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.


Introduction
The notion of -convergence was introduced by Kostyrko et al. for real sequences (see [1]) and then extended to metric spaces by Kostyrko et al. (see [2]).Fast [3] introduced statistical convergence and -convergence, which is based on using ideals of N to define sets of density 0; is a natural extension of Fast's definition.Definition 1.A family of sets  ⊆ 2 N is called an ideal if and only if (i) 0 ∈ ; (ii) for each ,  ∈  we have  ∪  ∈ ; (iii) for each  ∈  and each  ⊆  we have  ∈ .
An ideal is called nontrivial if N ∉  and a nontrivial ideal is called admissible if {} ∈  for each  ∈ N (see [2]).Definition 2. A family of sets  ⊂ 2 N is a filter in N if and only if (i) 0 ∉ ; (ii) for each ,  ∈  we have  ∩  ∈ ; (iii) for each  ∈  and each  ⊇  we have  ∈  (see [2]).
Proposition 3.  is a nontrivial ideal in N if and only if is a filter in N (see [2]).Definition 4. A real sequence  = (  ) is said to be convergent to  ∈ R if and only if for each  > 0 the set belongs to .The number  is called the -limit of the sequence  (see [2]).
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 [4]).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 [5]).
Theorem 5. Let  = (  ) be a nonnegative matrix. is asymptotic regular if and only if, for each , (see [5]).
The frequency of terms having zero values makes a termby-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 [6]).
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 [4]).
Definition 8. Two nonnegative sequences  and  are said to be asymptotically statistically equivalent provided that, for every  > 0, In this case we write   ∼  (see [4]).
Having introduced these ideas, Patterson then offered characterizations of (i) in Theorem 6 when a nonnegative summability matrix  maps bounded sequences into the absolutely convergent sequences and has the property that if  ∈  0 ,  ∈   , and   ∼ , then   ∼  and (ii) 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 (see [4]).
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 N.

Main Results
Observe that  ∈  and that for each  ∉ .By (ii), Observe that, for  ∈ , That is, On the other hand, By condition (ii), there is a set  ∈ () such that, for each  ∈ , the first and third terms of the above expression can be made small in relation to 1 − (/2) and, in particular, for each  ∈ .By ( 15) and ( 17), we have and hence (ii) implies (i).
The proof is completed by showing that statement (i) implies statement (ii).Assume that if  and  are bounded sequences,  ∈  0 ,  ∈   , and   ∼ , then   ∼ .Let  be a member of  and define the sequences  and  as follows: Observe that and, as for every  ∈ .
Proof.Suppose that  is asymptotic -regular.Let  ∈  and define the sequences  and  as follows: Observe that  and  are bounded sequences,  ∈  0 , and  ∈   .Hence.