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On asymptotically double lacunary statistical equivalent sequences in ideal context
Journal of Inequalities and Applications volume 2013, Article number: 543 (2013)
Abstract
An ideal ℐ is a family of subsets of positive integers which is closed under taking finite unions and subsets of its elements. In this paper, we present some definitions which are a natural combination of the definition of asymptotic equivalence, statistical convergence, lacunary statistical convergence, double sequences and an ideal. In addition, we also present asymptotically ℐ-equivalent double sequences and study some properties of this concept.
MSC:40A35, 40G15.
1 Introduction
Pobyvancts [1] introduced the concept of asymptotically regular matrices which preserve the asymptotic equivalence of two nonnegative numbers sequences. Marouf [2] presented definitions for asymptotically equivalent and asymptotic regular matrices. Patterson [3] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Patterson and Savaş [4] introduced the concept of an asymptotically lacunary statistical equivalent sequences of real numbers.
The idea of statistical convergence was formerly given under the name ‘almost convergence’ by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 [5]. The concept was formally introduced by Steinhaus [6] and Fast [7], and later, it was introduced by Schoenberg [8] and also independently by Buck [9]. A lot of developments have been made in this area after the works of S̆alát [10] and Fridy [11]. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Fridy and Orhan [12] introduced the concept of lacunary statistical convergence. Mursaleen and Mohiuddine [13] introduced the concept of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. Savaş and Patterson [14, 15] introduced the concept of lacunary statistical convergence for double sequences. Recently Mohiuddine et al. [16] introduced statistical convergence of double sequences in locally solid Riesz spaces. For details related to lacunary statistical convergence, we refer to [12, 17–24].
Kostyrko et al. [25] introduced the notion of I-convergence with the help of an admissible ideal, I denotes the ideal of subsets of ℕ, which is a generalization of statistical convergence. Quite recently, Das et al. [26] unified these two approaches to introduce new concepts of I-statistical convergence, I-lacunary statistical convergence and investigated some of their consequences. The notion of lacunary ideal convergence of real sequences was introduced in [27, 28]. Hazarika [29, 30] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some basic properties of this notion. Kumar and Sharma [31] studied asymptotically generalized statistical equivalent sequences using ideals. Recently Savas [32] and Savas and Gumus [33] studied ideal asymptotically lacunary statistical equivalent single sequences. For more applications of ideals, we refer to [34–47].
In this paper, we define asymptotically lacunary statistical equivalent double sequences using an ideal and establish some basic results regarding this notion.
2 Definitions and preliminaries
In this section, we recall some definitions and notations, which form the base for the present study.
A family of sets (power sets of ℕ) is called an ideal if and only if for each , we have , and for each and each , we have . A non-empty family of sets is a filter on ℕ if and only if , for each , we have and each and each , we have . An ideal I is called non-trivial ideal if and . Clearly, is a non-trivial ideal if and only if is a filter on ℕ. A non-trivial ideal is called admissible if and only if . A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal containing I as a subset. Further details on ideals of can be found in Kostyrko et al. [25]. Recall that a sequence of points in ℝ is said to be I-convergent to a real number ℓ if for every [25]. In this case, we write .
By a lacunary sequence , where , we mean an increasing sequence of non-negative integers with as . The intervals determined by θ will be denoted by , and the ratio will be defined by (see [48]).
The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of ℕ. A subset of ℕ is said to have natural density if
Definition 2.1 A real or complex number sequence is said to be statistically convergent to L if for every ,
In this case, we write or , and S denotes the set of all statistically convergent sequences.
Definition 2.2 [12]
A sequence is said to be lacunary statistically convergent to the number L if for every ,
Let denote the set of all lacunary statistically convergent sequences. If , then is the same as S.
Definition 2.3 [26]
Let be a non-trivial ideal. A sequence is I-statistically convergent to L if for each and ,
In this case, we write .
Definition 2.4 [26]
Let be a non-trivial ideal. A sequence is said to be I-lacunary statistically convergent to L if for each and ,
In this case, we write . If , then is the same as .
Definition 2.5 [2]
Two nonnegative sequences and are said to be asymptotically equivalent if
denoted by .
Definition 2.6 [3]
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 .
Patterson and Savas [4] defined the asymptotically lacunary statistical equivalent sequences as follows.
Definition 2.7 Two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ,
denoted by and simply asymptotically lacunary statistical equivalent if . If we take , then we get Definition 2.6.
By the convergence of a double sequence, we mean the convergence in Pringsheim’s sense [49]. A double sequence has a Pringsheim limit L (denoted by ) provided that given an , there exists an such that , whenever . We describe such an more briefly as `P-convergent’. We denote the space of all P-convergent sequences by . The double sequence is bounded if there exists a positive number M such that for all k and l. We denote all bounded double sequences by .
Let and denote the number of in K such that and (see [50]). Then the lower natural density of K is defined by . In case the sequence has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined by .
For example, let . Then
i.e., the set K has double natural density zero, while the set has double natural density .
Definition 2.8 [50]
A real double sequence is said to be P-statistically convergent to ℓ provided that for each ,
We denote the set of all statistical convergent double sequences by .
The double sequence is called double lacunary sequence if there exist two increasing sequences of integers such that (see [15])
and
Notations: , and is determined by
Definition 2.9 A double sequence is said to be double lacunary convergent to L if
In this case, we write . We denote the set of all double lacunary convergent sequences.
Definition 2.10 [51]
Let be a non-trivial ideal. A double sequence is said to be ℐ-convergent to L if for each ,
In this case, we write .
Throughout the paper, we denote ℐ as admissible ideal of subsets of , unless otherwise stated.
3 Asymptotically lacunary statistical equivalent double sequences using ideals
In this section, we define asymptotically ℐ-equivalent, asymptotically ℐ-statistical equivalent, asymptotically ℐ-lacunary statistical equivalent and asymptotically lacunary ℐ-equivalent double sequences and obtain some analogous results from these new definitions point of views.
Definition 3.1 Let be a non-trivial ideal. A double sequence is said to be ℐ-statistically convergent to L if for each and ,
In this case, we write .
Definition 3.2 Let be a non-trivial ideal. A double sequence is said to be ℐ-lacunary statistically convergent to L if for each and ,
In this case, we write .
Definition 3.3 Two nonnegative double sequences and are said to be P-asymptotically equivalent if
denoted by .
Definition 3.4 Two nonnegative double sequences and are said to be asymptotically statistical equivalent of multiple L provided that for every
denoted by and simply asymptotically statistical equivalent if .
Definition 3.5 Two nonnegative double sequences and are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ,
denoted by and simply asymptotically lacunary statistical equivalent if . If we take , then we get Definition 3.4.
Definition 3.6 Two non-negative double sequences and are said to be asymptotically ℐ-equivalent of multiple L provided that for every ,
denoted by and simply asymptotically ℐ-equivalent if .
Lemma 3.1 Let be an admissible ideal. Let , be two double sequences and with such that . Then there exists a sequence with such that .
Definition 3.7 Two non-negative double sequences and are said to be asymptotically ℐ-statistically equivalent of multiple L provided that for every and for every ,
denoted by and simply asymptotically ℐ-statistical equivalent if .
Definition 3.8 Two non-negative double sequences and are said to be Cesaro asymptotically ℐ-equivalent (or -equivalent) of multiple L provided that for every ,
denoted by and simply asymptotically -equivalent if .
Definition 3.9 Two non-negative double sequences and are said to be strongly Cesaro asymptotically ℐ-equivalent (or -equivalent) of multiple L provided that for every ,
denoted by and simply strongly Cesaro asymptotically -equivalent if .
Definition 3.10 Two non-negative double 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 3.11 Two non-negative double sequences and are said to be asymptotically ℐ-lacunary equivalent (or -equivalent) of multiple L provided that for every ,
denoted by and simply asymptotically -equivalent if .
Definition 3.12 Two non-negative double sequences and are said to be asymptotically ℐ-lacunary statistically equivalent (or -equivalent) of multiple L provided that for every , for every ,
denoted by and simply asymptotically -equivalent if .
Theorem 3.1 Let be a non-trivial ideal. Let be a double lacunary sequence. If and . Then .
Proof (a) Suppose that and . Then we can assume that
Let . Then we have
Consequently, if , δ and ε are independent, put , we have
This shows that . □
Corollary 3.1 Let be a non-trivial ideal. If and . Then .
Theorem 3.2 Let be a non-trivial ideal. Let be a double lacunary sequence. Then
-
(a)
.
-
(b)
is a proper subset of .
-
(c)
Let and , then .
-
(d)
.
Proof (a) Let and . Then we can write
Thus, for any ,
implies that
Therefore, we have
Since , so that
which implies that
This shows that .
(b) Suppose that . Let and be two sequences defined as follows:
and
It is clear that , and for ,
This implies that
By virtue of last part of (3.1), the set on the right side is a finite set, and so it belongs to ℐ. Consequently, we have
Therefore, .
On the other hand, we shall show that is not satisfied. Suppose that . Then for every , we have
Now,
It follows for the particular choice that
for some which belongs to ℱ as ℐ is admissible. This contradicts (3.2) for the choice . Therefore, .
(c) Suppose that and . We assume that and for all . Given , we get
If we put
and
where , (δ and ε are independent), then we have , and so . This shows that .
(d) It follows from (a), (b) and (c). □
Theorem 3.3 Let be an admissible ideal. Suppose that for given and every such that
then .
Proof Let be given. For every , choose , such that
It is sufficient to show that there exists , such that for , ,
Let ; . The relation (3.3) will be true for , . If , chosen fixed, then we get
Now, for , , we have
Thus, for sufficiently large n,
This established the result. □
Theorem 3.4 Let be a non-trivial ideal. Let be a double lacunary sequence with . Then .
Proof Suppose that , then there exists an such that for sufficiently large r, s. Then we have
If , then for every and for sufficiently large r, s, we have
Therefore, for any , we have
This completes the proof. □
Theorem 3.5 Let be a non-trivial ideal, and let be a double lacunary sequence with . Then .
Proof If . Then there exists a such that for all . Let . Then there exists and , we put
Since . Then for every and , we have
and, therefore, it is a finite set. We choose integers such that
Let and m, n be two integers with satisfying , , then we have
This completes the proof of the theorem. □
Definition 3.13 Let . Two non-negative double sequences and are said to be strongly asymptotically lacunary p-equivalent of multiple L if
denoted by and simply strongly asymptotically lacunary p-equivalent if .
Definition 3.14 Let . Two non-negative double sequences and are said to be asymptotically lacunary p-statistically equivalent of multiple L if for every ,
denoted by and simply asymptotically lacunary p-statistical equivalent if .
Theorem 3.6 Let be a double lacunary sequence. Then
-
(a)
.
-
(b)
is a proper subset of .
-
(c)
Let and , then .
-
(d)
.
The proof of the above theorem is similar to Theorem 3.2 for .
Definition 3.15 Let . We say that two non-negative double sequences and are strongly asymptotically ℐ-lacunary p-equivalent of multiple L if for every ,
denoted by and simply strongly asymptotically ℐ-lacunary p-equivalent if .
Definition 3.16 Let . We say that two non-negative double sequences and are asymptotically ℐ-lacunary p-statistically equivalent of multiple L if for every , for every
denoted by and simply asymptotically ℐ-lacunary p-statistical equivalent if .
Theorem 3.7 Let be a non-trivial ideal, and let be a double lacunary sequence. Then
-
(a)
.
-
(b)
is a proper subset of .
-
(c)
Let and , then .
-
(d)
.
Proof The proof of the theorem follows from the proofs of the Theorems 3.2 and 3.6. □
For , this theorem reduces to Theorem 3.6.
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The authors thank all three reviewers for their useful comments that led to the improvement of the original manuscript.
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BH drafted the manuscript. VK checked and organized the manuscript to be its final form. BH also makes the revision as the corresponding author.
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Hazarika, B., Kumar, V. On asymptotically double lacunary statistical equivalent sequences in ideal context. J Inequal Appl 2013, 543 (2013). https://doi.org/10.1186/1029-242X-2013-543
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DOI: https://doi.org/10.1186/1029-242X-2013-543