Asymptotically lacunary equivalent sequences defined by ideals and modulus function

Abstract

Purpose: For a non-trivial ideal I ⊂ P(N), a lacunary sequence θ = (kr), and a modulus function f ,t he purpose of present paper is to introduce certain new notions: I(f )-asymptotically equivalent, I(wf )-asymptotically equivalent, and I(N f )-asymptotically equivalent sequences of numbers. Methods: We use an analytic method to obtain our results. Results: Certain theorems on generalized equivalent sequences by the use of ideals, lacunary sequences, and a modulus function are obtained. Conclusions: We observe that if the modulus function f satisfies limt→∞ f (t) t > 0, then the notions I(w) and I(Nθ ) respectively coincide with the notions I(wf ) and I(N f ). However, if f is bounded, the notions I(wf ) and I(N f )