Research Article | Open Access
Asymptotic I-Equivalence of Two Number Sequences and Asymptotic I-Regular Matrices
We study I-equivalence of the two nonnegative sequences and . Also we define asymptotic I-regular matrices and obtain conditions for a matrix to be asymptotic I-regular.
The notion of -convergence was introduced by Kostyrko et al. for real sequences (see ) and then extended to metric spaces by Kostyrko et al. (see ). Fast  introduced statistical convergence and -convergence, which is based on using ideals of to define sets of density 0; is a natural extension of Fast’s definition.
Definition 1. A family of sets is called an ideal if and only if(i);(ii)for each we have ;(iii)for each and each we have .
An ideal is called nontrivial if and a nontrivial ideal is called admissible if for each (see ).
Definition 2. A family of sets is a filter in if and only if(i);(ii)for each we have ;(iii)for each and each we have (see ).
Proposition 3. is a nontrivial ideal in if and only if
is a filter in (see ).
Definition 4. A real sequence is said to be -convergent to if and only if for each the set
belongs to . The number is called the -limit of the sequence (see ).
Let and be real sequences. Pobyvanets introduced asymptotic equivalence for and as follows: if
then and are called asymptotic equivalent; this is denoted by (see ). Pobyvanets also introduced asymptotic regular matrices which preserve the asymptotic equivalence of two nonnegative number sequences; that is, for the nonnegative matrix if then (see ).
Theorem 5. Let be a nonnegative matrix. is asymptotic regular if and only if, for each ,
The frequency of terms having zero values makes a term-by-term ratio inapplicable in many cases, which motivated Fridy to introduce some related notions. In particular, he analyzed the asymptotic rates of convergence of the tails and partial sums of series as well as the supremum of the tails of bounded sequences (see ).
Define , , , and spaces as follows:
Following Fridy, given a sequence , the sequence is defined to be where, for each , .
Theorem 6. If is a nonnegative summability matrix, then the following statements are equivalent:(i)if and are bounded sequences such that and , for some , then ;(ii)for each , (see ).
In 2003, Patterson extended these concepts by introducing asymptotically statistical equivalent sequences, an analog of the above definitions, and investigated natural regularity conditions for nonnegative summability matrices (see ).
Definition 7. The sequence has statistical limit , denoted by - provided that for every , (see ).
Definition 8. Two nonnegative sequences and are said to be asymptotically statistically equivalent provided that, for every ,
In this case we write (see ).
Definition 9. A summability matrix is asymptotically statistically regular provided that whenever , , and for some (see ).
Having introduced these ideas, Patterson then offered characterizations of in Theorem 6 when a nonnegative summability matrix maps bounded sequences into the absolutely convergent sequences and has the property that if , , and , then and when a summability matrix is asymptotically statistical regular summability matrices.
Theorem 10. In order for a summability matrix to be asymptotically statistical regular it is necessary and sufficient that(i) is bounded for each ;(ii)for any fixed and , (see ).
The main results of this paper have a similar focus, where statistical convergence is replaced by convergence with respect to an admissible ideal of subsets of .
2. Main Results
Definition 11. Let and be nonnegative real sequences and let be an admissible ideal in . If
belongs to , for every , then and are called asymptotically -equivalent sequences; this is denoted by .
Theorem 12. Let be a nonnegative summability matrix and let and be admissible ideals in . Then the following statements are equivalent:(i)if and are bounded sequences such that , , and , for some , then ;(ii) for each .
Proof. The proof that statement implies statement is given first. Assuming that , let be given and set
Observe that and that
for each . By ,
Observe that, for ,
On the other hand,
By condition , there is a set such that, for each , the first and third terms of the above expression can be made small in relation to and, in particular,
for each . By (15) and (17), we have
and hence implies .
The proof is completed by showing that statement implies statement . Assume that if and are bounded sequences, , , and , then . Let be a member of and define the sequences and as follows:
and, as , it follows that
Definition 13. Let be a nonnegative summability matrix and let be an admissible ideal in . Further assume that and are nonnegative real sequences with and for some . The summability matrix is said to be asymptotic -regular if implies .
Lemma 14. Let and belong to and let be an admissible ideal in . Then necessary and sufficient condition for is .
Proof. Assuming that , then
and, for each , we have
Now suppose that . Then
For each , we have
and hence .
Definition 15. Let be an admissible ideal in , a real sequence, and a nonnegative summability matrix. If implies , then is called an -regular matrix.
Theorem 17. Let be an admissible ideal in . Then a nonnegative summability matrix is asymptotic -regular if and only if
for every .
Proof. Suppose that is asymptotic -regular. Let and define the sequences and as follows:
Observe that and are bounded sequences, , and . Hence.
Since , it follows that
Next we show that condition (27) is sufficient for to be asymptotic -regular. Define the matrix by . Note that, since the row sums equal and condition (27) yields criteria of Theorem 16, is an -regular matrix. Also observe that, since the row sums equal 1, the matrix maps members of to . Observe that
Since , it follows that .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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