( f , p )-Asymptotically Lacunary Equivalent Sequences with Respect to the Ideal I

In this study, we define (f, p)-Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a non-trivial ideal ( ) I P N ⊂ , a lacunary sequence ( ) r k = θ , a strictly positive sequence ( ) k p p = , and a modulus function f, and obtain some revelent connections between these notions.


Introduction
Let , , s c ∞  denote the spaces of all real sequences, bounded, and convergent sequences,respectively.Any subspace of s is called a sequence space.
Following Freedman et al. [1], we call the sequence q k k − = .These notations will be used troughout the paper.The sequence space of lacunary strongly convergent sequences N θ was defined by Freedman et al. [1], as follows: ( ) 1 : lim 0 for some The notion of modulus function was introduced by Nakano [2].We recall that a modulus f is a function from for , 0 x y ≥ , 3) f is increasing and 4) f is continuous from the right at 0. Hence f must be continuous everywhere on [ ) 0, ∞ .
Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in [9].Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices in [10].Subsequently, many authors have shown their interest to solve different problems arising in this area (see [11]- [13]).
Recently, Bilgin [20] used modulus function to define some notions of asymptotically equivalent sequences and studied some of their connections.Kumar and Sharma extended these concepts by presenting a non-trivial ideal I This paper presents introduce some new notions, (f, p)-asymptotically equivalent of multiple L, strong (f, p)asymptotically equivalent of multiple L, and strong (f, p)-asymptotically lacunary equivalent of multiple L with respect to the ideal I which is a natural combination of the definition for asymptotically equivalent, a non-trivial ideal I, Lacunary sequence, a strictly positive sequence ( ) , and Modulus function.In addition to these definitions, we obtain some revelent connections between these notions.

Definition 2.1. A sequence [ ]
x and [ ] y are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every 0 ε > , Definition 2.9.Let f be any modulus; the two nonnegative sequences [ ] x and [ ] y are said to be fasymptotically equivalent of multiple L provided that, lim 0 (denoted by y x f  ) and simply strong f-asymptotically equivalent, if 1 L = .
Definition 2.10.Let f be any modulus; the two nonnegative sequences [ ] x and [ ] y are said to be strong f- Definition 2.11.Let f be any modulus and θ be a lacunary sequence; the two nonnegative sequences [ ] x and [ ] y are said to be strong f-asymptotically lacunary equivalent of multiple L provided that ) and simply strong f-asymptotically lacunary equivalent, if 1 L = .
For any non-empty set X, let ( ) P X denote the power set of X. Definition 2.12.A family ( ) imply B I ∈ .Definition 2.13.A non-empty family ( ) : .
In this case, we write x and [ ] y are said to be strongly asymptotically equivalent of multiple L with respect to the ideal I provided that for each 0 .
and simply strongly asymptotically equivalent with respect to the ideal I, if 1 L = .Definition 2.17.Let : x y θ ∼ and simply asymptotically lacunary statistical equivalent with respect to the ideal I, if 1 L = .y are said to be strongly asymptotically lacunary equivalent of multiple L with respect to the ideal I provided that for 0 x y θ ∼ and simply strongly asymptotically lacunary equivalent with respect to the ideal I, if 1 L = .y are said to be f-asymptotically equivalent ofmultiple L with respect to the ideal I provided that for each 0 and simply f-asymptotically equivalent with respect to the ideal y are said to be strongly f-asymptotically equivalent of multiple L with respect to the ideal I provided that for each 0 and simply strongly f-asymptotically equivalent with respect to the ideal I, if 1 L = .y are said to be strongly f-asymptotically lacunary equivalent of multiple L with respect to the ideal I provided that for each 0 and simply strongly f-asymptotically lacunary equivalent with respect to the ideal I, if 1 L = .

Main Results
We now consider our main results.We begin with the following definitions.y are said to be strongly (f, p)asymptotically equivalent of multiple L with respect to the ideal I provided that for each 0 and simply strongly (f, p)-asymptotically equivalent with respect to the ideal I, if 1 L = .
If we take ( ) and simply strongly p-asymptotically equivalent with respect to the ideal I, if 1 L = .
If we take k p p = for all k N ∈ , we write x and [ ] y are said to be strongly (f, p)-asymptotically lacunary equivalent of multiple L with respect to the ideal I provided that for each 0 ∼ and simply strongly (f, p)-asymptotically lacunary equivalent with respect to the ideal I, if 1 L = .
If we take k p p = for all k N ∈ , we write Note that,we put 1 p = , we write ( ) is the same as the ( ) x y θ ∼ of Kumar and Sharma [15].Also if we put ( ) is the same as the  and ε > 0. We choose 0 1 δ < < such that ( ) We can write where the first summation is over k k x L y δ − ≤ and the second summation over > .
, it follows the later set, and hence, the first set in above expression belongs to I.This proves that ( ) ( ) , it follows that the later set belongs to I, and therefore, the theorem is proved.Theorem 3.2.Let ( ) be a non-trivial ideal in N, f be a modulus function, ( ) r k θ = be a lacunary sequence and 0 inf sup q < ∞ then there exists 0 K > such that r q K < for every r.Now suppose that ( ) Hence, for all j A ∈ we have 1 .
Let n be any integer with . Because for any set ( ) and lim inf 1 r r q > .There exists 0 δ > such that ( ) r ≥ .We have, for sufficiently large r, that ( ) .

∑
We have ( ) , which is the filter of the ideal I, For each r k A ∈ , we have Part (iii): This immediately follows from (i) and (ii).Now we give relation between asymptotically statistical equivalence and strong (f, p)-asymptotically equivalence with respect to the ideal I. Also we give relation between asymptotically lacunary statistical equivalence and strong (f, p)-asymptotically lacunary equivalence with respect to the ideal I.
Theorem 3.4.Let ( ) be a non-trivial ideal in N , f be a modulus function, ( ) r k θ = be a lacunary sequence and 0 inf sup Proof.Part 1): Suppose Consequently, for any 0 γ > , we have Therefore we have 2) Suppose f is bounded and x y ∼ .Since f is bounded, there exists an integer T such that ( ) Moreover, for 0 ε > , We split the sum for k n Therefore we have Proof.Part 1): Take 0 ε > and let 1 ∑ denote the sum over But then, by definition of an ideal, later set belongs to I, and therefore ( ) x y θ ∼ Part 2): Suppose that f is bounded and x y θ ∼ .Since f is bounded, there exists an integer T such that ( ) for all k, k t t = for all k and 0 p t < ≤ .Then it follows following Theorem.Theorem 3.6.Let . Thus we have   ≤ .There- fore ( ) Now for each r; and so Thus we have r → ∞ .The intervals determined by θ will be denoted by

Definition 2 . 3 .Definition 2 . 4 .Definition 2 . 5 .
Let f be any modulus; the sequence [ ] x is strongly (Cesaro) summable to L with respect to a Two nonnegative sequences [ ] x and [ ] y are said to be asymptotically equivalent if Two nonnegative sequences [ ] x and [ ] y are said to be asymptotically statistical equivalent of multiple L provided that for every 0

Definition 2 . 6 .
Two nonnegative sequences [ ] x and [ ] y are said to be strong asymptotically equivalent of multiple L

Definition 2 . 8 .
Let θ be a lacunary sequence; the two nonnegative sequences [ ] x and [ ] y are said to be T. Bilgin strong asymptotically lacunary equivalent of multiple L provided that 1 ⊂ be a non-trivial ideal in N and ( ) r k θ = be a lacunary sequence.The two nonnegative sequences [ ] x and [ ]y are said to be asymptotically lacunary statistical equivalent of multiple L with respect to the ideal I provided that for each 0 sequence.The two non-negative sequences [ ] x and [ ] ⊂ be a non-trivial ideal in N and f be a modulus function.The two nonnegative sequences [ ] x and [ ] -trivial ideal in N and f be a modulus function.The two nonnegative sequences [ ] x and [ ] Definition 2.21.Let ( ) I P N ⊂ be a non-trivial ideal in N, f be a modulus function and ( ) sequence.The two non-negative sequences [ ] x and [ ] ⊂ be a non-trivial ideal in N, f be a modulus function, and real numbers.Two number sequences [ ]x and [ ] ⊂ be a non-trivial ideal in N, f be a modulus function, of positive real numbers.Two number sequences [ ] Gumus[25] We start this section with the following theorem to show that the relation between (f, p)-asymptotically equivalence and strong p-asymptotically equivalence with respect to the ideal I Theorem 3-trivial ideal in N, f be a modulus function, The proof of Theorem 3.2 is very similar to the Theorem 3.1.Then, we omit it.The next theorem shows the relationship between the strong (f, p)-asymptotically equivalence and the strong (f, p)-asymptotically lacunary equivalence with respect to the ideal I.Theorem 3.3.Let( )I P N ⊂ be a non-trivial ideal in N, f be a modulus function, of positive real numbers.Then -trivial ideal in N, f be a modulus function, -trivial ideal in N, f be a modulus function, and -trivial ideal in N, f be a modulus function,