New Classes of Generalized Seminormed Difference Sequence Spaces

M Mursaleen,Sunil K Sharma,A Alotaibi

doi:10.1155/2014/461081

M Mursaleen, Sunil K Sharma + Show 1 more

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https://doi.org/10.1155/2014/461081

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Abstract

The purpose of this paper is to introduce new classes of generalized seminormed difference sequence spaces defined by a Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations between resulting sequence spaces.

Highlights

Introduction and PreliminariesLet l0 denote the space of all real sequences x = {xk}
The notion of the difference sequence space was introduced by Kızmaz [6] which was generalized by Mursaleen [7]
The main goal of the present paper is to introduce new classes of generalized seminormed difference sequence spaces defined by Musielak-Orlicz function

Summary

Introduction and Preliminaries

The notion of the difference sequence space was introduced by Kızmaz [6] which was generalized by Mursaleen [7] It was further generalized by Et and Colak [8] as follows: Z(Δμ) = {x = (xk) ∈ ω : (Δμxk) ∈ z} for z = l∞, c, and c0, where μ is a nonnegative integer and. In [11], Dutta introduced the sequence spaces c(‖⋅, ⋅‖, Δμ(η), p), c0(‖⋅, ⋅‖, Δμ(η), p), l∞(‖⋅, ⋅‖, Δμ(η), p), m(‖⋅, ⋅‖, Δμ(η), p), and m0(‖⋅, ⋅‖, Δμ(η), p), where η, μ ∈ N and Δμ(η)xk = (Δμ(η)xk) = (Δμ(η−)1xk − Δμ(η−)1xk−η) and Δ0(η)xk = xk for all k, η ∈ N, which is equivalent to the following binomial representation: Δμ(η)xk (μV) xk−ηV. For a given Musiclak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows: tM = {x ∈ ω : IM (cx) < ∞ for some c > 0} , (15).

IM x k

Main Results

Then the

This implies inf