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A generalization on ℐ-asymptotically lacunary statistical equivalent sequences
Journal of Inequalities and Applications volume 2013, Article number: 270 (2013)
Abstract
In this study we extend the concept of ℐ-asymptotically lacunary statistical equivalent sequences by using the sequence which is the sequence of positive real numbers where θ is a lacunary sequence and ℐ is an ideal of the subset of positive integers.
MSC:40G15, 40A35.
1 Introduction
The concept of ℐ-convergence was introduced by Kostyrko et al. in a metric space [1]. Later it was further studied by Dems [2], Das and Savaş [3], Savaş [4–7] and many others. ℐ-convergence is a generalization form of statistical convergence, which was introduced by Fast (see [8]) and that is based on the notion of an ideal of the subset of positive integers ℕ.
Definition 1.1 A family is said to be an ideal of ℕ if the following conditions hold:
-
(a)
implies ,
-
(b)
, implies .
An ideal is called non-trivial if , and a non-trivial ideal is called admissible if for each .
Definition 1.2 A family of sets is a filter in ℕ if and only if:
-
(i)
.
-
(ii)
For each , we have .
-
(iii)
For each and each , we have .
Proposition 1.1 ℐ is a non-trivial ideal in ℕ if and only if
is a filter in ℕ (see [1]).
Definition 1.3 A real sequence is said to be ℐ-convergent to if and only if for each the set
belongs to ℐ. The number L is called the ℐ-limit of the sequence x (see [1]).
Remark 1 If we take . Then is a non-trivial admissible ideal of N and the corresponding convergence coincides with the usual convergence.
A lacunary sequence is an increasing integer sequence such that and as . The intervals determined by θ are denoted and the ratio is denoted by .
In 1993, Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices.
Definition 1.4 [9]
Two nonnegative sequences and are said to be asymptotically equivalent if
and it is denoted by .
Definition 1.5 (Fridy [10])
The sequence has statistic limit L, denoted by , provided that for every ,
In 2003, Patterson defined asymptotically statistical equivalent sequences by using the definition of statistical convergence as follows.
Definition 1.6 (Patterson [11])
Two nonnegative sequences and are said to be asymptotically statistical equivalent of multiple L provided that for every ,
(denoted by ), and simply asymptotically statistical equivalent if .
In 2006, Patterson and Savaş presented definitions for asymptotically lacunary statistical equivalent sequences (see [12]).
Definition 1.7 Let be a lacunary sequence, two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every
where the vertical bars indicate the number elements in the enclosed set.
Definition 1.8 Let be a lacunary sequence, two number sequences and are said to be strong asymptotically lacunary equivalent of multiple L provided that
In 2008, Savaş and Patterson gave an extension on asymptotically lacunary statistical equivalent sequences, and they investigated some relations between strongly asymptotically lacunary equivalent sequences and strongly Cesáro asymptotically equivalent sequences. More applications of the asymptotically statistical equivalent sequences can be seen in [13–16].
Definition 1.9 [17]
Let be a lacunary sequence and let be a sequence of positive real numbers. Two number sequences and are said to be strongly asymptotically lacunary equivalent of multiple L provided that
(denoted by ) and simply strongly asymptotically lacunary equivalent if .
Definition 1.10 Let be a sequence of positive real numbers. Two number sequences and are said to be strongly Cesáro asymptotically equivalent to L provided that
(denoted by ) and simply strongly Cesáro asymptotically equivalent if .
The following definitions are given in [3].
Definition 1.11 A sequence is said to be ℐ-statistically convergent to L or -convergent to L if, for any and ,
In this case, we write . The class of all ℐ-statistically convergent sequences will be denoted by .
Definition 1.12 Let θ be a lacunary sequence. A sequence is said to be ℐ-lacunary statistically convergent to L or -convergent to L if, for any and ,
In this case, we write . The class of all ℐ-lacunary statistically convergent sequences will be denoted by .
Definition 1.13 Let θ be a lacunary sequence. A sequence is said to be strong ℐ-lacunary convergent to L or -convergent to L if, for any ,
In this case, we write . The class of all strong ℐ-lacunary statistically convergent sequences will be denoted by .
Throughout ℐ will stand for a proper admissible ideal of ℕ, and by sequence we always mean sequences of real numbers.
Recently, Savaş defined ℐ-asymptotically lacunary statistical equivalent sequences by using the definitions ℐ-convergence and asymptotically lacunary statistical equivalent sequences together.
Definition 1.14 [18]
Let be a lacunary sequence; the two nonnegative sequences and are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every and ,
In this case we write .
2 Main results
In this section we shall give some new definitions and also examine some inclusion relations.
Definition 2.1 Let be a lacunary sequence and let be a sequence of positive real numbers. Two number sequences and are said to be strongly ℐ-asymptotically lacunary equivalent of multiple L for the sequence p provided that
In this situation we write .
If we take for all , we write instead of .
Definition 2.2 Let be a lacunary sequence and let be a sequence of positive real numbers. Two number sequences and are said to be strongly Cesáro ℐ-asymptotically equivalent of multiple L provided that
(denoted by ) and simply strongly Cesáro ℐ-asymptotically equivalent if .
Theorem 2.1 Let be a lacunary sequence. Then:
-
(a)
If , then ;
-
(b)
If and , then ;
-
(c)
.
Proof (a) Let and be given. Then
and so
Then, for any ,
Therefore .
-
(b)
Let x and y be bounded sequences and . Then there is an M such that for all k. For each ,
Then, for any ,
Therefore .
-
(c)
This follows from (a) and (b). □
Theorem 2.2 Let be a lacunary sequence, and . Then
Proof Assume that and . Then
and
Thus we have . □
Theorem 2.3 Let x and y be bounded sequences, and . Then
Proof Suppose that x and y are bounded and . Then there is an integer K such that for all k,
and
Thus we have . □
Theorem 2.4 Let be a lacunary sequence with , then
Proof If , then there exists such that for all . Since , we have and . Let and define the set
We can easily say that , which is the filter of the ideal ℐ,
for each . Choose . Therefore,
and it completes the proof. □
For the next result, we assume that the lacunary sequence θ satisfies the condition that for any set , .
Theorem 2.5 Let be a lacunary sequence with , then
Proof If , then there exists such that for all . Let and define the sets T and R such that
and
Let
for all . It is obvious that . Choose n is any integer with , where ,
Choose and in view of the fact that , where , it follows from our assumption on θ that the set R also belongs to and this completes the proof of the theorem. □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
We would like to thanks the referees for their valuable comments.
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Savaş, E., Gumuş, H. A generalization on ℐ-asymptotically lacunary statistical equivalent sequences. J Inequal Appl 2013, 270 (2013). https://doi.org/10.1186/1029-242X-2013-270
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DOI: https://doi.org/10.1186/1029-242X-2013-270