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A new variant of statistical convergence
Journal of Inequalities and Applications volume 2013, Article number: 309 (2013)
Abstract
In this paper we study the notion of statistical -summability, which is a generalization of statistical A-summability. We study here many other related concepts and its relations with statistical convergence and λ-statistical convergence and provide some interesting examples.
1 Introduction and preliminaries
The concept of statistical convergence was first introduced by Fast [1]. In 1953 the concept arose as an example of convergence in density as introduced by Buck [2]. Schoenberg [3] studied statistical convergence as a summability method and Zygmund [4] established a relation between it and strong summability. This idea has grown a little faster after the papers of Šalát [5] , Fridy [6] , Connor [7, 8], Kolk [9], Mursaleen [10], Mursaleen and Edely [11, 12], Mursaleen and Mohiuddine [13–17] and many others. Its various generalizations, extensions and variants have been studied by various authors so far. For example, lacunary statistical convergence [18], λ-statistical convergence [10, 19–21], A-statistical convergence [9], statistical summability [22–24]; statistical λ-summability [25], statistical lacunary summability [26], statistical A-summability [27]etc. For more details, related concepts and applications, we refer to [28–41] and references therein. Here we define the notion of statistical -summability as a λ-statistical convergence of A-transform of x and prove some results on some related sets of sequences. The results of this paper extend several ones obtained up to now and establish several inclusion relations, implications and other properties.
Let , the set of natural numbers. Then the natural density of K is defined by
if the limit exists, where the vertical bars denote the cardinality of the enclosed set.
The idea of λ-statistical convergence was introduced in [10] as follows:
Let be a non-decreasing sequence of positive numbers tending to ∞ such that
The generalized de la Vallée-Poussin mean is defined by
where .
Let . Then
is said to be λ-density of K.
In case , λ-density reduces to the natural density. Also, since , for every .
A sequence is said to be λ-statistically convergent to L if for every the set has λ-density zero, i.e., . That is,
In this case we write .
Let be an infinite matrix of real or complex numbers and be a sequence of real or complex numbers. Then we write , which is called the A-transform of the sequence whenever the series on the right converges for each .
We assume throughout this paper that the symbols ω and c denote the spaces of all sequences (real or complex numbers) and the space of all convergent sequences, respectively. Let X and Y be two nonempty subsets of the space ω. If implies , then we say that A defines a matrix transformation from X into Y, and we denote by the class of matrices A which transform X into Y. By we denote the subset of for which limit or sum is preserved.
A matrix is said to be conservative if for , and we denote this by .
A matrix is said to be regular if it is conservative and , and we denote this by .
The following are well-known Silverman-Toeplitz [42] conditions for the regularity of A.
A matrix is regular, i.e., if and only if
(i) ;
(ii) , for each k;
(iii) .
Let be a non-negative regular matrix. A sequence x is said to be statistically A-summable to L if, for every , , i.e.,
where . Thus x is statistically A-summable to L if and only if Ax is statistically convergent to L. In this case we write .
2 Statistical -summability
In [43], Malafosse and Rakočević presented the following definition of statistically -summable.
Definition 2.1 A sequence x is said to be statistically -summable to L if for every , , i.e.,
Thus x is statistically -summable to L if and only if Ax is λ-statistically convergent to L. In this case we write . By we denote the set of all statistically -summable sequences.
We define the following.
Definition 2.2 A sequence is said to be strongly -convergent () to the limit L if , and we write it as . In this case L is called the -limit of x.
Remarks 2.3
(i) If (the unit matrix), then the statistical -summability is reduced to the λ-statistical convergence.
(ii) If , then the statistical -summability is reduced to the statistical A-summability.
(iii) If and
then the statistical -summability is reduced to the statistical -summability due to Moricz [22].
(iv) If and
then the statistical -summability is reduced to the statistical -summability due to Moricz and Orhan [44], where is a sequence of nonnegative numbers such that and
(v) If and
where , then the statistical -summability is reduced to the statistical -summability due to Moricz [45].
3 Main results
In this section, we establish the relation between statistical -summability and A-statistical convergence.
Theorem 3.1 If a bounded sequence is A-statistically convergent to ℓ and , then it is A summable to ℓ, statistically A-summable to ℓ, and hence statistically -summable to ℓ but not conversely.
Proof Let x be bounded and A-statistically convergent to L, and . Then
By using the definition of A-statistical convergence and the conditions of regularity of A, we get
and hence , i.e., x is statistically A-summable to L. Now, using Theorem 3.1 of [10], we get , i.e., x is statistically -summable to L.
To see that the converse does not hold, we construct the following example.
Let and A be a Cesàro matrix, i.e.,
Let
Then x is A-summable to (and hence statistically -summable to ) but not A-statistically convergent.
This completes the proof of the theorem. □
Theorem 3.2 If and x is statistically -summable to L, then x is statistically A-summable to L.
Proof Let . Then there exists such that for all n. Since and
we have
Now, taking the limit as , we get the desired result. □
Theorem 3.3 Statistical -summability implies statistical A-summability if and only if
Proof For , we have
Therefore
Taking the limit as and using (3.1), we get that statistical -summability implies statistical A-summability.
Conversely, suppose that
Choose a subsequence such that . Define a sequence such that
Then, as in Theorem 3.1 of [10], we get that is not λ-statistically convergent, i.e., x is not statistically -summable. Hence (3.1) is necessary.
This completes the proof of the theorem. □
Theorem 3.4 (a) If and a sequence is strongly -convergent to the limit L, then x is statistically -convergent to L.
(b) If is bounded and statistically -convergent to L, then .
Proof (a) It follows easily from the following:
The following example shows that the inclusion is proper. Let be such that its A-transform is given by
Then and for ,
i.e., x is statistically -convergent to 0. But
i.e., x is not strongly -convergent to the limit 0.
(b) Suppose is bounded and statistically -convergent to L. Then for all k, where . For , we have
Hence if x is statistically -convergent to L.
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. (334/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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Mohiuddine, S.A., Alotaibi, A. & Mursaleen, M. A new variant of statistical convergence. J Inequal Appl 2013, 309 (2013). https://doi.org/10.1186/1029-242X-2013-309
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DOI: https://doi.org/10.1186/1029-242X-2013-309