Open Access

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 convergence λ-statistical convergence difference 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
(2)
Department of Mathematics, Aligarh Muslim University

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© Alotaibi and Mursaleen; licensee Springer 2013

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