On Wijsman asymptotically lacunary I-statistical equivalence of weight g of sequence of sets

Abstract

This paper presents the following definition which is a natural combination of the definitions of asymptotically equivalence, I-convergence, statistical limit, lacunary sequence, and Wijsman convergence of weight g; where g : N → [0,∞) is a function satisfying limn→∞ g (n) = ∞ and n g(n) 9 0 as n → ∞ for sequence of sets. Let (X, ρ) be a metric space, θ = {kr} be a lacunary sequence and I ⊆ 2N be an admissible ideal. For any non-empty closed subsets Ak, Bk ⊆ X such that d(x,Ak) > 0 and d(x,Bk) > 0 for each x ∈ X , we say that the sequences {Ak} and {Bk} are Wijsman I-asymptotically lacunary statistical equivalent of multiple L of weight g if for every ε > 0, δ > 0 and for each x ∈ X, { r ∈ N : 1 g (hr) ∣∣∣∣{k ∈ Ir : ∣∣∣∣d(x,Ak) d(x,Bk) − L ∣∣∣∣ ≥ ε}∣∣∣∣ ≥ δ} ∈ I (denoted byAk S θ (IW ) g ∼ Bk ). We mainly investigate their relationship and also make some observations about these classes.