Generalized Lacunary Statistical Difference Sequence Spaces of Fractional Order

Ugur Kadak

doi:10.1155/2015/984283

Abstract

We generalize the lacunary statistical convergence by introducing the generalized difference operatorΔναof fractional order, whereαis a proper fraction andν=(νk)is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator and investigate the topological structures of related sequence spaces. Furthermore, we introduce some properties of the strongly Cesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relations of these spaces. We also determine the relationship between lacunary statistical and strong Cesaro difference sequence spaces of fractional order.

Highlights

By ω, we denote the space of all real valued sequences and any subspace of w is called a sequence space
We introduce some properties of the strongly Cesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relations of these spaces
Let l∞, c, and c0 be the linear spaces of bounded, convergent, and null sequences x = with real or complex terms, respectively, normed by ‖x‖∞ = supk|xk|, where k ∈ N, the set of positive integers

Summary

Introduction

We denote the space of all real valued sequences and any subspace of w is called a sequence space. Let l∞, c, and c0 be the linear spaces of bounded, convergent, and null sequences x = (xk) with real or complex terms, respectively, normed by ‖x‖∞ = supk|xk|, where k ∈ N, the set of positive integers. With this norm, it is proved that these are all Banach spaces. The main focus of the present paper is to generalize strong Cesaro and lacunary statistical difference sequence spaces and investigate their topological structures as well as some interesting results concerning the operator Δα]. We present some theorems related to the lacunary statistical convergence of difference sequences of fractional order and examine some inclusion relations of these spaces

Some New Difference Sequence Spaces with Fractional Order

Cesaro Difference Sequence Spaces of Fractional Order

Lacunary Statistical Convergence of Difference Sequences

Concluding Remarks