Asymptotically I-Lacunary statistical equivalent of order α for sequences of sets

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of order α, where 0 < α 6 1, I-statistically limit, and I-lacunary statistical convergence for sequences of sets. Let (X, ρ) be a metric space and θ be a lacunary sequence. 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 asymptotically I-lacunary statistical equivalent of order α to multiple L, where 0 < α 6 1, provided that for each ε > 0 and each x ∈ X, {r ∈N : 1 hr |{k ∈ Ir : |d(x;Ak,Bk) − L| > }| > δ} ∈ I, (denoted by {Ak} Sθ(IW) α ∼ {Bk}) and simply asymptotically I-lacunary statistical equivalent of order α if L = 1. In addition, we shall also present some inclusion theorems. The study leaves some interesting open problems. c ©2017 All rights reserved.