On deferred statistical convergence of sequences of sets

1.   Introduction

The idea of statistical convergence was introduced by Fast ^[1] and Steinhaus ^[2] independently in the same year 1951 and since then several generalizations and applications of this concept have been investigated by various authors namely Bhardwaj and Dhawan ^[3,4], Cakalli ^[5], Cinar et al. ^[6], Caserta et al. ^[7], Colak ^[8], Connor ^[9], Et et al. ^[10,11,12], Esi et al. ^[13], Fridy ^[14], Hazarika et al. ^[15], Isik et al. ^[16,17], Mursaleen ^[18], Nuray and Rhoades ^[19], Salat ^[20], Savas and Et ^[21], Srivastava and Et ^[22], Sengul et al. ^[23,24], Yilmazturk and Kucukaslan ^[25] and many others.

The idea of statistical convergence depends upon the density of subsets of the set $\mathbb{N}$ of natural numbers. The density of a subset $\mathbb{E}$ of $\mathbb{N}$ is defined by

provided that the limit exists, where $\chi _{\mathbb{E}}$ is the characteristic function of the set $\mathbb{E}$. It is clear that any finite subset of $\mathbb{N}$ has zero natural density and that

A sequence $x = \left(x_{k}\right) _{k\in \mathbb{N}}$ is said to be statistically convergent to $L$ if, for every $\varepsilon > 0,$ we have

In this case, we write

Agnew ^[26] introduced the concept of deferred Cesàro mean of real (or complex) valued sequences $x = (x_{k})$ defined by

where $a = \left(a_{n}\right)$ and $b = \left(b_{n}\right)$ are two sequences of non-negative integers satisfying

Deferred density of $\mathbb{K}\subset \mathbb{N}$ defined by

A sequence $x = (x_{k})$ is said to be deferred statistically convergent to $L$ provided that

for each $\varepsilon > 0$ and it is written by $S_{a}^{b}-\lim x_{k} = L$ ^[27].

Let $(X, \rho)$ be a metric space. The distance $d(x, A)$ from a point $x$ to a non-empty subset $A$ of $(X, \rho)$ is defined to be

If $\sup_{k}d\left(x, A_{k}\right) < \infty$ $\left(\text{for each }x\in X\right),$ then we say that the sequence $\left \{ A_{k}\right \}$ is bounded.

2.   Main results

In the present section we shall give the definitions of Wijsman deferred statistical convergence and Wijsman strong deferred Cesàro summability and examine some inclusion properties regarding these concepts.

Definition 1. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$, $A$ and $A_{k}$ be non-empty closed subsets of $X$ for each $k.$ A sequence $\left \{ A_{k}\right \}$ is said to be Wijsman deferred statistically convergent to $A$ (or $WS_{d}-$convergent) provided that

for each $\varepsilon > 0$ and for each $x\in X,$ and it is written by $A_{k}\longrightarrow A\left(WS_{d}\right)$ or $WS_{d}-\lim A_{k} = A.$ The set of all $WS_{d}-$convergent sequences will be denoted by $WS_{d}$. If $b_{n} = n,$ $a_{n} = 0,$ then we write $WS$ instead of $WS_{d}.$

If we take $b_{n} = k_{n},$ $a_{n} = k_{n-1},$ where $(k_{n})$ is a lacunary sequence, then $WS_{d}-$convergence is the same as Wijsman lacunary statistical convergence given by Bhardwaj and Dhawan ^[4].

As an example, consider the following sequence:

Let $b_{n} = k_{n},$ $a_{n} = k_{n-1},$ where $(k_{n})$ is a lacunary sequence and consider a sequence of sets defined by

For $X = \mathbb{R},$ $\rho \left(x, y\right) = \left \vert x-y\right \vert, A = \left \{ 1\right \}$ and $x > 1,$ we have

so $WS_{d}-\lim A_{k} = \left \{ 1\right \}.$

Definition 2. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$, $A$ and $A_{k}$ non-empty closed subsets of $X$ for each $k.$ We say that the sequence $\left \{ A_{k}\right \}$ is Wijsman strong deferred Cesàro convergent to $A$ (or $WN_{d}-$convergent) if for each $x\in X,$

and we write $A_{k}\longrightarrow A\left(WN_{d}\right)$ or $WN_{d}-\lim A_{k} = A.$ The set of all $WN_{d}-$convergent sequences is denoted by $WN_{d}$. If $b_{n} = n,$ $a_{n} = 0,$ then we write $WN$ instead of $WN_{d}.$

If we take $b_{n} = k_{n},$ $a_{n} = k_{n-1},$ where $(k_{n})$ is a lacunary sequence, then $WN_{d}-$convergence coincides with Wijsman lacunary strong Cesàro convergence given by Bhardwaj and Dhawan ^[4].

As an example, consider the following sequence:

Let $b_{n} = k_{n},$ $a_{n} = k_{n-1},$ where $(k_{n})$ is a lacunary sequence and consider a sequence defined by

Let $X = \mathbb{R},$ $x, y\in X,$ $\rho \left(x, y\right) = \left \vert x-y\right \vert,$ $A = \left \{ 1\right \}$ and $x > 1.$ Since

the sequence $\left \{ A_{k}\right \}$ is $WN_{d}-$convergent to $\left \{ 1\right \}.$

Here and in what follows we suppose that the sets $A$ and $A_{k}$ $\left(\text{for each }k\in \mathbb{N}\right)$ are non-empty closed subsets of $X$ for each $k.$

Theorem 1. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X$, then every Wijsman strong deferred Cesàro convergent sequence is Wijsman deferred statistically convergent, but the converse is not true.

Proof. First part of the proof is easy. For the converse, take $b_{n} = k_{n},$ $a_{n} = k_{n-1},$ where $(k_{n})$ is a lacunary sequence, $\rho \left(x, y\right) = \left \vert x-y\right \vert,$ $X = \mathbb{R}$ and define $\left \{ A_{k}\right \}$ by

Note that $\left \{ A_{k}\right \}$ is not a bounded sequence and for each $x\in X$, we have

so $\left \{ A_{k}\right \} \notin WN_{d},$ but

and so $\left \{ A_{k}\right \} \in WS_{d}.$

From Theorem 1 we have the following:

Corollary 1. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X$, then every Wijsman strong Cesàro convergent sequence is Wijsman statistically convergent, but the converse is not true.

Theorem 2. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X.$ If the sets $A$ and $A_{k}$ $\left(\text{for each }k\in \mathbb{N}\right)$ are bounded, then every Wijsman deferred statistically convergent sequence is Wijsman strong deferred Cesàro convergent.

Proof. Let $\left \{ A_{k}\right \}$ be Wijsman deferred statistically convergent to $A$ and $\varepsilon > 0$ be given. Then there exists $x\in X$ such that

Since $A, A_{k}$ (for each $k\in \mathbb{N)}$ are bounded$,$ we can write (for each $x\in X$ and $\varepsilon > 0)$

Taking limit $n\rightarrow \infty,$ we get $WN_{d}-\lim A_{k} = A.$

From Theorem 2 we have the following:

Corollary 2. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X.$ If the sets $A$ and $A_{k}$ $\left(\text{for each }k\in \mathbb{N}\right)$ are bounded, then every Wijsman statistically convergent sequence is Wijsman strong Cesàro convergent.

Theorem 3. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X,$ If $\lim \limits_{n}\dfrac{b_{n}-a_{n}}{n} = a > 0, \left(a\in \mathbb{R}\right)$ and $b_{n} < n$, then every $WS-$convergent sequence to $A$ is $WS_{d}-$convergent to $A.$

Proof. Let $\left \{ A_{k}\right \}$ be a $WS-$convergent sequence to $A$, $\lim \limits_{n}\dfrac{b_{n}-a_{n}}{n} = a > 0$ and $b_{n} < n.$ For a given $\varepsilon > 0,$ we have

therefore

So $\left \{ A_{k}\right \}$ is $WS_{d}-$convergent to $A.$

Theorem 4. Let $\left(a_{n}\right), \left(a_{n}^{^{\prime }}\right), \left(b_{n}^{^{\prime }}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the following condition and $\left(2\right)$

$A, A_{k}\subset X$ and suppose that the sets $\left \{ k:a_{n} < k\leq a_{n}^{^{\prime }}\right \}$ and $\left \{ k:b_{n}^{^{\prime }} < k\leq b_{n}\right \}$ are finite for all $n\in \mathbb{N},$ then every $WS_{d^{^{\prime }}}-$convergent sequence is $WS_{d}-$convergent, where

Proof. Let us assume that the sequence $\left \{ A_{k}\right \}$ is $WS_{d^{^{\prime }}}-$convergent. Then for any $\varepsilon > 0$ we have

Taking limit when $n\rightarrow \infty,$ we get

Theorem 5. Let $\left(a_{n}\right), \left(a_{n}^{^{\prime }}\right), \left(b_{n}^{^{\prime }}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying $\left(2\right)$ and $\left(3\right)$ such that

and $A, A_{k}\subset X,$ then every $WS_{d}-$convergent sequence is $WS_{d^{^{\prime }}}-$convergent.

Proof. It is easy to see that the inclusion

holds and so the following inequality too

Therefore we have

Taking limits when $n\rightarrow \infty,$ we get

Theorem 6. Let $\left(a_{n}\right), \left(a_{n}^{^{\prime }}\right), \left(b_{n}^{^{\prime }}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying $\left(2\right), \left(3\right), \left(4\right)$ and $A, A_{k}\subset X$, then every $WN_{d}-$ convergent sequence is $WN_{d^{^{\prime }}}-$convergent.

Proof. Proof follows from the inequality

Theorem 7. Let $\left(a_{n}\right), \left(a_{n}^{^{\prime }}\right), \left(b_{n}^{^{\prime }}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying $\left(2\right)$ and $\left(3\right),$ $A_{k}$ $\left(\text{for each }k\in \mathbb{N}\right)$ and $A$ are bounded ($A, A_{k}\subset X)$ and suppose that the sets $\left \{ k:a_{n} < k\leq a_{n}^{^{\prime }}\right \}$ and $\left \{ k:b_{n}^{^{\prime }} < k\leq b_{n}\right \}$ are finite for all $n\in \mathbb{N}$, then every $WN_{d^{^{\prime }}}-$convergent sequence is $WN_{d}-$convergent$.$

Proof. Since $A_{k}$ $\left(\text{for each }k\in \mathbb{ N}\right)$ and $A$ are bounded, we have $\left \vert d\left(x, A_{k}\right) -d\left(x, A\right) \right \vert \leq M$ for some $M > 0$. So we have

Taking limit when $n\rightarrow \infty,$ we get

In the following theorem, by changing the conditions on the sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ we give the same relation with Theorem 3.

Theorem 8. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X,$ and let $\lim \inf_{n}\dfrac{b_{n}}{ a_{n}} > 1$. If the sequece $\left \{ A_{k}\right \}$ is Wijsman statistically convergent to $A,$ then it is Wijsman deferred statistically convergent to $A.$

Proof. Let $\lim \inf_{n}\dfrac{b_{n}}{a_{n}} > 1,$ then we can find a number $r > 0$ such that $\dfrac{b_{n}}{a_{n}} > 1+r$ for sufficiently large $n,$ which implies that

If $WS-\lim A_{k} = A,$ then we have

So we have $WS_{d}-\lim A_{k} = A.$

Theorem 9. Let $A, A_{k}\subset X$ and $sup_{n}\left(\dfrac{b_{n}}{b_{n}-a_{n}}\right) < \infty.$ Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying $\left(2\right)$ such that

$i)$ $\underset{n\rightarrow \infty }{\lim }a_{n} = \infty$,

$ii)\, \underset{n\rightarrow \infty }{\lim }b_{n}-a_{n} = \infty.$

If $\left \{ A_{k}\right \}$ is Wijsman strong Cesàro convergent to $A,$ then it is Wijsman strong deferred Cesàro convergent to $A.$

Proof. Suppose that $sup_{n}\dfrac{b_{n}}{b_{n}-a_{n}} < \infty,$ then $sup_{n}\left(\dfrac{a_{n}}{b_{n}-a_{n}}\right) < \infty$. In this case we can find positive numbers $M$ and $K$ such that $\dfrac{b_{n}}{ b_{n}-a_{n}}\leq M$ and $\dfrac{a_{n}}{b_{n}-a_{n}}\leq K.$ Then we have

So $\left \{ A_{k}\right \}$ is $WN_{d}-$convergent to $A.$

In the following theorems, by changing the conditions on the sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ we give the same relation with Theorem 9.

Theorem 10. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X,$ If $\lim \inf_{n}\dfrac{b_{n}}{a_{n}} > 1$, then $WN\subset WN_{d}.$

Theorem 11. Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of non-negative integers satisfying the conditions $(2)$ and $A, A_{k}\subset X,$ if $\lim \inf_{n}\dfrac{\left(b_{n}-a_{n}\right) }{n} > 0$ and $b_{n} < n$ then $WN\subset WN_{d}.$

3.   Conclusion

The concepts of Wijsman statistical convergence and Wijsman strong Cesàro summability for sequences of sets were introduced and studied by Nuray and Rhoades ^[19] in 2012 and then the concepts were improved by Bhardwaj et al. ^[4], Esi et al. ^[13], Hazarika et al. ^[15] and Sengul ^[24]. In this paper we study the concepts of Wijsman deferred statistical convergence and Wijsman strong deferred Cesàro summability for sequences of sets. The results which we obtained in this study are much more general than those obtained by others. To get these general results, we introduce some of fairly wide classes of sequences of sets using two sequences of non-negative integers satisfying the conditions $a_{n} < b_{n}\,$and$\, \underset{n\rightarrow \infty }{\lim }b_{n} = \infty.$ Researchers who are working in this area can study the concepts of Wijsman deferred statistical convergence of order $\alpha$ and Wijsman strong deferred Cesàro summability of order $\alpha$ for sequences of sets, where $0 < \alpha \leq 1.$

Acknowledgments

This research was supported by FUBAP (The Management Union of the Scientific Research Projects of Firat University) under the Project Number: FUBAB FF.19.05.

Conflict of interest

The authors declare that they have no conflict of interests.