# Generalized statistical convergence of difference sequences

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## Abstract

In this paper we define the $λ(u)$-statistical convergence that generalizes, in a certain sense, the notion of λ-statistical convergence. We find some relations with sets of sequences which are related to the notion of strong convergence.

MSC:40A05, 40H05.

## 1 Introduction and preliminaries

The notion of statistical convergence (see Fast ) has been studied in various setups, and its various generalizations, extensions and variants have been studied by various authors so far. For example, lacunary statistical convergence , A-statistical convergence [3, 4], statistical summability $(C,1)$ [5, 6], statistical λ-summability , statistical σ-convergence , statistical A-summability , λ-statistical convergence with order α , lacunary and λ-statistical convergence in a solid Riesz space [11, 12], lacunary statistical convergence and ideal convergence in random 2-normed spaces [13, 14], generalized weighted statistical convergence etc. In this paper we define the notion of λ-statistical convergence as a matrix domain of a difference operator , which is obtained by replacing the sequence x by $uΔx$, where $Δx= ( x k − x k + 1 ) k = 1 ∞$ and $u= ( u k ) k = 1 ∞$ is another sequence with $u k ≠0$ for all k. We find some relations with sets of sequences which are related to the notion of strong convergence .

Let K be a subset of the set of natural numbers . Then the asymptotic density of K denoted by $δ(K)$ is defined as $δ(K)= lim n 1 n |{k≤n:k∈K}|$, where the vertical bars denote the cardinality of the enclosed set.

A number sequence $x=( x k )$ is said to be statistically convergent to the number L if for each $ϵ>0$, the set $K(ϵ)={k≤n:| x k −L|>ϵ}$ has asymptotic density zero, i.e.,

$lim n 1 n | { k ≤ n : | x k − L | } | =0.$

In this case, we write $S-limx=L$.

Let $λ=( λ n )$ be a non-decreasing sequence of positive numbers tending to ∞ such that

$λ n + 1 ≤ λ n +1, λ 1 =0.$

The generalized de la Vallée-Poussin mean is defined by

$t n (x)=: 1 λ n ∑ j ∈ I n x j ,$

where $I n =[n− λ n +1,n]$.

A sequence $x=( x j )$ is said to be $(V,λ)$-summable to a number L if

In this case, L is called the λ-limit of x.

Let $K⊆N$. Then the λ-density of K is defined by

$δ λ (K)= lim n 1 λ n | { n − λ n + 1 ≤ j ≤ n : j ∈ K } | .$

In case $λ n =n$, λ-density reduces to the asymptotic density. Also, since $( λ n /n)≤1$, $δ(K)≤ δ λ (K)$ for every $K⊆N$.

A sequence $x=( x k )$ is said to be λ-statistically convergent (see ) to L if for every $ϵ>0$ the set $K ϵ :={k∈N:| x k −L|≥ϵ}$ has λ-density zero, i.e., $δ λ ( K ϵ )=0$. That is,

$lim n 1 λ n | { k ∈ I n : | x k − L | ≥ ϵ } | =0.$

In this case, we write $S λ -limx=L$ and we denote the set of all λ-statistically convergent sequences by $S λ$.

## 2 $λ(u)$-Statistical convergence

We consider the infinite matrix of first difference $Δ= ( a n m ) n , m ≥ 1$ defined by $a n n =1$, $a n , n + 1 =−1$ and $a n m =0$ otherwise. Let $D u$ be the diagonal matrix defined by $[ D u ] n n = u n$ for all n and consider the set U of all sequences such that $u n ≠0$ for all n. Then we write $Δ(u)= D u Δ$ for $u∈U$.

From the generalized de la Vallée-Poussin mean defined by

we are led to define the following sets:

$[ V , λ ] 0 ( Δ ( u ) ) = { x = ( x k ) : lim n → ∞ 1 λ n ∑ k ∈ I n | Δ ( u ) x k | = 0 } [ V , λ ] 0 ( Δ ( u ) ) = { x = ( x k ) : lim n → ∞ 1 λ n ∑ k ∈ I n | u k ( x k − x k + 1 ) | = 0 } , [ V , λ ] ∞ ( Δ ( u ) ) = { x = ( x k ) : sup n 1 λ n ∑ k ∈ I n | Δ ( u ) x k | < ∞ } [ V , λ ] ∞ ( Δ ( u ) ) = { x = ( x k ) : sup n 1 λ n ∑ k ∈ I n | u k ( x k − x k + 1 ) | = 0 } .$

In the case when $λ n =n$, we write the previous sets $[ V ] 0 (Δ(u))$ and $[ V ] ∞ (Δ(u))$, respectively. Now we can state the definition of $λ(u)$-statistical convergence to zero.

A sequence $x= ( x k ) k ≥ 1$ is said to be $λ(u)$-statistically convergent to zero if for every $ε>0$,

$lim n → ∞ 1 λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } |=0.$

In this case, we write $x k →0 S λ (Δ(u))$. If $λ n =n$ for all n, we then write $x k →0S(Δ(u))$.

## 3 Main results

We are ready to prove the following result.

Theorem 1 Let $u∈U$. Then

1. (a)

$[ V , λ ] 0 (Δ(u))⊂ S λ 0 (Δ(u))$ and the inclusion is proper,

2. (b)

if $x∈ l ∞$ and $x k →0 S λ (Δ(u))$, then $x∈ [ V , λ ] 0 (Δ(u))$,

3. (c)

$S λ 0 (Δ(u))∩ l ∞ = [ V , λ ] 0 (Δ(u))∩ l ∞$.

Proof (a) Let $ε>0$ be given and $x∈ [ V , λ ] 0 (Δ(u))$. Then we have

$1 λ n ∑ k ∈ I n |Δ(u) x k |≥ 1 λ n ∑ k ∈ I n | x k − L | ≥ ε |Δ(u) x k |≥ ε λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } |.$

Therefore $x∈ S λ 0 (Δ(u))$. The following example shows that the inclusion is proper: Let $x=( x k )$ be defined by

Then $x∉ l ∞$ and for $0<ε≤1$,

$1 λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } |= [ λ n ] λ n →0(n→∞),$

i.e., $x∈ S λ 0 (Δ(u))$. But

$1 λ n ∑ k ∈ I n |Δ(u) x k |↛0(n→∞),$

i.e., $x∉ [ V , λ ] 0 (Δ(u))$.

1. (b)

Let $x∈ l ∞$ and $x k →0 S λ (Δ(u))$. Then $|Δ(u) x k |≤M$ for all k, where $M>0$. For $ε>0$, we have

$1 λ n ∑ k ∈ I n | Δ ( u ) x k | = 1 λ n ∑ k ∈ I n | x k − L | ≥ ϵ | Δ ( u ) x k | + 1 λ n ∑ k ∈ I n | x k − L | < ϵ | Δ ( u ) x k | ≤ M λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } | + ε .$

Hence, $x∈ [ V , λ ] 0 (Δ(u))$.

1. (c)

This immediately follows from (a) and (b).

This completes the proof of the theorem. □

Theorem 2 $S 0 (Δ(u))⊆ S λ 0 (Δ(u))$ if and only if where by $x∈ S 0 (Δ(u))$ (or $x∈ S λ 0 (Δ(u))$) we mean $x k →0S(Δ(u))$ (or $x k →0 S λ (Δ(u))$).

Proof For $ε>0$, we have

${ k ∈ I n : | Δ ( u ) x k | ≥ ε } ⊂ { k ≤ n : | Δ ( u ) x k | ≥ ε } .$

Therefore

$1 n | { k ≤ n : | Δ ( u ) x k | ≥ ε } | ≥ 1 n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } | ≥ λ n n ⋅ 1 λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } | .$

Taking the limit as $n→∞$ and using (), we get the inclusion.

Conversely, suppose that

$lim inf n → ∞ λ n n =0.$

Choose a subsequence $( n ( j ) ) j ≥ 1$ such that $λ n ( j ) n ( j ) < 1 j$. Define a sequence $x= ( x k ) k ≥ 1$ such that

Then $Δx∈[C,1]$ and hence, by Theorem 2.1 of , $x∈ S 0 (Δ(u))$. On the other hand, $x∉ [ V , λ ] 0 (Δ(u))$ and Theorem 1(b) implies that $x∉ S λ 0 (Δ(u))$. Hence, () is necessary.

This completes the proof of the theorem. □

Presently, for the reverse inclusion, we have only one way condition.

Theorem 3 If $lim sup n (n− λ n )<∞$, then $S λ 0 (Δ(u))⊆ S 0 (Δ(u))$.

Proof Let $lim sup n (n− λ n )<∞$. Then there exists $M>0$ such that $n− λ n ≤M$ for all n. Since $1 n ≤ 1 λ n$ and ${1≤k≤n:|Δ(u) x k |≥ε}$ $⊆{k∈ I n :|Δ(u) x k |≥ε}∪{1≤k≤n− λ n :|Δ(u) x k |≥ε}$, we have

$1 n | { 1 ≤ k ≤ n : | Δ ( u ) x k | ≥ ε } | ≤ 1 λ n | { 1 ≤ k ≤ n : | Δ ( u ) x k | ≥ ε } | ≤ 1 λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } | + 1 λ n | { k ≤ n − λ n : | Δ ( u ) x k | ≥ ε } | ≤ 1 λ n | { k ∈ I n : | Δ ( u ) x k | ≥ ε } | + M λ n .$

Now, taking the limit as $n→∞$, we get $S λ 0 (Δ(u))⊆ S 0 (Δ(u))$.

This completes the proof of the theorem. □

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## Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under Grant No. (174/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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### Competing interests

The authors declare that they have no competing interests.

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Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

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Alotaibi, A., Mursaleen, M. Generalized statistical convergence of difference sequences. Adv Differ Equ 2013, 212 (2013). https://doi.org/10.1186/1687-1847-2013-212 