Abstract
This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐstatistically limit and ℐlacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences and are said to be ℐasymptotically lacunary statistical equivalent of multiple L provided that for every , and ,
(denoted by ) and simply ℐasymptotically lacunary statistical equivalent if .
MSC: 40A99, 40A05.
Keywords:
asymptotical equivalent; ideal convergence; lacunary sequence; statistical convergence1 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 ℐstatistical convergence and ℐlacunary statistical convergence by using ideal.
In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐasymptotically statistical equivalent and ℐasymptotically 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
Definition 2.2 (Fridy [6])
The sequence has statistic limitL, 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:
Definition 2.6 ([10])
A nonempty family is said to be a filter of ℕ if the following conditions hold:
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 ℐconvergent to if for each the set .
Following these results, we introduce two new notions ℐasymptotically lacunary statistical equivalent of multiple L and strong ℐasymptotically lacunary equivalent of multiple L.
The following definitions are given in [5].
Definition 2.9 A sequence is said to be ℐstatistically convergent to L or convergent to L if, for any and ,
In this case, we write . The class of all ℐstatistically convergent sequences will be denoted by .
Definition 2.10 Let θ be a lacunary sequence. A sequence is said to be ℐlacunary statistically convergent to L or convergent to L if, for any and ,
In this case, we write . The class of all ℐlacunary statistically convergent sequences will be denoted by .
Definition 2.11 Let θ be a lacunary sequence. A sequence is said to be strong ℐlacunary convergent to L or convergent to L if, for any
In this case, we write . The class of all strong ℐlacunary 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 ℐasymptotically statistical equivalent of multiple L provided that for every and ,
(denoted by ) and simply ℐasymptotically statistical equivalent if .
For , ℐasymptotically 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 ℐasymptotically lacunary statistical equivalent of multiple L provided that for every and ,
(denoted by ) and simply ℐasymptotically lacunary statistical equivalent if .
For , ℐasymptotically 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 ℐasymptotically lacunary equivalent of multiple L provided that
(denoted by ) and strong simply ℐasymptotically lacunary equivalent if .
4 Main result
In this section, we state and prove the results of this article.
Theorem 4.1Letbe a lacunary sequence then
(1)
wheredenote the set of bounded sequences.
and so
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
Consequently, we have
Part (3): Follows from (1) and (2). □
Theorem 4.2Let ℐ is an ideal andis 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
This completes the proof. □
For the next result we assume that the lacunary sequence θ satisfies the condition that for any set , .
Theorem 4.3Let ℐ is an ideal andis 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. □
Competing interests
The author declares that they have no competing interests.
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