(σ,f)-Asymptotically Lacunary Equivalent Sequences

Tunay Bilgin

doi:10.1155/2014/945902

Abstract

We introduce the strong (σ,f)-asymptotically equivalent and strong (σ,f)-asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ-convergence, and lacunary sequences. Then we use these definitions to prove strong (σ,f)-asymptotically equivalent and strong (σ,f)-asymptotically lacunary equivalent analogues of Connor’s results in Connor, 1988, Fridy and Orhan’s results in Fridy and Orhan, 1993, and Das and Patel’s results in Das and Patel, 1989.

Highlights

Let s, ∞, c denote the spaces of all real sequences, bounded, and convergent sequences,respectively
The sequence space of lacunary strongly convergent sequences Nθ was defined by Freedman et al.[4], as follows: Nθ = {x = (xi) ∈ s : h−r 1 |xi − s| = 0 for some s}
The notion of modulus function was introduced by Nakano [11]

Summary

Introduction

Let s, ∞, c denote the spaces of all real sequences, bounded, and convergent sequences,respectively. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in [10] Patterson extended these concepts by presenting an. This paper presents introduce some new notions, f -asymptotically equivalent of multiple L, strong f -asymptotically equivalent of multiple L, and strong f -asymptotically lacunary equivalent of multiple L which is a natural combination of the definition for asymptotically equivalent, Statistically limit, Lacunary sequence, and Modulus function. In addition to these definitions, natural inclusion theorems shall be presented

Definitions and Notations

Main Theorems