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Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology
Journal of Inequalities and Applications volume 2021, Article number: 134 (2021)
Abstract
In this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of \(\mathbb{N}^{l}\) is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.
1 Introduction
The theory of probabilistic metric space (PM-space) as a generalization of ordinary metric space was introduced by Menger in [12]. In this space, distribution functions are considered as the distance of a pair of points in statistics rather than deterministic.
The concept of the generalized metric space (briefly G-metric space) was introduced by Mustafa and Sims in 2006 [16]. Then, in 2014, Zhou et al. [26] generalized the notion of PM-space to the G-metric spaces and defined the probabilistic generalized metric space which is denoted by PGM-space.
In [3], Choi et al. proposed a generalization of G-metric space named g-metric space with degree l, in which the distance function with degrees \(l=1,2\) is equivalent to ordinary and G-metric, respectively.
The idea of statistical convergence was first introduced by Steinhaus [25] for real sequences and developed by Fast [7], then reintroduced by Shoenberg [22]. Many authors, such as [4, 6, 8, 9, 17, 21], have discussed and developed this concept. The theory of statistical convergence has many applications in various fields such as approximation theory [5], finitely additive set functions [4], trigonometric series [27], and locally convex spaces [11].
In 2008, Sencimen and Pehlivan [24] introduced the concepts of statistically convergent sequence and statistically Cauchy sequence in the probabilistic metric space endowed with strong topology.
The purpose of this paper is to develop a concept to generalize the probabilistic G-metric space to the probabilistic g-metric space with degree l. Here, the notation of the generalized space is still referred as PGM-space. The l-dimensional asymptotic density of a subset A of \(\mathbb{N}^{l}\) defined previously by the author in [1] is used to introduce the statistically convergent and Cauchy sequences with respect to strong topology, and some basic facts are studied. Note that in this definition \(l=1\) and \(l=2\) values coincide exactly with the statistical convergence in PM-space and PGM-space (related to G-metric), respectively. Thus, the definitions and the obtained results show that this study is more comprehensive.
2 Preliminaries
In this section, some basic definitions and results related to PM-space, PGM-space, and statistical convergence are presented and discussed. First, recall the definition of triangular norm (t-norm) as follows.
Definition 2.1
([23])
AÂ mapping \(T:[0,1]\times [0,1] \to [0,1]\) is called a continuous t-norm if T satisfies the following conditions:
-
(i)
T is commutative and associative, i.e., \(T(a,b)=T(b,a)\) and \(T(a,T(b,c))=T(T(a,b),c)\) for all \(a,b,c\in [0,1]\);
-
(ii)
T is continuous;
-
(iii)
\(T(a,1)=a\) for all \(a\in [0,1]\);
-
(iv)
\(T(a,b)\leq T(c,d)\) whenever \(a\leq c\) and \(b\leq d\) for all \(a,b,c,d\in [0,1]\).
A distribution function F is a map from extended reals \(\mathbb{R}_{\infty }:=\mathbb{R}\cup \{-\infty , \infty \}\) into \([0, 1]\) such that it is nondecreasing, left-continuous at every real number, and \(F(-\infty )=0\) and \(F(\infty )=1\). The set of all distribution functions is denoted by Δ and \(\Delta ^{+}:=\{F\in \Delta : F(0)=0\}\).
Definition 2.2
([23])
AÂ Menger probabilistic metric space (PM-space) is a triple \((X,F,T)\), where X is a nonempty set, T is a continuous t-norm. and F is a mapping from \(X\times X\to \Delta ^{+}\) satisfying the following conditions:
(\(F_{(x,y)}\) denotes the value of F at the pair \((x,y)\))
-
(i)
\(F_{(x,y)}(t)=1\) for all \(x,y\in X\) and \(t>0\) if and only if \(x=y\);
-
(ii)
\(F_{(x,y)}(t)=F_{(y,x)}(t)\);
-
(iii)
\(F_{(x,y)}(t+s)\geq T (F_{(x,z)}(t), F_{(z,y)}(s) )\) for all \(x,y,z\in X\) and \(t,s\geq 0\).
Definition 2.3
([16])
Let X be a nonempty set and \(G:X\times X \times X \to \mathbb{R}^{+}\), be a function satisfying:
-
1)
\(G(x,y,z)=0\) if \(x=y=z\);
-
2)
\(0< G(x,x,y)\) for all \(x,y \in X\) with \(x\neq y\);
-
3)
\(G(x,x,y)\leq G(x,y,z)\) for all \(x,y,z\in X\) with \(z\neq y\);
-
4)
\(G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots\) (symmetry in all three variables);
-
5)
\(G(x,y,z)\leq G(x,a,a)+G(a,y,z)\) for all \(x,y,z,a \in X\).
Then the pair \((X, G)\) is called G-metric space.
The following definition is a developing of PM-space on G-metric.
Definition 2.4
([26])
AÂ Menger probabilistic G-metric space (PGM-space) is a triple \((X, G^{*}, T)\), where X is a nonempty set, T is a continuous t-norm, and \(G^{*}\) is a mapping from \(X\times X \times X\) into \(\Delta ^{+}\), satisfying the following conditions:
-
(i)
\(G^{*}_{(x,y,z)}(t)=1\) for all \(x, y,z \in X\) and \(t>0\) if and only if \(x=y=z\);
-
(ii)
\(G^{*}_{(x,x,y)}(t)\geq G^{*}_{(x,y,z)}(t)\) for all \(x,y\in X\) with \(z\neq y\) and \(t>0\);
-
(iii)
\(G^{*}_{(x,y,z)}(t)=G^{*}_{(x,z,y)}(t)=G^{*}_{(y,x,z)}(t)=\cdots\) (symmetry in all three variables);
-
(iv)
\(G^{*}_{(x,y,z)}(t+s)\geq T(G^{*}_{(x,a,a)}(t), G^{*}_{(a,y,z)}(s))\) for all \(x,y,z,a\in X\) and \(s,t\geq 0\).
Definition 2.5
([26])
Let \((X, G^{*}, T)\) be a PGM-space and \(x_{0}\in X\). For \(\epsilon >0\) and \(0<\delta <1\), the \((\epsilon , \delta )\)-neighborhood of \(x_{0}\) is defined as follows:
Definition 2.6
([26])
-
(i)
AÂ sequence \(\{x_{n}\}\) in a PGM-space \((X, G^{*}, T)\) is said to be convergent to a point \(x\in X\) if, for every \(\epsilon >0\) and \(0<\delta <1\), there exists a positive integer \(M_{\epsilon ,\delta }\) such that \(x_{n}\in N_{x}(\epsilon , \delta )\) whenever \(n>M_{\epsilon ,\delta }\).
-
(ii)
AÂ sequence \(\{x_{n}\}\) in a PGM-space \((X, G^{*}, T)\) is called a Cauchy sequence if, for every \(\epsilon >0\) and \(0<\delta <1\), there exists a positive integer \(M_{\epsilon ,\delta }\) such that \(G^{*}_{(x_{m}, x_{n},x_{l})}(\epsilon )>1-\delta \) whenever \(m, n, l>M_{\epsilon ,\delta }\).
-
(iii)
A PGM-space \((X, G^{*}, T)\) is said to be complete if every Cauchy sequence in X converges to a point in X.
In the following, some basic concepts of statistical convergence are discussed.
Definition 2.7
([7])
Let \(A\subset \mathbb{N}\) and \(A(n)=\{k\in A ; k\leq n\}\). Then the asymptotic density of A is defined as follows:
For a subset A of \(\mathbb{N}\), if \(\delta (A)=1\), then it is said to be statistically dense. It is clear that \(\delta (\mathbb{N}-A)=1-\delta (A)\).
Definition 2.8
([7])
AÂ sequence \(\{x_{n}\}\) in \(\mathbb{R}\) is said to be statistically convergent to a point x in \(\mathbb{R}\) if, for each \(\epsilon >0\),
For more information about statistical convergence, the references [2, 4, 7–10, 13–15, 18–20] can be addressed.
3 Main results
In this section the main definitions and results are introduced and discussed. First of all, consider the following definition which is a generalization of a G-metric space to an l-dimensional case, where \(l\in \mathbb{N}\).
Definition 3.1
([3])
Let X be a nonempty set. AÂ function \(g:X^{l+1}\longrightarrow \mathbb{R}_{+}\) is called a g-metric with degree l on X if it satisfies the following conditions:
-
g1)
\(g(x_{0}, x_{1},\ldots, x_{l})=0 \) if and only if \(x_{0}=x_{1}=\cdots=x_{l}\),
-
g2)
\(g(x_{0}, x_{1},\ldots, x_{l})=g(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{ \sigma (l)})\) for permutation σ on \(\{0, 1,\ldots, l\}\),
-
g3)
\(g(x_{0}, x_{1},\ldots, x_{l})\leq g(y_{0}, y_{1},\ldots, y_{l})\) for all \((x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}\) with \(\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}\),
-
g4)
For all \(x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t}, w\in X\) with \(s+t+1=l\),
$$ g(x_{0}, x_{1},\ldots, x_{s}, y_{0}, y_{1},\ldots, y_{t})\leq g(x_{0}, x_{1},\ldots, x_{s}, w, w,\ldots, w)+g(y_{0}, y_{1},\ldots, y_{t}, w, w,\ldots, w). $$
The pair \((X,g)\) is called a g-metric space. It is noteworthy that, if \(l=1\) (resp. \(l=2\)), then it is equivalent to an ordinary metric space (resp. G-metric space).
Definition 3.2
([3])
Let \((X,g)\) be a g-metric space, \(x\in X\) be a point, and \(\{x_{k}\}\subseteq X\) be a sequence.
-
1)
\(\{x_{k}\}\) g-converges to x if for all \(\epsilon >0\) there exists \(N\in \mathbb{N}\) such that
$$ i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow \quad g(x, x_{1},\ldots, x_{l})< \epsilon . $$ -
2)
\(\{x_{k}\}\) is said to be g-Cauchy if for all \(\epsilon >0\) there exists \(N\in \mathbb{N}\) such that
$$ i_{0}, i_{1},\ldots, i_{l}\geq N\quad \Longrightarrow\quad g(x_{i_{0}}, x_{i_{1}},\ldots, x_{i_{l}})< \epsilon . $$ -
3)
\((X, g)\) is complete if every g-Cauchy sequence in \((X, g)\) is g-convergent in \((X, g)\).
Now, by equipping Definition 2.4 with g-metric, we introduce the following definition that is a generalization.
Definition 3.3
AÂ Menger probabilistic g-metric space (still is denoted as PGM-space) is a triple \((X, F, T)\), where X is a nonempty set, T is a continuous t-norm, and F is a mapping from \(X^{l+1}\) into \(\Delta ^{+}\), satisfying the following conditions:
-
(i)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=1\) for all \(x_{0}, x_{1},\ldots, x_{l} \in X\) and \(t>0\) if and only if \(x_{0}= x_{1}= \cdots= x_{l}\);
-
(ii)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)\geq F_{(y_{0}, y_{1},\ldots, y_{l})}(t)\) for all \((x_{0}, x_{1},\ldots, x_{l}), (y_{0}, y_{1},\ldots, y_{l})\in X^{l+1}\) with \(\{x_{i}: i=0, 1,\ldots, l\}\subseteq \{y_{i}: i=0, 1,\ldots, l\}\);
-
(iii)
\(F_{(x_{0}, x_{1},\ldots, x_{l})}(t)=F_{(x_{\sigma (0)}, x_{\sigma (1)},\ldots,\ldots,x_{\sigma (l)})}(t)\) for permutation σ on \(\{0, 1,\ldots, l\}\);
-
(iv)
For all \(x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n}, w\in X\) with \(m+n+1=l\),
$$ F_{(x_{0}, x_{1},\ldots, x_{m}, y_{0}, y_{1},\ldots, y_{n})}(t+s)\geq T\bigl(F_{(x_{0}, x_{1},\ldots, x_{m}, w, w,\ldots, w)}(t), F_{(y_{0}, y_{1},\ldots, y_{n}, w, w,\ldots, w)}(s)\bigr). $$
In the following, according to the generalization of asymptotic density given in [1], statistically convergent and Cauchy sequences in a PGM-space are introduced.
Definition 3.4
Let \((X, F,T)\) be a PGM-space. For any \(\epsilon >0\), \(0<\delta <1\) and \(x\in X\), the strong \((\epsilon , \delta )\)-vicinity of x is defined by the subset \(M_{x}(\epsilon , \delta )\) of \(X^{l}\) as follows:
Next, we generalize the concept of asymptotic density of a set in an l-dimensional case.
Definition 3.5
Let \(K\subset \mathbb{N}^{l}\), the l-dimensional asymptotic density of K is defined by
Definition 3.6
Let \((X, F,T)\) be a PGM-space.
-
(i)
AÂ sequence \(\{x_{n}\}\) in X is statistically convergent to a point x in X w.r.t. strong topology if, for any \(\epsilon >0\) and \(0<\delta <1\),
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x)}(\epsilon )\leq 1- \delta \bigr\} \bigr)=0, $$and is denoted by \(x_{n} \overset{st}{\longrightarrow } x\) or \(st-\lim_{n\to \infty }x_{n}=x\).
-
(ii)
\(\{x_{n}\}\) is said to be statistically Cauchy w.r.t. strong topology if, for all \(\epsilon >0\) and \(0<\delta <1\), there exists \(i_{\epsilon }\in \mathbb{N}\) such that
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: F_{(x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}}, x_{i_{\epsilon }})}(\epsilon )\leq 1- \delta \bigr\} \bigr)=0. $$
We can restate part \((i)\) of the above definition as follows:
- (\(i^{\prime }\)):
-
\(x_{n} \overset{st}{\longrightarrow } x\) if and only if, for any \(\epsilon >0\) and \(0<\delta <1\),
$$ \delta _{l}\bigl(\bigl\{ (i_{1}, i_{2},\ldots, i_{l})\in \mathbb{N}^{l}: (x_{i_{1}}, x_{i_{2}},\ldots, x_{i_{l}})\notin M_{x}(\epsilon , \delta )\bigr\} \bigr)=0. $$
The following theorem shows that if a sequence is statistically convergent to a point in X, then that point is unique.
Theorem 3.7
Let \(\{x_{n}\}\) be a sequence in a PGM-space \((X, F, T)\) such that \(x_{n} \overset{st}{\longrightarrow } x\) and \(x_{n} \overset{st}{\longrightarrow } y\), then \(x=y\).
Proof
Let \(\epsilon >0\) and \(0<\delta <1\), by the continuity of T, there exists \(0<\delta _{0}<1\) such that
Set
and
Since \(x_{n} \overset{st}{\longrightarrow } x\) and \(x_{n} \overset{st}{\longrightarrow } y\), so \(\delta _{l}(A(\epsilon ,\delta ))=\delta _{l}(B(\epsilon ,\delta ))=0\) and hence \(\delta _{l}(C(\epsilon ,\delta ))=0\), therefore \(\delta _{l}(C^{c}(\epsilon ,\delta ))=1\). Suppose \((i_{1}, i_{2},\ldots, i_{l})\in C^{c}(\epsilon ,\delta )\), then by parts (ii) of Definition 3.3 and (iv) of Definition 2.1 we have
Since \(\delta >0\) is arbitrary, we conclude that \(F_{(x, y, y,\ldots, y)}(\epsilon )=1\), and therefore \(x=y\). □
Theorem 3.8
Every convergent sequence in a PGM-space is statistically convergent.
Proof
Let \(\{x_{n}\}\) be a sequence in the PGM-space \((X,F, T)\) that converges to a point \(x\in X\). For \(\epsilon >0\) and \(0<\delta <1\), there exists \(n_{0}\in \mathbb{N}\) such that, for all \(i_{1}, i_{2},\ldots, i_{l}\geq n_{0}\),
Set
then
and
so
 □
Example 3.9 shows that the converse of the above theorem is not valid.
Example 3.9
Let \(X=\mathbb{R}\) and \(G:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}\) be a G-metric on \(\mathbb{R}\) defined by
\((T= \min )\) Define a function \(F:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}^{+}\) as follows:
where \(H(t)\) and \(\mathcal{D}(t)\) are distribution functions as follows:
Now, consider the following sequence in \(\mathbb{R}\):
It is clear that \(\{x_{n}\}\) statistically converges to 1 but it is not convergent normally.
Definition 3.10
AÂ set \(A=\{n_{k}: k\in \mathbb{N}\}\) is said to be statistically dense in \(\mathbb{N}\) if the set
has asymptotic density 1, i.e.,
Theorem 3.11
Let \(\{x_{n}\}\) be a sequence in the PGM-space \((X, F, T)\). Then the following are equivalent:
-
(i)
\(\{x_{n}\}\) statistically converges to a point \(x \in X\).
-
(ii)
There is a sequence \(\{y_{n}\}\) in X such that \(x_{n}=y_{n}\) for almost all n, and \(\{y_{n}\}\) converges to x.
-
(iii)
There is a statistically dense subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) is convergent.
-
(iv)
There is a statistically dense subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) is statistically convergent.
Proof
\((i\Longrightarrow \mathit{ii})\) Let \(\{x_{n}\}\) be a sequence that converges to x, so
For each \(k\in \mathbb{N}\), we can choose an increasing sequence \(\{n_{k}\}\) such that, for every \(n>n_{k}\),
Define the sequence \(\{y_{n}\}\) as follows:
Choose \(k\in \mathbb{N}\) such that \(\frac{1}{2^{k}}<\delta \). It is clear that \(\{y_{m}\}\) converges to x. Fix \(n\in \mathbb{N}\), for \(n_{k}< n\leq n_{k+1}\), we have
Hence,
so
\((\mathit{ii} \Longrightarrow \mathit{iii})\) Let \(\{y_{n}\}\) be a convergent sequence in X and \(A=\{n\in \mathbb{N}: y_{n}\neq x_{n}\}\). We have \(\delta _{l}(A)=1\), so the sequence \(\{y_{n}\}\) is a statistical dense subsequence of \(\{x_{n}\}\) that is convergent.
\((\mathit{iii} \Longrightarrow \mathit{iv})\) It is obvious from Theorem 3.8.
\((\mathit{iv} \Longrightarrow i)\) Let \(\{x_{n_{k}}\}\) be a statistically dense subsequence of \(\{x_{n}\}\) that is statistically convergent to a point \(x\in X\). Set \(A=\{n_{k}: k\in \mathbb{N}\}\), so \(\delta _{l}(A)=1\). For ϵ> and \(0<\delta <1\),
Hence,
So,
Therefore \(\{x_{n}\}\) statistically converges to x. □
The following corollary is a direct consequence of the above theorem.
Corollary 3.12
Every statistically convergent sequence in a PGM-space has a convergent subsequence.
Theorem 3.13
Every statistically convergent sequence in a PGM-space is statistically Cauchy.
Proof
Suppose that \(\{x_{n}\}\) is a sequence that statistically converges to a point x. Let \(\epsilon >0\) and \(0<\delta <1\). Since T is continuous, there are \(0<\delta _{1}<1\) and \(0<\delta _{2}<1\) such that \(T(1-\delta _{1}, 1-\delta _{2})>1-\delta \). On the other hand, there exists \(i_{\epsilon }\) such that
Since
so
Hence
Since \(\{x_{n}\}\) is statistically convergent, so the right-hand side of the previous inequality is zero. Therefore it shows that the sequence \(\{x_{n}\}\) is statistically Cauchy. □
Definition 3.14
Let \((X, F,T)\) be a PGM-space. If every statistically Cauchy sequence is statistically convergent, then \((X, F,T)\) is said to be statistically complete.
Corollary 3.15
Every statistically complete PGM-space is complete.
Proof
Let \((X,F,T)\) be a statistically complete PGM-space. Suppose that \(\{x_{n}\}\) is a Cauchy sequence in \((X,F,T)\), so it is a statistically Cauchy sequence. Since X is statistically complete, so \(\{x_{n}\}\) is statistically convergent. By Corollary 3.12, there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) that converges to a point \(x\in X\). By the continuity of T, for \(0<\delta <1\), there exist \(0<\delta _{1}, \delta _{2}, \delta _{3}, \delta _{4}<1\) such that
Let \(\delta _{5}:=\max \{\delta _{2}, \delta _{3}\}\), then we have
For \(\epsilon >0\), since \(\{x_{n}\}\) is Cauchy, then there exist \(N_{1}\in \mathbb{N}\) and \(x_{i_{\epsilon }}\in \{x_{n}\}\) such that, for all \(i_{1}, i_{2},\ldots, i_{l}\geq N_{1}\),
and since \(x_{n_{k}}\longrightarrow x\), there exists \(N_{2}\geq N_{1}\) such that, for \(i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}\),
For \(i_{1}, i_{2},\ldots, i_{l}, i_{n_{1}}, i_{n_{2}},\ldots, i_{n_{l}}\geq N_{2}\), we have
The third inequality arises from part \((\mathit{ii})\) of Definition 3.3 and the nondecreasing property of F. So, \(\{x_{n}\}\) is convergent and therefore \((X,F,T)\) is complete. □
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Abazari, R. Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology. J Inequal Appl 2021, 134 (2021). https://doi.org/10.1186/s13660-021-02669-w
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DOI: https://doi.org/10.1186/s13660-021-02669-w