for some x in X, then x is called a coincidence point of f and g, and w is called point of coincidence of f and g.
In the sequel, we will denote by \(C(f,F)\) the set of all coincidence points of f and F.
We recall the definitions of property \((E.A)\) for a hybrid pair of mappings and common property \((E.A)\) for two hybrid pairs of mappings in Menger PM-spaces.
Definition 1.9
[20]
Let \((X, \mathscr{F}, \Delta)\) be a Menger PM-space, \((\Omega,\tilde{\mathscr{F}},\Delta)\) be the induced Menger PM-space, \(f:X\rightarrow X\) be a self-mapping and \(F:X\rightarrow\Omega\) be a multivalued mapping. A pair of mappings \((f,F)\) is said to satisfy the property \((E.A)\), if there exist a sequence \(\{x_{n}\}\) in X, some \(a\in X\), and \(A\in\Omega\), such that \(\lim_{n\rightarrow\infty}fx_{n}=a\in A=\lim_{n\rightarrow\infty}Fx_{n}\).
Definition 1.10
[20]
Let \((X, \mathscr{F}, \Delta)\) be a Menger PM-space and \((\Omega,\tilde{\mathscr{F}},\Delta)\) be the induced Menger PM-space, \(f,g:X\rightarrow X\), and \(F,G:X\rightarrow\Omega\). Two pairs of mappings \((f,F)\) and \((g,G)\) are said to satisfy the common property \((E.A)\) if there exist two sequences \(\{x_{n}\}\), \(\{y_{n}\}\) in X, some \(u\in X\) and \(A,B\in\Omega\), such that
2 Menger-Hausdorff metric in Menger PGM-spaces
In this section, we first introduce some new concepts in Menger PGM-spaces, and then establish some useful results in Menger PGM-spaces.
Definition 2.1
Let A be a nonempty subset of X. The function \(D^{*}_{A}\) defined by
is called the generalized probabilistic diameter of A.
Definition 2.2
A nonempty subset A of X is said to be
-
(1)
generalized probabilistically bounded, if \(\sup_{t>0}D^{*}_{A}(t)=1\);
-
(2)
generalized probabilistically semi-bounded, if \(0<\sup_{t>0}D^{*}_{A}(t)<1\);
-
(3)
generalized probabilistically unbounded, if \(\sup_{t>0}D^{*}_{A}(t)=0\).
Lemma 2.1
If A and B are two nonempty subsets of X, then
Proof
Let x, y be given, for (2.1), we first prove that
Case (1):
For any \(p,q,r\in X\), we have
Taking the infimum on both sides of this inequality as p ranges over A, a ranges over \(A\cap B\), and r, q range over B, and using (2.3), we have
So, (2.2) is proved.
Case (2): \(\inf_{p,q,r\in A\cup B}G^{*}_{p,q,r}(x+y)<\inf_{\substack{p\in A,\\q,r\in B}}G^{*}_{p,q,r}(x+y)\). Then one of the following equalities:
-
(a)
\(\inf_{p,q,r\in A\cup B}G^{*}_{p,q,r}(x+y)=\inf_{p,q,r\in A}G^{*}_{p,q,r}(x+y)\),
-
(b)
\(\inf_{p,q,r\in A\cup B}G^{*}_{p,q,r}(x+y)=\inf_{p,q,r\in B}G^{*}_{p,q,r}(x+y)\)
and
-
(c)
\(\inf_{p,q,r\in A\cup B}G^{*}_{p,q,r}(x+y)=\inf_{\substack{p\in B\\q,r\in A}}G^{*}_{p,q,r}(x+y)\)
holds.
If (a) holds, we have
Then (2.2) is proved.
Similarly, we can prove that (2.2) is satisfied if (b) or (c) holds.
Finally, by (2.2) and the continuity of Δ, we have
This completes the proof. □
Lemma 2.2
Let \((X,G^{*},\Delta)\) be a Menger PGM-space with a continuous t-norm.
-
(1)
If A is a generalized probabilistically bounded set, then \(D^{*}_{A}\) is a distribution function.
-
(2)
If \(A,B\subseteq X\) are two generalized probabilistically bounded sets, then \(A\cup B\) is also a generalized probabilistically bounded set.
Proof
(1) Since A is a generalized probabilistically bounded set, by Definition 2.1, it is easy to see that \(D^{*}_{A}(t)\) is nondecreasing in t, \(D^{*}_{A}(0)=0\), \(\sup_{t>0}D^{*}_{A}(t)=1\) and \(D^{*}_{A}(t)\) is left-continuous in t. This shows that \(D^{*}_{A}(t)\) is a distribution function.
(2) Since A and B are generalized probabilistically bounded sets, from Lemma 2.1 and the continuity of Δ, we have \(\sup_{t>0}D^{*}_{A\cup B}(t)\geq\Delta(\sup_{t>0}D^{*}_{A}(\frac{t}{2}),\sup_{t>0}D^{*}_{B}(\frac {t}{2}))=\Delta(1,1)=1\). This completes the proof. □
Remark 2.1
By Lemma 2.2(2), we claim that if A, B, C are generalized probabilistically bounded sets, then \(A\cup B\cup C\) is also a generalized probabilistically bounded set.
In the remainder of this paper, we always assume that \((X,G^{*},\Delta)\) is a Menger PGM-space with a continuous t-norm Δ and \(\Omega^{*}\) be the family of all nonempty \(\mathscr {T}\)-closed generalized probabilistically bounded sets.
Definition 2.3
For \(A,B,C\in\Omega^{*}\), define the mapping \(\widetilde{G}^{*}:\Omega^{*}\times\Omega^{*}\times\Omega^{*}\rightarrow \mathscr{D} \) by
where \(g_{A,B}(t)=\sup_{s< t}\Delta(\inf_{x\in A}\sup_{y\in B}g_{x,y}(s),\inf_{y\in B}\sup_{x\in A}g_{x,y}(s))\), \(g_{x,y}(s)=\Delta(G^{*}_{x,x,y}(s), G^{*}_{x,y,y}(s))\).
Then \(\widetilde{G}^{*}\) is called the Menger-Hausdorff metric induced by \(G^{*}\).
Definition 2.4
Let \(A,B,C\in\Omega^{*}\) and \(x,y,z\in X\).
-
(1)
The generalized probabilistic distance between two points x, y and a set C is the function \(\widetilde{G}^{*}_{x,y,C}(t)\) defined by
$$\widetilde{G}^{*}_{x,y,C}(t)=\sup_{s< t}\sup _{z\in C}G^{*}_{x,y,z}(s),\quad t\geq0. $$ -
(2)
The generalized probabilistic distance between a point x and two sets B, C is the function \(\widetilde{G}^{*}_{x,B,C}(t)\) defined by
$$\widetilde{G}^{*}_{x,B,C}(t)=\min\bigl\{ g_{x,B}(t),g_{B,C}(t),g_{x,C}(t) \bigr\} , \quad t\geq0, $$where \(g_{x,B}(t)=\sup_{s< t}\sup_{y\in B}g_{x,y}(s)\).
Lemma 2.3
For any \(A,B,D\in\Omega^{*}\) and \(a,b>0\), we have
Proof
For any \(x,y,z\in X\) and \(s,t>0\), we have
and
for all \(z\in D\). Using the continuity and monotonicity of Δ, we have the following inequalities:
and
Thus, we have
Similarly, we can get
Since Δ is associative, by combining (2.4), (2.5), (2.6), and (2.7), we obtain
This completes the proof. □
Theorem 2.1
\((\Omega^{*},\widetilde{G}^{*},\Delta)\) is a Menger PGM-space.
Proof
First, we prove that \(\widetilde{G}^{*}\) is a distribution function. By the definition of \(\widetilde{G}^{*}(t)\), it is easy to see that \(\widetilde{G}^{*}(t)\) is nondecreasing and left-continuous in t and \(\widetilde{G}^{*}(0)=0\). Now, we prove
In fact, since \(A,B,C\in\Omega^{*}\), we know \(A\cup B\cup C \in \Omega^{*}\). By the continuity of Δ, we have
and
Similarly, we have \(\sup_{t>0}g_{B,C}(t)=1\) and \(\sup_{t>0}g_{C,A}(t)=1\). This shows that \(\widetilde{G}^{*}\) is a mapping from \(\Omega^{*}\times\Omega^{*}\times\Omega^{*}\) into \(\mathscr{D}\).
Next, we will show that \(\widetilde{G}^{*}(t)\) satisfies the following:
-
(1)
\(\widetilde{G}^{*}_{A,B,C}(t)=1\) for all \(t>0\) if and only if \(A=B\);
-
(2)
\(\widetilde{G}^{*}_{A,A,B}(t)\geq\widetilde{G}^{*}_{A,B,C}(t)\) for all \(A,B,C\in\Omega^{*}\) with \(B\neq C\) and \(t>0\);
-
(3)
\(\widetilde{G}^{*}_{A,B,C}(t)=\widetilde{G}^{*}_{A,C,B}(t)=\widetilde{G}^{*}_{B,A,C}(t)=\cdots\) (symmetry in all three variables);
-
(4)
\(\widetilde{G}^{*}_{A,B,C}(t+s)\geq\Delta(\widetilde{G}^{*}_{A,D,D}(t),\widetilde{G}^{*}_{D,B,C}(t))\) for all \(A,B,C\in \Omega^{*}\) with \(t>0\).
• (i) If \(\widetilde{G}^{*}_{A,B,C}(t)=1\) for all \(t>0\), then for any \(\epsilon >0\), we have
By \(g_{A,B}(\epsilon )=1\), we have
From (2.10), it follows that \(\sup_{s<\epsilon }\sup_{y\in B}\Delta(G^{*}_{x,x,y}(s),G^{*}_{x,y,y}(s))=1\) for all \(x\in A\). Therefore, for any \(a\in A\) and \(\lambda>0\), there exists \(b^{*}\in B\), such that
So, we have
This shows that the point a is an accumulation point of B and hence \(a\in B\), i.e., \(A\subseteq B\).
From (2.11), we can prove that \(B\subseteq A\). Therefore, we have \(A=B\).
Similarly, we can also prove that \(B=C\), \(C=A\). So, we have \(A=B=C\).
Conversely, if \(A=B=C\), then for any \(t>0\), we have
For any \(s\in(0,1)\),
Therefore (1) is satisfied.
• (ii) \(\widetilde{G}^{*}_{A,A,B}(t)=\min\{ g_{A,A}(t),g_{A,B}(t),g_{A,B}(t)\}=g_{A,B}(t)\geq\min\{ g_{A,B}(t),g_{B,C}(t),g_{C,A}(t)\} = \widetilde{G}^{*}_{A,B,C}(t)\). So, (2) is satisfied.
• (iii) It is obvious that (3) holds.
• (iv) From Definition 2.3, we have
We just need to show
In fact,
So, (4) is also satisfied. This completes the proof. □
Remark 2.2
By the proof process of Lemma 2.3 and Theorem 2.1, we can also prove that \((X,g,\Delta)\) and \((\Omega^{*},g,\Delta)\) are Menger PM-spaces. We call \((X,g,\Delta)\) the PM-space induced by \((X,G^{*},\Delta)\), and \((\Omega^{*},g,\Delta)\) is the PM-space induced by \((X,g,\Delta)\). So, the properties in Lemma 1.1 can be applied to \((X,g,\Delta)\) and \((\Omega^{*},g,\Delta)\).
Example 2.1
Let \((X,d)\) be a metric space and \(x,y,z\in X\), \(G^{*}_{x,y,z}(t)=\frac{t}{t+\max\{d(x,y),d(y,z),d(x,z)\}}\) for all \(t\geq0\), then \(( X,G^{*},\Delta_{m})\) is a Menger PGM-space. In fact, \(G^{*}_{x,y,z}(0)=0\), \(\sup_{t>0}G^{*}_{x,y,z}(t)=1\), and \(G^{*}_{x,y,z}(t)\) is nondecreasing and continuous in t, so \(G^{*}_{x,y,z}(t)\) is a distribution function. Obviously, \(G^{*}_{x,y,z}(t)\) satisfy (PGM-1), (PGM-2), and (PGM-3). Next, we will show that (PGM-4) is also satisfied. Since \(d(x,y)\leq d(x,a)+d(a,y)\) and \(d(x,z)\leq d(x,a)+d(a,z)\), we have
which implies that (PGM-4) is satisfied. So \(( X,G^{*},\Delta_{m})\) is a Menger PGM-space. Then
and
Thus,
where \(\delta(A,B)=\max\{\inf_{x\in A}\sup_{y\in B}d(x,y), \inf_{y\in B}\sup_{x\in A}d(x,y)\}\). Then \((\Omega^{*},\widetilde{G}^{*}_{A,B,C}(t),\Delta_{m})\) is a Menger PGM-space induced by \(( X,G^{*},\Delta_{m})\).
Theorem 2.2
Let \((\Omega^{*},\widetilde{G}^{*},\Delta)\) be a Menger PGM-space. Then for any \(A,B,C,D\in\Omega^{*}\) and \(x,y,z\in X\), we have the following:
-
(1)
\(\widetilde{G}^{*}_{x,B,C}(t)=1\) if and only if \(x\in B=C\);
-
(2)
\(\widetilde{G}^{*}_{x,x,B}(t)\geq\widetilde{G}^{*}_{x,B,B}(t)\geq\widetilde{G}^{*}_{x,B,C}(t)\geq\widetilde{G}^{*}_{A,B,C}(t)\) for all \(x\in A\) and \(t\geq0\);
-
(3)
\(\widetilde{G}^{*}_{x,B,C}(t+s)\geq\Delta(\widetilde{G}^{*}_{x,D,D}(t),\widetilde{G}^{*}_{D,B,C}(s))\) for all \(s,t\geq0\);
-
(4)
\(\widetilde{G}^{*}_{x,y,C}(t+s)\geq\Delta(\widetilde{G}^{*}_{x,a,a}(t),\widetilde{G}^{*}_{a,y,C}(s))\) for all \(s,t\geq0\) and \(a\in X\).
Proof
(1) If \(\widetilde{G}^{*}_{x,B,C}(t)=\min\{ g_{x,B}(t),g_{B,C}(t),g_{x,C}(t)\}=1\), then we have \(g_{x,B}(t)=g_{B,C}(t)=g_{x,C}(t)=1\), which implies that \(x\in B\), \(x\in C\), \(B=C\), that is, \(x\in B=C\).
Conversely, it is obvious that \(\widetilde{G}^{*}_{x,B,C}(t)=1\) holds.
(2) From Definition 2.4 and Lemma 1.1, we have
So, (2) is proved.
(3) By Definition 2.4 and Lemma 1.1, we have \(\widetilde{G}^{*}_{x,B,C}(t+s)=\min\{g_{x,B}(t+s),g_{B,C}(t+s),g_{x,C}(t+s)\}\),
So, (3) is proved.
(4) By Lemma 1.1, we have
□
Remark 2.3
By (1), (2), and the proof of Lemma 2.2, it is easy to prove that \(\widetilde{G}^{*}_{x,x,B}(t)=1\) if and only if \(x\in B\), and \(\widetilde{G}^{*}_{x,B,B}(t)=1\) if and only if \(x\in B\).
3 Common fixed point theorems in Menger PGM-spaces
In this section, we will give some common fixed point theorems in Menger probabilistic G-metric spaces. To this end, we first introduce the concept of common property \((E.A)\) for three hybrid pairs of mappings in Menger probabilistic G-metric spaces.
Definition 3.1
Let \((X,G^{*},\Delta)\) be a Menger PM-space and \((\Omega^{*},\widetilde{G}^{*},\Delta)\) be the induced Menger PM-space, \(f,h,r:X\rightarrow X\) and \(F,H,R:X\rightarrow \Omega^{*}\). Three pairs of mappings \((f,F)\), \((h,H)\), and \((r,R)\) are said to satisfy the common property \((E.A)\) if there exist three sequences \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{z_{n}\}\) in X, some \(u\in X\) and \(A,B,C\in \Omega^{*}\), such that
We are now ready to give the common fixed point theorems in Menger probabilistic G-metric spaces.
Theorem 3.1
Let \((X,G^{*},\Delta)\) be a Menger PGM-space with a continuous t-norm on \([0,1]\times[0,1]\) and \((\Omega^{*},\widetilde{G}^{*},\Delta)\) be the induced Menger PGM-space. Suppose that \(f,h,r:X\rightarrow X\) and \(F,H,R:X\rightarrow\Omega^{*}\) are mappings satisfying the following conditions:
-
(1)
\((f,F)\), \((h,H)\), and \((r,R)\) satisfy the common property \((E.A)\);
-
(2)
\(f(X)\), \(h(X)\), and \(r(X)\) are \(\mathscr{T}\)-closed subsets of X;
-
(3)
for any \(x,y,z\in X\) with Fx, Hy, and Rz not all equal and some \(1\leq k\leq3\),
$$ \widetilde{G}^{*}_{Fx,Hy,Rz}> \min\bigl\{ G^{*}_{fx,hy,rz}, {}_{\frac {3}{k}}\bigl[\widetilde{G}^{*}_{Fx,hy,rz} \oplus\widetilde{G}^{*}_{fx,Hy,rz}\oplus\widetilde{G}^{*}_{fx,hy,Rz} \bigr]\bigr\} , $$(3.1)where \({}_{\frac{3}{k}}[\widetilde{G}^{*}_{Fx,hy,rz}\oplus\widetilde{G}^{*}_{fx,Hy,rz}\oplus\widetilde{G}^{*}_{fx,hy,Rz}](t)\) means \([\widetilde{G}^{*}_{Fx,hy,rz}\oplus \widetilde{G}^{*}_{fx,Hy,rz}\oplus\widetilde{G}^{*}_{fx,hy,Rz}](\frac{3}{k}t)\).
Then \((f,F)\), \((h,H)\), and \((r,R)\) each has a coincidence point. Moreover, if \(ffv=fv\) for \(v\in C(f,F)\), \(hhv=hv\) for \(v\in C(h,H)\), and \(rrv=rv\) for \(v\in C(r,R)\), then f, h, r, F, H, and R have a common fixed point in X.
Proof
Since \((f,F)\), \((h,H)\), and \((r,R)\) satisfy the common property \((E.A)\), there exist \(\{x_{n}\},\{y_{n}\},\{z_{n}\}\subset X\), some \(u\in X\) and \(A,B,C\in\Omega^{*}\), such that
Since \(f(X)\) is \(\mathscr{T}\)-closed, there exists some \(v\in X\), such that \(u=fv\). We claim that \(fv\in Fv\). Suppose this is not true, then \(fv\notin Fv\). By \(u=fv\in B\), we have \(B\neq Fv\). Thus, there exists some \(t_{0}>0\), such that
(Otherwise, for all \(t>0\), \(\widetilde{G}^{*}_{Fv,B,C}(t)=\widetilde{G}^{*}_{Fv,B,C}(\frac {3t}{k})=\cdots=\widetilde{G}^{*}_{Fv,B,C}((\frac {3}{k})^{n}t)\rightarrow 1\) as \(n\rightarrow\infty\), that is, \(\widetilde{G}^{*}_{Fv,B,C}(t)=1\), for all \(t>0\), which is a contradiction.)
Without loss of generality, we can assume that \(t_{0}\) is a continuous point of \(\widetilde{G}^{*}_{Fv,B,C}(\cdot)\). In fact, by the left continuity of the distribution function, we know that there exists some \(\delta>0\), such that
Since the distribution function is nondecreasing, the discontinuous points are at most a countable set. Thus, when \(t_{0}\) is not a continuous point of \(\widetilde{G}^{*}_{Fv,B,C}(\cdot)\), we can always choose a point \(t_{1}\) in \((t_{0}-\delta, t_{0}]\) to replace \(t_{0}\).
Noting that \(\lim_{n\rightarrow\infty}fx_{n}=u\notin Fv\) and \(u\in B=\lim_{n\rightarrow\infty}Hy_{n}\), we have \(Fv\neq\lim_{n\rightarrow\infty}Hy_{n}\), so there exists some \(n_{0}\in\mathbb{Z^{+}}\), such that for all \(n\geq n_{0}\), \(Hy_{n}\neq Fv\).
From (3.1) we know that
It is easy to verify that
In fact, for any \(\delta_{1},\delta_{2}\in(0,\frac{3}{k}t_{0})\), we have
Since \(fv=u\in[(B=\lim_{n\rightarrow\infty}Hy_{n})\cap (C=\lim_{n\rightarrow\infty}Ry_{n})]\), by Lemma 1.3 and Theorem 2.2(1), we get
Letting \(\delta_{1},\delta_{2}\rightarrow0\), by the left continuity of the distribution function, we obtain (3.5).
Noting that \(t_{0}\) is the continuous point of \(\widetilde{G}^{*}_{Fv,B,C}(\cdot)\), by Lemma 1.3, we have
Thus, letting \(n\rightarrow\infty\) in (3.4) and using (3.5), we obtain
that is,
But since \(fv\in B\), by Theorem 2.2(3) and (3.3), we obtain
which is a contradiction. So, we get \(fv\in Fv\).
On the other hand, since \(h(X)\) is \(\mathscr{T}\)-closed, there exists some \(w\in X\), such that \(u=hw\). We claim that \(hw\in Hw\). Suppose this is not true, that is, \(hw\notin Hw\). Noting that \(u=hw\in C\), we have \(C\neq Hw\). Similarly, we know that there exists some \(t_{1}>0\), such that
Similarly, without loss of generality, we can assume that \(t_{1}\) is a continuous point of \(\widetilde{G}^{*}_{Fv,Hw,C}(\cdot)\).
Noting that \(\lim_{n\rightarrow\infty}rz_{n}=u\notin Hw\) and \(u\in C=\lim_{n\rightarrow\infty}Rz_{n}\), there exists some \(n_{1}\in\mathbb{Z^{+}}\), such that for all \(n\geq n_{1}\), \(Rz_{n}\neq Hw\).
From (3.1) we know that
Similarly, we can verify that
Noting that \(t_{1}\) is a continuous point of \(\widetilde{G}^{*}_{Fv,Hw,C}(\cdot)\), by Lemma 1.3, we have
Thus, letting \(n\rightarrow\infty\) in (3.7) and using (3.8), we obtain
which is a contradiction. So, we get \(hw\in Hw\).
Since \(r(X)\) is \(\mathscr{T}\)-closed, there exists some \(a\in X\), such that \(u=ra\). We claim that \(ra\in Ra\). Suppose this is not true, that is, \(ra\notin Ra\). Noting that \(u=ra\in A\), we have \(A\neq Ra\). Similarly, we know that there exists some \(t_{2}>0\), such that
Similarly, without loss of generality, we can assume that \(t_{2}\) is a continuous point of \(\widetilde{G}^{*}_{A,Hw,Ra}(\cdot)\).
Noting that \(\lim_{n\rightarrow\infty}fx_{n}=u\notin Ra\) and \(u\in A=\lim_{n\rightarrow\infty}Fx_{n}\), there exists some \(n_{2}\in\mathbb{Z^{+}}\), such that for all \(n\geq n_{2}\), \(Fx_{n}\neq Ra\).
From (3.1), we know that
Similarly, it is easy to prove that \(u=ra\in Ra\). This implies that v is a coincidence point of \((f,F)\), w is a coincidence point of \((h,H)\), and a is a coincidence point of \((r,R)\).
Since \(v\in C(f,F)\), \(w\in C(h,H)\), and \(a\in C(r,R)\), we have \(u=fv=ffv=fu\in Fv\), \(u=hw=hhw=hu\in Hw\), and \(u=ra=rra=ru\in Rw\). Next, we prove that \(Fv=Fu\), \(Hw=Hu\), and \(Ra=Ru\).
(1) First, we assert that \(Fv=Hw\). In fact, suppose that \(Fv\neq Hw\). Then, by (3.1), there exists some \(t_{3}>0\), such that
This implies that
which is a contradiction, and thus we have \(Fv=Hw\).
(2) Next, we assert that \(Fu=Hw\). In fact, suppose that \(Fu\neq Hw\). Then, by (3.1), there exists some \(t_{4}>0\), such that
This implies that
which is a contradiction, and thus we have \(Fu=Hw\). Combining these two facts yields \(Fv=Fu\). Similarly, we can prove that \(Hw=Ra=Hu\) and \(Ra=Fv=Ru\). Thus, we have \(u=fu\in Fu\), \(u=hu\in Hu\), and \(u=ru\in Ru\), that is, u is the common fixed point of f, h, r, F, H, and R. This completes the proof. □
Setting \(f=h=r\) and \(F=H=R\), we obtain the following result.
Theorem 3.2
Let \((X,G^{*},\Delta)\) be a Menger PGM-space with a continuous t-norm on \([0,1]\times[0,1]\) and \((\Omega^{*},\widetilde{G}^{*},\Delta)\) be the induced Menger PGM-space. Suppose that \(f:X\rightarrow X\) and \(F:X\rightarrow\Omega^{*}\) are mappings satisfying the following conditions:
-
(1)
\((f,F)\) satisfies the property \((E.A)\);
-
(2)
\(f(X)\) is a \(\mathscr{T}\)-closed subset of X;
-
(3)
for any \(x,y,z\in X\) with Fx, Fy, and Fz not all equal and some \(1\leq k\leq3\),
$$ \widetilde{G}^{*}_{Fx,Fy,Fz}(t)> \min\bigl\{ G^{*}_{fx,fy,fz}, {}_{\frac {3}{k}}\bigl[\widetilde{G}^{*}_{Fx,fy,fz}\oplus \widetilde{G}^{*}_{fx,Fy,fz}\oplus\widetilde{G}^{*}_{fx,fy,Fz} \bigr]\bigr\} , $$where \({}_{\frac{3}{k}}[\widetilde{G}^{*}_{Fx,fy,fz}\oplus\widetilde{G}^{*}_{fx,Fy,fz}\oplus\widetilde{G}^{*}_{fx,fy,Fz}](t)\) means \([\widetilde{G}^{*}_{Fx,fy,fz}\oplus \widetilde{G}^{*}_{fx,Fy,fz}\oplus\widetilde{G}^{*}_{fx,fy,Fz}] (\frac{3}{k}t)\).
Then f and F have a coincidence point. Moreover, if \(ffv=fv\) for \(v\in C(f,F)\), then f and F have a common fixed point in X.
4 An example
In this section, we will provide an example to show the validity of Theorem 3.1.
Example 4.1
Let \(X=(-2,2)\) and define
for all \(x,y,z\in X\), \(A,B,C\in\Omega^{*}\), and \(t\geq0\). Then, by Example 2.1, \((X,G^{*},\Delta_{m})\) and \((\Omega^{*},\widetilde{G}^{*},\Delta _{m})\) are PGM-spaces. Define \(f,h,r:X\rightarrow X\) and \(F,H,R:X\rightarrow\Omega^{*}\) as follows:
Consider the sequences \(\{x_{n}=\frac{1}{n+1}\}\) and \(\{y_{n}=-\frac {1}{n+1}\}\) in X. Then
which shows that \((f,F)\), \((h,H)\), and \((r,R)\) satisfy the common property \((E.A)\). Also \(f(X)\), \(h(X)\), and \(r(X)\) are \(\mathscr {T}\)-closed subsets of X. By a routine calculation, one can verify that (3.1) holds for all \(x,y,z\in X\), \(t>0\), and some \(1\leq k< 3\).
In fact, if \(x,y,z\in(-2,-1)\cup(1,2)\), for any \(t>0\),
So, we have
Similarly, if \(x,y,z\in[-1,0]\), or \(x,y,z\in[0,1]\), we also have
If \(x,y\in(-2,-1)\cup(1,2)\), \(z\in[0,1]\), we have
So, we have
Similarly, it is easy to verify (3.1) for the other cases. Thus, all the conditions of Theorem 3.1 are satisfied and 0 is the unique coincidence point of \((f,F)\), \((h,H)\), and \((r,R)\). Furthermore, noting that \(ff0=f0\), \(hh0=h0\), and \(rr0=r0\), 0 remains the common fixed point of \((f,F)\), \((h,H)\), and \((r,R)\).
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Acknowledgements
The authors would like to thank the editor and the referees for their constructive comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11326099, 11461045, 11071108) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 20142BAB211016, 2010GZS0147).
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Tu, Q., Zhu, C. & Wu, Z. Menger-Hausdorff metric and common fixed point theorems in Menger probabilistic G-metric spaces. Fixed Point Theory Appl 2015, 130 (2015). https://doi.org/10.1186/s13663-015-0383-5
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DOI: https://doi.org/10.1186/s13663-015-0383-5