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Axioms 2018, 7(3), 54; doi:10.3390/axioms7030054
A New Type of Generalization on W—Asymptotically λ—Statistical Equivalence with the Number of α
Department of Math Education, Faculty of Eregli Education, Necmettin Erbakan University, Konya 42060, Turkey
İkizli Middle School, Konya 42060, Turkey
Author to whom correspondence should be addressed.
Received: 19 June 2018 / Accepted: 31 July 2018 / Published: 2 August 2018
In our paper, by using the concept of asymptotically statistical equivalence of order which has been previously defined, we present the definitions of asymptotically statistical equivalence of order , strongly asymptotically statistical equivalence of order , and strongly Cesáro asymptotically statistical equivalence of order where . We also extend these notions with a sequence of positive real numbers, , and we investigate how our results change if is constant.
Keywords:-statistical convergence; asymptotic equivalence; set sequences; λ=(λn) sequence
1. Introduction and Background
To make it easier to understand, we prefer to give introduction section in five parts. In first part, we give the main definitions related to statistical convergence: statistical convergence, —convergence, —statistical convergence, and statistical convergence. In the second part, we mention asymptotic equivalence and asymptotic equivalence. In the third part, we explain set sequences, and we give some important definitions for set sequences in the Wijsman sense. In the fourth part, we explain how statistical convergence and convergence were expanded using the number of Finally, in last part, we explain the purpose and innovations of our study.
1.1. Statistical Convergence
Statistical convergence is a concept which was formally introduced by Fast  and Steinhaus , independently. Later on, Schoenberg reintroduced this concept in his own study . This new type of convergence has been used in different areas by several authors in references [4,5,6,7,8]. Statistical convergence is based on the definition of the natural density of the set and we define the natural density of K by . In this definition, and gives the number of elements in .
Using this information, we say that a sequence (x) of real numbers is statistically convergent to the number L if In this case we write , and usually, S denotes the set of all statistical convergent sequences.
Let be a positive number sequence which is non-decreasing and tending to ∞. Also, for this sequence We denote the set of this kind of sequence by , and we have the interval Mursaleen  defined statistical convergence such thatfor any , and he denoted this new method by . On the other hand, Kostyrko, Šalát and Wilezyński  introduced a new type of convergence which is defined in a metric space and is called —convergence. This type of convergence is based on the definition of an ideal in
A family of sets, , is an ideal if the following properties are provided:
; implies ; and for each and each implies .
We say that is non-trivial if and is admissible if for every
A family of sets, , is a filter if the following properties are provided:
; if then we have ; and for each and each , we have .
If is an ideal in , then we have,is a filter in .
() A sequence of reals is convergent to if and only if the setfor each . In this case, we say that L is the limit of the sequence (x).
—convergence generalizes many types of convergence such as usual convergence and statistical convergence. If we choose the ideals and , then we obtain usual convergence and statistical convergence, respectively.
Based on the statistical convergence and convergence, an important role was located in this area, —statistical convergence, which was introduced by Das, Savaş and Ghosal  as follows:
() A sequence is statistically convergent to L iffor every and
1.2. Asymptotic Equivalence
Asymptotic equivalence was first introduced by Pobyvanets  and some main definitions and asymptotic reguler matrices were given by Marouf . Bilgin  defined Asymptotically equivalent sequences, and on the other hand, asymptotically statistically equivalent sequences were presented by Patterson . Gümüş and Savaş  gave the definition of —asymptotically statistically equivalent sequences by using the sequence, and they were also interested in some inclusion relations between other related spaces.
According to Marouf, if and are two non-negative sequences, we say that they are asymptotically equivalent if
This is denoted by .
() Let be an admissible ideal and Two number sequences and are asymptotically equivalent of multiple L (or — asymptotically statistically equivalent) if every
1.3. Set Sequences
In recent years, studies on set sequences has become popular. Firstly, usual convergence has been extended to convergence of sequences of sets. The first definitions of this subject were based on Baronti and Papini’s  work in 1986. Now, we revisit the definitions of convergence, boundedness, and the Cesáro summability of set sequences. Throughout the paper, is a metric space, and represents non-empty closed subsets of X for all
is a metric, is a point in X, and A is any non-empty subset of The distance from x to is defined by
() In any metric space, the set sequence is Wijsman convergent to A iffor each We write for this case .
We would like to give a well known example of this subject.
In the -plane, consider the sequence of circles. We can easily see that for , this sequence is Wijsman convergent to the y-axis, i.e.,
() In any metric space, the set sequence is bounded iffor each This is shown as
() In any metric space, the set sequence is Wijsman Cesáro summable to A iffor each , and is Wijsman strongly Cesáto summable to A iffor each
Nuray and Rhodes  introduced Wijsman statistical convergence for set sequences by combining statistical convergence with this new concept. Similarly, Kisi and Nuray  defined Wijsman convergence for set sequences with an ideal
() Let be a metric space. For any non-empty closed subsets, , we say that the sequence is Wijsman statistically convergent to A if is statistically convergent to , i.e., andIn this case, we write or We denote the set of all Wijsman statistically convergent sequences by .
() Let be a metric space and be a proper ideal in . For any non-empty closed subsets, , we say that the sequence is Wijsman convergent to if for each , and , the set is
In this case, we write or , and we denote the set of all Wijsman convergent sequences by .
Let and be a sequence as follows:and
The sequence is not Wijsman convergent to the set Howeverm if we choose the ideal , then is Wijsman convergent to set , where , and where d is the natural density.
() In any metric space, let be a non-trivial ideal and . The sequence is said to be Wijsman statistically convergent to A or -convergent to A iffor each and each and , and we write The class of all Wijsman statistically convergent sequences is denoted by
1.4. The Number
In recent years, many concepts that are considered essential in this area has been reworked using the alpha number. In references [22,23], by using the natural density of order the statistical convergence of order was introduced. The new definition is not exactly parallel to that of statistical convergence. Some other applications of this concept are the statistical convergence of order by Çolak and Bektaş , the lacunary statistical convergence of order by Şengül and Et , the weighted statistical convergence of order and its applications by Ghosal , and the almost statistical convergence of order by Et, Altın and Çolak . statistical convergence and lacunary statistical convergence of order were introduced by Das and Savaş in 2014 . In all of these studies, n was replaced by in the denominator in the definition of natural density, and a different direction was given.
In 2017, Savas  gave a new definition about Wijsman asymptotically statistical equivalence of order as follows:
() In any metric space, let be any non-empty closed subsets such that and for all We say that the sequences and are Wijsman asymptotically statistically equivalent of order α to multiple L if for each , and
In this case, we write . It is obvious that denotes the the set of all sequences such that
1.5. Present Study
It should be mentioned that the generalization of the concept of asymptotically statistical equivalence of order for sequences has not been studied until now. So, this brings to mind the question of how our new results will be if we use and p sequences. This makes the study interesting. In this study, we searched for the answer to this question. We generalized asymptotically statistical equivalence of order and compared the properties of this new concept with the other type of convergences without .
2. Main Results
Following this information, we now consider our main definitions and results. Throughout the paper, is a metric space, is an admissible ideal, and is the power of of that is , and is a positive real number sequence. We use the W symbol since our expressions are defined for set sequences.
Let be non-empty closed subsets such that and for all Then, the sequences and are strongly Cesáro asymptotically statistically equivalent of order α to multiple L if for each and
For this situation, we write , and denotes the set of all sequences and such that .
Now let us give our definitions with the sequence.
Let be non-empty closed subsets such that and for all Then, the sequences and are asymptotically statistical equivalent of order α to multiple L and denoted by if for each , and
We denote the set of all sequences of and such that by .
Let be non-empty closed subsets such that and for all Then, the sequences and are strongly asymptotically statistically equivalent of order α to multiple L if for each and each
We denote the set of this kind of sequence by .
The next theorem examines the relation between Savas’ definition and our second definition.
- If then .
- If then .
(i) Assume that and . Then, there exists a such that for sufficiently large For every , we have,
If we think about the number of elements of the sets that provide this relation,we get, for any ,
Then, we have the proof.
(ii) Let Since , we have such that for all . Forfor all Hence,
We know that the right side belongs to the ideal because of the theorem expression. So we have the proof. ☐
Now let us investigate how the sequence affects the previous definitions. Initially, we use the constant sequence of positive real numbers in the following two theorems.
- If then .
- If , and then .
(i) Let and For eachand so,
Then, for any we have,
(ii) Assume that , and There is an M such that for each and all For eachand then for any☐
Now let us examine the above theorems for a non-constant sequence of positive real numbers.
Let be a positive real number sequence, and . implies .
Let and be bounded sequences, , and . Then, implies .
(i) Assume that and Then, we can writeand so for , we have
(ii) From the theorem’s statement there is an integer (M) such that for each and all For eachand☐
Finally, in the last theorem we investigate the relationship between strongly asymptotically statistical equivalence of order and strongly Cesáro asymptotically statistical equivalence of order .
If , then .
Now, assume that and .☐
According to these operations,
3. Conclusions and Future Developments
In our paper, we obtained some different results by defining the asymptotically statistical equivalence of order for sequences. Later on, we generalized our results by using a positive real number sequence . Firstly, we compared the asymptotically statistical equivalence of order and the asymptotically statistical equivalence of order for set sequences. These results are important to understand the role of . In other theorems, we investigated the relations between asymptotically statistical equivalence and strongly asymptotically statistically equivalence of order according to whether p is constant or not. Then, we searched for the relation between strongly Cesáro asymptotically statistically equivalent sequences of order and asymptotically statistical equivalent sequences of order
We know that the p sequence mentioned in this article is a sequence of positive integers. It is a matter of curiosity as to how the results will be obtained if the p sequence does not provide these conditions. On the other hand, it would be interesting to compare the results obtained using a different sequence to with the results in this article.
Conceptualization, H.G. and N.D.; Investigation, H.G. and N.D.; Writing—original draft, H.G.
This research received no external funding.
The authors are grateful to the referees and the editor for their corrections and suggestions, which have greatly improved the readability of the paper.
Conflicts of Interest
The authors declare no conflict of interest.
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