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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 [1]) 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 [2], A-statistical convergence [3, 4], statistical summability (C,1) [5, 6], statistical λ-summability [7], statistical σ-convergence [8], statistical A-summability [9], λ-statistical convergence with order α [10], 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 [15]etc. In this paper we define the notion of λ-statistical convergence as a matrix domain of a difference operator [16], 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 [17].

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 |{kn:kK}|, 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)={kn:| 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

t n (x)Las n.

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

Let KN. 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 KN.

A sequence x=( x k ) is said to be λ-statistically convergent (see [12]) to L if for every ϵ>0 the set K ϵ :={kN:| 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 uU.

From the generalized de la Vallée-Poussin mean defined by

t n (x)= 1 λ n k I n x k for x= ( x k ) k ,

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 uU. 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

x k = { j = k j , for  n [ λ n ] + 1 k n , 0 , otherwise.

Then x l and for 0<ε1,

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

Δ x k = { 1 , for  k I n ( j ) , j = 1 , 2 , 3 , , 0 , otherwise.

Then Δx[C,1] and hence, by Theorem 2.1 of [18], 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 {1kn:|Δ(u) x k |ε} {k I n :|Δ(u) x k |ε}{1kn 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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Correspondence to Mohammad Mursaleen.

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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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Keywords

  • statistical convergence
  • λ-statistical convergence
  • difference sequences