Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 123798, 13 pages
http://dx.doi.org/10.1155/2013/123798
Research Article
Generalized Difference -Sequence Spaces Defined by Ideal
Convergence and the Musielak-Orlicz Function
Awad A. Bakery
1,2
1
Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200,
Khulais 21589, Saudi Arabia
2
Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt
Correspondence should be addressed to Awad A. Bakery; awad bakery@yahoo.com
Received 2 June 2013; Accepted 22 September 2013
Academic Editor: Abdelghani Bellouquid
Copyright © 2013 Awad A. Bakery. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduced the ideal convergence of generalized diference sequence spaces combining an infnite matrix of complex numbers
with respect to -sequences and the Musielak-Orlicz function over -normed spaces. We also studied some topological properties
and inclusion relations between these spaces.
1. Introduction
Troughout the paper , ℓ
∞
, ,
0
, and ℓ
denote the classes
of all, bounded, convergent, null, and p-absolutely summable
sequences of complex numbers. Te sets of natural numbers
and real numbers will be denoted by N and R, respectively.
Many authors studied various sequence spaces using normed
or seminormed linear spaces. In this paper, using an infnite
matrix of complex numbers and the notion of ideal, we
aimed to introduce some new sequence spaces with respect to
generalized diference operator Δ
on -sequences and the
Musielak-Orlicz function in -normed linear spaces. By an
ideal we mean a family ⊂2
of subsets of a nonempty
set satisfying the following: (i) ∈ ; (ii) , ∈
imply ∪∈ ; (iii) ∈, ⊂ imply ∈,
while an admissible ideal of further satisfes {} ∈ for
each ∈. Te notion of ideal convergence was introduced
frst by Kostyrko et al. [1] as a generalization of statistical
convergence. Te concept of 2-normed spaces was initially
introduced by G¨ ahler [2] in the 1960s, while that of -normed
spaces can be found in [3]; this concept has been studied
by many authors; see for instance [4–7]. Te notion of ideal
convergence in a 2-normed space was initially introduced by
G¨ urdal [8]. Later on, it was extended to -normed spaces by
G¨ urdal and S ¸ahiner [9]. Given that ⊂2
N
is a nontrivial
ideal in N, the sequence (
)
∈N
in a normed space (; ‖⋅‖) is
said to be -convergent to ∈, if, for each >0,
() = { ∈ N :
−
≥ } ∈ . (1)
A sequence (
) in a normed space (,‖ ⋅ ‖) is said to
be -bounded if there exists >0 such that
{ ∈ N :
> } ∈ . (2)
A sequence (
) in a normed space (,‖ ⋅ ‖) is said to
be -Cauchy if, for each > 0, there exists a positive
integer = () such that
{ ∈ N :
−
≥ } ∈ . (3)
In paper [10], the notion of -convergent and bounded
sequences is introduced as follows: let = (
)
∞
=1
be a
strictly increasing sequence of positive real numbers tending
to infnity; that is,
0<
1
<
2
<⋅⋅⋅,
→ ∞ as → ∞. (4)
We say that a sequence = (
)∈ is -convergent
to the number ∈ C, called the -limit of , if Λ
() →
as →∞, where
Λ
() =
1
∑
=1
(
−
−1
)
, ∈ N. (5)