LACUNARY INVARIANT STATISTICAL EQUIVALENCE FOR DOUBLE SET SEQUENCES

In this paper, we introduce the notions of asymptotical strong σ2-equivalence, asymptotical σ2-statistical equivalence, asymptotical lacunary strong σ2-equivalence and asymptotical lacunary σ2-statistical equivalence in the Wijsman sense for double set sequences. Also, we investigate some relations between these new asymptotical equivalence notions.


Introduction
Long after the notion of convergence for double sequences was introduced by Pringsheim [1], this notion was extended to the notion of statistical convergence by Móricz [2] and Mursaleen and Edely [3] in the same year, to the notion of lacunary statistical convergence by Patterson and Savaş [4] and to the notion of double σconvergent lacunary statistical sequence by Savaş and Patterson [5]. Moreover, for double sequences, the notion of asymptotical equivalence was introduced by Patterson [6].
Over the years, on the various convergence notions for set sequences have been studied by many authors (see, [7][8][9]). One of them, discussed in this paper, is the notion of convergence in the Wijsman sense [10]. Using the notions of statistical convergence, double lacunary sequence and invariant mean, this notion was extended to the notions of convergence for double set sequences by some authors [11][12][13]. Furthermore, for double set sequences, the notions of asymptotical equivalence in the Wijsman sense were introduced by Nuray et al. [14] and then these notions were studied by some authors [15][16][17]. In this paper, using the notion of invariant 2 U. ULUSU, E. DÜNDAR, N. PANCAROGLU AKIN mean, we study on new asymptotical equivalence notions in the Wijsman sense for double set sequences. More information on the notions of asymptotical equivalence for set sequences can be found in [18,19].

Basic Definitions and Notations
In this section, let us remind the basic notions necessary for a better understanding of our paper.
for any y ∈ Y and any nonempty B ⊆ Y . Throughout this study, (Y, d) will be considered as a metric space and B, B jk , D jk will be considered as any nonempty closed subsets of Y . Let σ be a mapping such that σ : N → N (the set of positive integers). A continuous linear functional ψ on ℓ ∞ , the space of real bounded sequences, is called an invariant mean (or a σ-mean) if it satisfies the following conditions: (1) ψ(x s ) ≥ 0, when the sequence (x s ) has x s ≥ 0 for all s, (2) ψ(e) = 1, where e = (1, 1, 1, . . .) and The mapping σ is assumed to be one-to-one and such that σ j (s) ̸ = s for all j, s ∈ N, where σ j (s) denotes the j th iterate of the mapping σ at s. Thus ψ extends the limit functional on c, the space of convergent sequences, in the sense that ψ(x s ) = lim x s for all (x s ) ∈ c.
uniformly in s, t.
A double sequence θ 2 = {(j r , k u )} is called a double lacunary sequence if there exist increasing sequences (j r ) and (k u ) of the integers such that In general, the following notations is used for any double lacunary sequence: Throughout this study, θ 2 = {(j r , k u )} will be considered as a double lacunary sequence.

Definition 7.
[12] A double set sequence {B jk } is called lacunary invariant convergent to the set B in the Wijsman sense if for each y ∈ Y , Definition 9.
[12] A double set sequence {B jk } is called lacunary invariant statistically convergent to the set B in Wijsman sense if for every ε > 0 and each y ∈ Y , is defined as follows: Definition 10. [14] Two double set sequences {B jk } and {D jk } are called asymptotically equivalent of multiplicity λ in the Wijsman sense if for each y ∈ Y , It is denoted by B jk W λ 2 ∼ D jk and simply called asymptotically equivalent in the Wijsman sense if λ = 1.
As an example to asymptotically equivalent double set sequences, the following sequences can be considered: for every y ∈ R 2 , the double set sequences {B jk } and {D jk } are asymptotically equivalent in the Wijsman sense, i.e., B jk W2 ∼ D jk .
Definition 11. Two double set sequences {B jk } and {D jk } are said to be asymptotically σ 2 -equivalent of multiplicity λ in the Wijsman sense if for each y ∈ Y , This type of equivalence is denoted by B jk This type of equivalence is denoted by B jk W λ θσ 2 ∼ D jk and simply called asymptotically lacunary σ 2 -equivalent in the Wijsman sense if λ = 1.
Definition 15. Two double set sequences {B jk } and {D jk } are said to be asymptotically lacunary strong σ 2 -equivalent of multiplicity λ in the Wijsman sense if for each y ∈ Y , This type of equivalence is denoted by B jk ∼ D jk and simply called asymptotically lacunary strong σ 2 -equivalent in the Wijsman sense if λ = 1. Example 1. Let Y = R 2 and double set sequences {B jk } and {D jk } be defined as following: uniformly in s, t. This type of equivalence is denoted by B jk W S λ θσ 2 ∼ D jk and simply called asymptotically lacunary σ 2 -statistical equivalent in the Wijsman sense if λ = 1.
Example 2. Let Y = R 2 and double set sequences {B jk } and {D jk } be defined as following: ; if (j, k) ∈ I ru , j and k are square integers, and ; if (j, k) ∈ I ru , j and k are square integers, In this case, the double set sequences {B jk } and {D jk } are asymptotically lacunary σ 2 -statistical equivalent in the Wijsman sense.
The set of all asymptotically lacunary σ 2 -statistical equivalent double set sequences of multiplicity λ in the Wijsman sense is denoted by {W S λ θσ2 }.
∼ D jk . For every ε > 0 and each y ∈ Y , we have for all s, t, which gives the result.
for all j, k and s, t. Thus, for every ε > 0 and each y ∈ Y we have for all s, t, which gives the result. With a technique similar to that of Theorem 1, the following theorem can be proved.
Thus, for every ε > 0 and each y ∈ Y we have for all s, t, which gives the result. □ Theorem 4. If lim sup r q r < ∞ and lim sup u q u < ∞ for any Proof. Let lim sup r q r < ∞ and lim sup u q u < ∞. Then, there exist α, β > 0 such that q r < α, q u < β for all r, u > 1. Also, suppose that B jk W S λ θσ 2 ∼ D jk and δ > 0. Then, there exist n 0 , m 0 ∈ N such that for every ε > 0, each y ∈ Y and all j ≥ n 0 , k ≥ m 0 for all s, t. We can also find an M > 0 such that S jk < M for all j, k = 1, 2, . . .. Now, let n and m be any integers satisfying j r−1 < n ≤ j r , k u−1 < m ≤ k u where r > n 0 , u > m 0 . Then, for every y ∈ Y we have for all s, t, which gives the result. □ Theorem 5. If Proof. The proof is obvious from Theorem 3 and Theorem 4. □ With techniques similar to that of Theorem 3, Theorem 4 and Theorem 5, the following theorems can be respectively proved.
Theorem 6. If lim inf r q r > 1 and lim inf u q u > 1 for any θ 2 = {(j r , k u )}, then

Conclusion
When (σ(s), σ(t)) = (s + 1, t + 1), from Definitions 11-16 we get the definitions of asymptotical almost equivalence, asymptotical strong almost equivalence, asymptotical almost statistical equivalence, asymptotical lacunary almost equivalence, asymptotical lacunary strong almost equivalence and asymptotical lacunary almost statistical equivalence in the Wijsman sense for double set sequences. So, the analogues of Theorem 1-8 can also be obtained between these definitions, which have not been appeared anywhere by this time.
Author Contribution Statements The authors contributed equally to this work. All authors read and approved the final copy of this paper.

Declaration of Competing Interests
The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.