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Generalized statistical convergence of difference sequences

Advances in Difference Equations20132013:212

https://doi.org/10.1186/1687-1847-2013-212

Received: 16 March 2013

Accepted: 26 June 2013

Published: 13 July 2013

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.

Keywords

statistical convergenceλ-statistical convergencedifference sequences

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 u Δ x , where Δ x = ( 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 | { 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 - lim x = 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 ) L as  n .

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 [12]) 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 - lim x = 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
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 0 S ( 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
x k = { j = k j , for  n [ λ n ] + 1 k n , 0 , otherwise.
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 0 S ( 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 Δ x [ 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 { 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. □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, Aligarh Muslim University, Aligarh, India

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Copyright

© Alotaibi and Mursaleen; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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