Wijsman Asymptotic Lacunary I 2 -Invariant Equivalence for Double Set Sequences

In this study, for double set sequences, we present the notions of Wijsman asymptotic lacunary invariant equivalence, Wijsman asymptotic lacunary I 2 -invariant equivalence and Wijsman asymptotic lacunary I (cid:3) 2 -invariant equivalence. Also, we examine the relations between these notions and Wijsman asymptotic lacunary invariant statistical equivalence studied in this (cid:12)eld before.


Introduction
The notion of convergence for double sequences was firstly introduced by Pringsheim [34]. Then, this notion was extended to the notion of lacunary statistical convergence by Patterson and Savaş [32] and the notion of I-convergence by Das et al. [5].
The notion of asymptotic equivalence for double sequences was introduced by Patterson [31]. Then, this notion was extended to the notion of asymptotic double lacunary statistical equivalence by Esi [8] and the notion of asymptotic I-equivalence by Hazarika and Kumar [12].
Over the years, many authors have studied on the notions of various convergence for set sequences. One of them, discussed in this study, is the notion of convergence in the Wijsman sense [1][2][3]56,57]. Using the notions of lacunary statistical convergence, invariant mean and I-convergence, the notion of convergence in the Wijsman sense was extended to new convergence notions for double set sequences by some authors [24,25,47].
The notions of asymptotic equivalence in the Wijsman sense for double set sequences were firstly introduced by Nuray et al. [26] and studied by many authors. In this paper, using invariant mean, we study new asymptotic equivalence notions for double set sequences.
Let σ be a mapping such that σ : N + → N + (the set of positive integers). A continuous linear functional ψ on ℓ ∞ is said to be an invariant mean or a σ-mean if it satisfies the following conditions: The mappings σ are assumed to be one to one and such that σ m (n) ̸ = n for all m, n ∈ N + , where σ m (n) denotes the m th iterate of the mapping σ at n. Thus ψ extends the limit functional on c, in the sense that ψ(x n ) = lim x n for all (x n ) ∈ c.
A family of sets I ⊆ 2 N is said to be an ideal if it satisfies the following conditions: An ideal I ⊆ 2 N is said to be nontrivial if N / ∈ I and a nontrivial ideal is said to be admissible if {n} ∈ I for each n ∈ N.
A nontrivial ideal I 2 ⊆ 2 N×N is said to be strongly admissible if {n} × N and N × {n} belong to I 2 for each n ∈ N. Obviously a strongly admissible ideal is admissible.
Throughout the study, I 2 ⊆ 2 N×N will be considered as a strongly admissible ideal. Let . A family of sets F ⊆ 2 N is said to be a filter if it satisfies the following conditions: For any ideal I ⊆ 2 N , there is a filter F(I) corresponding with I such that A double sequence θ 2 = {(k r , j u )} is said to be double lacunary sequence if there exist two increasing sequences of integers (k r ) and (j u ) such that For any double lacunary sequence θ 2 = {(k r , j u )}, the following notations are used in general: Throughout the study, θ 2 = {(k r , j u )} will be considered as a double lacunary sequence.
Let θ = {(k r , j u )} be a double lacunary sequence, A ⊆ N × N and If the following limits exist then they are said to be a lower lacunary σ-uniform density and an upper lacunary σ-uniform density of the set A, respectively.
Two nonnegative double sequences (x kj ) and (y kj ) are said to be asymptotically equivalent if lim k,j→∞ For a nonempty set X, let a function f : , which are the codomain elements of f , is said to be set sequences.
for any x ∈ X and any nonempty set U ⊆ X.
Throughout the study, (X, ρ) will be considered as a metric space and U, U kj , V kj as any nonempty closed subsets of X.
The double sequence {U kj } is said to be Wijsman strongly lacunary invari- uniformly in m and n. The double sequence {U kj } is said to be Wijsman lacunary I 2 -invariant convergent or I σθ 2 -convergent to U if every ε > 0 and each x ∈ X, the set is defined as follows: The double sequences {U kj } and {V kj } are said to be Wijsman asymptot- It is denoted by U kj L ∼ V kj and simply is said to be Wijsman asymptotically equivalent if L = 1.
As an example, consider the following sequences of circles in the (x, y)plane: uniformly in m and n and it is denoted by U kj W (S σθ 2(L) ) ∼ V kj , and simply is said to be Wijsman asymptotically lacunary invariant statistical equivalent if L = 1.

Main Results
In this section, for double set sequences, we present the notions of Wijsman asymptotic lacunary invariant equivalence uniformly in m and n. In this case, we write U kj W (N σθ 2(L) ) ∼ V kj and simply say Wijsman asymptotically lacunary invariant equivalent if L = 1.

Definition 2 Two double set sequences {U kj } and {V kj } are Wijsman asymptotically lacunary I 2 -invariant equivalent of multiple L if every ε > 0 and each
x ∈ X, the set In this case, we write U kj W (I σθ 2(L) ) ∼ V kj and simply say Wijsman asymptotically lacunary I 2 -invariant equivalent if L = 1.
Proof Let m, n ∈ N be arbitrary and ε > 0 is given. Also, we assume that For every m, n = 1, 2, . . . and each x ∈ X, we have For every m, n = 1, 2, . . . and each x ∈ X, it is obvious that S 2 (m, n) < ε.
, there exists a λ > 0 such that for each x ∈ X and ((k, j) ∈ I ru , m, n = 1, 2, . . .), so we have Hence, due to our assumption, U kj uniformly in m and n. In this case, we write U kj W [N σθ 2(L) ] p ∼ V kj and simply say Wijsman asymptotically strongly p-lacunary invariant equivalent if L = 1.
Proof Let 0 < p < ∞ and ε > 0 is given. Also, we assume that For every m, n = 1, 2, . . . and each x ∈ X, we have Hence, due to our assumption, U kj and ε > 0 is given. Also, we assume that , there exists a λ > 0 such that for each x ∈ X and ((k, j) ∈ I ru , m, n = 1, 2, . . .), so we have Hence, due to our assumption, U kj Proof The proof is obvious from Theorem 2 and Theorem 3.
Now, without proof, we shall present a theorem that gives a relationship between the notions of Wijsman asymptotic lacunary I 2 -invariant equivalence of multiple L and Wijsman asymptotic lacunary invariant statistical equivalence of multiple L.

Theorem 5 For any double set sequences {U kj } and {V kj },
In this case, we write U kj W (I * σθ 2(L) ) ∼ V kj and simply say Wijsman asymptotically lacunary I * 2 -invariant equivalent if L = 1.
Proof Let U kj W (I * σθ 2(L) ) ∼ V kj and ε > 0 is given. Then, there exists a set M 2 ∈ F(I σθ 2 ) (N × N \ M 2 = H ∈ I σθ 2 ) such that for each x ∈ X, for all (k, j) ∈ M 2 where k ≥ k 0 , j ≥ j 0 . Hence, for every ε > 0 and each x ∈ X it is obvious that ) .
Since I σθ 2 ⊂ 2 N×N is a strongly admissible ideal, and so we have S(ε) ∈ I σθ 2 . Consequently, U kj If I σθ 2 has property (AP 2), then the converse of Theorem 6 is hold.

Theorem 7
If I σθ 2 has property (AP 2), then Proof Let I σθ 2 satisfies condition (AP 2) and ε > 0 is given. Also, suppose that U kj W (I σθ 2(L) ) ∼ V kj . Then, for every ε > 0 and each x ∈ X we have For every x ∈ X, denote E 1 , . . . , E n as follows where n ≥ 2 (n ∈ N). For each x ∈ X, note that E i ∩ E j = ∅ (i ̸ = j) and E i ∈ I σθ 2 (for each i ∈ N). Since I σθ 2 satisfies condition (AP 2), there exists a sequence of sets {F n } n∈N such that E i ∆F i is included in finite union of rows and columns in N × N (for each i ∈ N) and F = ∈ I σθ 2 . Now, to complete the proof, it is enough to prove that for each x ∈ X lim k,j→∞ (k,j)∈M2 where M 2 = N × N \ F . Let γ > 0 is given. Choose n ∈ N such that 1 n < γ. Then, for each x ∈ X we have Since E i ∆F i (i = 1, 2, . . .) are included in finite union of rows and columns, there exists n 0 ∈ N such that for each (2) If k, j > n 0 and (k, j) / ∈ F , then and by (2), for each x ∈ X Proof This is an immediate consequence of Theorem 6 and Theorem 7.

Conflict of interest
The authors declare that they have no conflict of interest.