Research Article
New Difference Sequence Spaces Defined by
Musielak-Orlicz Function
M. Mursaleen,
1
Sunil K. Sharma,
2
S. A. Mohiuddine,
3
and A. KJlJçman
4
1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2
Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal, Jammu and Kashmir 181122, India
3
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4
Department of Mathematics, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to A. Kılıc ¸man; akilic@upm.edu.my
Received 17 March 2014; Revised 11 July 2014; Accepted 11 July 2014; Published 22 July 2014
Academic Editor: Feyzi Bas ¸ar
Copyright © 2014 M. Mursaleen et al. 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 introduce new sequence spaces by using Musielak-Orlicz function and a generalized
∧
-diference operator on -normed space.
Some topological properties and inclusion relations are also examined.
1. Introduction and Preliminaries
Te notion of the diference sequence space was introduced
by Kızmaz [1]. It was further generalized by Et and C ¸ olak [2]
as follows: (Δ
)={=(
)∈:(Δ
)∈} for =ℓ
∞
,,
and
0
, where is a nonnegative integer and
Δ
=Δ
−1
−Δ
−1
+1
,Δ
0
=
∀∈ N (1)
or equivalent to the following binomial representation:
Δ
=
∑
V=0
(−1)
V
(
V
)
+V
. (2)
Tese sequence spaces were generalized by Et and Basarir [3]
taking =ℓ
∞
(), (), and
0
().
Dutta [4] introduced the following diference sequence
spaces using a new diference operator:
(Δ
()
)={=(
)∈:Δ
()
∈} for =ℓ
∞
,,
0
,
(3)
where Δ
()
=(Δ
()
)=(
−
−
) for all ,∈ N.
In [5], Dutta introduced the sequence spaces
(‖⋅,⋅‖,Δ
()
,),
0
(‖⋅,⋅‖,Δ
()
,), ℓ
∞
(‖⋅,⋅‖,Δ
()
,), (‖⋅,⋅‖,
Δ
()
,), and
0
(‖⋅,⋅‖,Δ
()
,), where , ∈ N and
Δ
()
= (Δ
()
)=(Δ
−1
()
−Δ
−1
()
−
) and Δ
0
()
=
for all ,∈ N, which is equivalent to the following binomial
representation:
Δ
()
=
∑
V=0
(−1)
V
(
V
)
−V
. (4)
Te diference sequence spaces have been studied by authors
[6–14] and references therein. Bas ¸ar and Altay [15] introduced
the generalized diference matrix =(
) for all ,∈ N,
which is a generalization of Δ
(1)
-diference operator by
=
{
{
{
{
{
, =
, =−1
0, (>) or (0≤<−1).
(5)
Bas ¸arir and Kayikc ¸i [16] defned the matrix
(
) which
reduced the diference matrix Δ
(1)
in case =1, = −1.
Te generalized
-diference operator is equivalent to the
following binomial representation:
=
(
)=
∑
V=0
(
V
)
−V
V
−V
. (6)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 691632, 9 pages
http://dx.doi.org/10.1155/2014/691632