Abstract
This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ 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 ,
(denoted by ) and simply ℐ-asymptotically lacunary statistical equivalent if .
MSC:40A99, 40A05.
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1 Introduction
In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.
In [3], asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences. Later on, the extension asymptotically lacunary statistical equivalent sequences is presented (see [4]).
Recently, Das, Savaş and Ghosal [5] introduced new notions, namely ℐ-statistical convergence and ℐ-lacunary statistical convergence by using ideal.
In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐ-asymptotically statistical equivalent and ℐ-asymptotically lacunary statistical equivalent. In addition to these definitions, natural inclusion theorems shall also be presented.
First, we introduce some definitions.
2 Definitions and notations
Definition 2.1 (Marouf [1])
Two nonnegative sequences and are said to be asymptotically equivalent if
(denoted by ).
Definition 2.2 (Fridy [6])
The sequence has statistic limit L, denoted by st- provided that for every ,
The next definition is natural combination of Definitions 2.1 and 2.2.
Definition 2.3 (Patterson [2])
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 .
By a lacunary ; , where , we shall mean an increasing sequence of nonnegative integers with as . The intervals determined by θ will be denoted by and . The ratio will be denoted by .
Definition 2.4 ([3])
Let θ be a lacunary sequence; the 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 .
More investigations in this direction and more applications of asymptotically statistical equivalent can be found in [7, 8] where many important references can be found.
The following definitions and notions will be needed.
Definition 2.5 ([9])
A nonempty family of subsets a nonempty set Y is said to be an ideal in Y if the following conditions hold:
-
(i)
implies ;
-
(ii)
, imply .
Definition 2.6 ([10])
A nonempty family is said to be a filter of ℕ if the following conditions hold:
-
(i)
;
-
(ii)
implies ;
-
(iii)
, imply .
If ℐ is proper ideal of ℕ (i.e., ), then the family of sets is a filter of ℕ. It is called the filter associated with the ideal.
A proper ideal ℐ is said to be admissible if for each .
Throughout ℐ will stand for a proper admissible ideal of ℕ, and by sequence we always mean sequences of real numbers.
Definition 2.8 ([9])
Let be a proper admissible ideal in ℕ.
The sequence of elements of ℝ is said to be ℐ-convergent to if for each the set .
Following these results, we introduce two new notions ℐ-asymptotically lacunary statistical equivalent of multiple L and strong ℐ-asymptotically lacunary equivalent of multiple L.
The following definitions are given in [5].
Definition 2.9 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 2.10 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 2.11 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 .
3 New definitions
The next definitions are combination of Definitions 2.1, 2.9, 2.10 and 2.11.
Definition 3.1 Two nonnegative sequences and are said to be ℐ-asymptotically statistical equivalent of multiple L provided that for every and ,
(denoted by ) and simply ℐ-asymptotically statistical equivalent if .
For , ℐ-asymptotically statistical equivalent of multiple L coincides with asymptotically statistical equivalent of multiple L, which is defined in [3].
Definition 3.2 Let θ 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 ,
(denoted by ) and simply ℐ-asymptotically lacunary statistical equivalent if .
For , ℐ-asymptotically lacunary statistical equivalent of multiple L coincides with asymptotically lacunary statistical equivalent of multiple L, which is defined in [3].
Definition 3.3 Let θ be a lacunary sequence; two number sequences and are strong ℐ-asymptotically lacunary equivalent of multiple L provided that
(denoted by ) and strong simply ℐ-asymptotically lacunary equivalent if .
4 Main result
In this section, we state and prove the results of this article.
Theorem 4.1 Let be a lacunary sequence then
-
(1)
-
(a)
If then ,
-
(b)
is a proper subset of ;
-
(a)
-
(2)
If and then ;
-
(3)
,
where denote the set of bounded sequences.
Proof Part (1a): If and then
and so
Then, for any ,
Hence, we have .
Part (1b): , let be defined as follows: to be at the first integers in and zero otherwise. for all k. These two satisfy the following , but the following fails .
Part (2): Suppose and are in and . Then we can assume that
Given , we have
Consequently, we have
Therefore, .
Part (3): Follows from (1) and (2). □
Theorem 4.2 Let ℐ is an ideal and is a lacunary sequence with , then
Proof Suppose first that , then there exists a such that for sufficiently large r, which implies
If , then for every and for sufficiently large r, we have
Then, for any , we get
This completes the proof. □
For the next result we assume that the lacunary sequence θ satisfies the condition that for any set , .
Theorem 4.3 Let ℐ is an ideal and is a lacunary sequence with , then
Proof If , then without any loss of generality, we can assume that there exists a such that for all . Suppose that and for define the sets
and
It is obvious from our assumption that , the filter associated with the ideal ℐ. Further observe that
for all . Let be such that for some . Now
Choosing and in view of the fact that where , it follows from our assumption on θ that the set T also belongs to and this completes the proof of the theorem. □
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Savaş, E. On ℐ-asymptotically lacunary statistical equivalent sequences. Adv Differ Equ 2013, 111 (2013). https://doi.org/10.1186/1687-1847-2013-111
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DOI: https://doi.org/10.1186/1687-1847-2013-111