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Hindawi Publishing Corporation
Abstr act and Appli ed Analys is
Volume 2013, Article ID 123798, 13 pages
http://dx.doi.org/10.1155/2013/123798
Research Article
Generalized Difference 𝜆-Sequence Spaces Defined by Ideal
Convergence and the Musielak-Orlicz Function
Awad A. Bakery1,2
1DepartmentofMathematics,FacultyofScienceandArts,KingAbdulazizUniversity(KAU),P.O.Box80200,
Khulais 21589, Saudi Arabia
2Department 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. is 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 dierence sequence spaces combining an innite 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
roughout the paper ,∞,,0,and 𝑝denote the classes
of all, bounded, convergent, null, and p-absolutely summable
sequences of complex numbers. e sets of natural numbers
and real numbers will be denoted by Nand R,respectively.
Many authors studied various sequence spaces using normed
or seminormed linear spaces. In this paper, using an innite
matrix of complex numbers and the notion of ideal, we
aimed to introduce some new sequence spaces with respect to
generalized dierence operator 𝑠
𝑚on -sequences and the
Musielak-Orlicz function in -normed linear spaces. By an
ideal we mean a family ⊂2𝑌of subsets of a nonempty
set satisfying the following: (i) ∈; (ii) , ∈
imply ∪∈; (iii) ∈,⊂imply ∈,
whileanadmissibleidealof further satises {}∈for
each ∈.enotionofidealconvergencewasintroduced
rst by Kostyrko et al. [1] as a generalization of statistical
convergence. e concept of 2-normed spaces was initially
introduced by G¨
ahler [2]inthe1960s,whilethatof -normed
spaces can be found in [3]; this concept has been studied
by many authors; see for instance [4–7]. e 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 ⊂2Nis a nontrivial
ideal in N,thesequence (𝑛)𝑛∈Nin a normed space (;⋅)is
said to be -convergent to ∈, if, for each >0,
()=∈N:𝑛−≥∈. (1)
Asequence(𝑘)in a normed space (,⋅)is said to
be -bounded if there exists >0such that
∈N:𝑘>∈. (2)
Asequence(𝑘)in a normed space (,⋅)is said to
be -Cauchy if, for each >0, there exists a positive
integer =()such that
∈N:𝑘−𝑚≥∈. (3)
In paper [10], the notion of -convergent and bounded
sequences is introduced as follows: let =(
𝑗)∞
𝑗=1 be a
strictly increasing sequence of positive real numbers tending
to innity; that is,
0<1<2<⋅⋅⋅, 𝑗→∞ as →∞. (4)
We say th a t a s e quen c e =(𝑗)∈is -convergent
to the number ∈C,calledthe-limit of ,ifΛ𝑗() →
as →∞,where
Λ𝑗()=1
𝑗
𝑗
𝑟=1 𝑟−𝑟−1𝑟,∈N.(5)
2Abstract and Applied Analysis
e class of all sequences (𝑗)satisfying this property is
denoted by Λ.
In particular, we say that is a -null sequence
if Λ𝑗() → 0as →∞. Further, we say that is -
bounded if sup𝑗|Λ𝑗()| <∞.Hereandinthesequel,we
will use the convention that any term with a zero subscript
is equal to naught; for example, 0=0and 0=0.Now,it
is well known [10]thatif lim
𝑗𝑗=in the ordinary sense of
convergence, then
lim
𝑗→∞1
𝑗
𝑗
𝑟=1 𝑟−𝑟−1𝑟−=0. (6)
is implies that
lim
𝑗Λ𝑗()−
=lim
𝑗→∞1
𝑗
𝑗
𝑟=1 𝑟−𝑟−1𝑟−=0, (7)
which yields that lim𝑗Λ𝑗()=and hence is -convergent
to . We therefore deduce that the ordinary convergence
implies the -convergence to the same limit.
An Orlicz function is a function :[0,∞)→
[0,∞)which is continuous, nondecreasing, and convex
with (0)= 0and ()> 0for >0and () →
∞,as→∞. If convexity of is replaced by (+
)≤()+(),thenitiscalledamodulusfunction,
introduced by Nakano [11]. Ruckle [12] and Maddox [13]used
the idea of a modulus function to construct some spaces of
complexsequences.AnOrliczfunctionis said to satisfy
the 2-condition for all values of ≥0, if there exists
aconstant>0,suchthat(2) ≤ ().e2-
condition is equivalent to () ≤ ()for all values
of and for >1. Lindentrauss and Tzafriri [14]usedthe
idea of an Orlicz function to dene the following sequence
spaces: 𝑀=∈:∞
𝑘=1|()|
<∞, (8)
whichisaBanachspacewiththeLuxemburgnormdened
by =inf >0:∞
𝑘=1|()|
≤1. (9)
e space 𝑀is closely related to the space 𝑝,whichis
an Orlicz sequence space with ()=𝑝for 1≤<∞.
Recently dierent classes of sequences have been introduced
using Orlicz functions. See [7,9,15–17].
Asequence=(
𝑘)of Orlicz functions 𝑘for
all ∈Nis called a Musielak-Orlicz function.
Kizmaz [18] dened the dierence sequences ∞(),(),
and 0()as follows.
()={=(𝑘):(𝑘)∈}.For=∞,,and
0,where=(𝑘−𝑘+1),forall∈N.eabovespaces
are Banach spaces, normed by =|1|+sup𝑘|𝑘|.e
notion of dierence sequence spaces was generalized by Et
and Colak [19] as follows: (𝑠)={=(𝑘):(𝑠𝑘)∈}.
For =∞,and 0,where∈N,(𝑠𝑘)=(𝑠−1𝑘−
𝑠−1𝑘+1)and so that 𝑠𝑘=∑𝑠
𝑛=0(−1)𝑛𝑠
𝑛𝑘+𝑛.Tripathy
and Esi [20] introduced the following new type of dierence
sequence spaces.
(𝑚)={=(𝑘):(𝑚𝑘)∈},=∞,,and 0,
where 𝑚𝑘=(𝑘−𝑘+𝑚),forall ∈N.Tripathyetal.[21],
generalized the previous notions and unied them as follows.
Let and be nonnegative integers, then for agiven
sequence space we have
𝑠
𝑚==𝑘:𝑠
𝑚𝑘∈,where
𝑠
𝑚𝑘=𝑠
𝑛=0(−1)𝑛𝑠
𝑛𝑘+𝑚𝑛 (forward dierence),
(𝑠)
𝑚==𝑘:(𝑠)
𝑚𝑘∈,where
(𝑠)
𝑚𝑘=𝑠
𝑛=0(−1)𝑛𝑠
𝑛𝑘−𝑚𝑛 (backward dierence),
(10)
where 𝑘=0,for <0.
2. Definitions and Preliminaries
Let ∈Nand be a linear space over the eld of
dimension ,where≥≥2and is the eld of
real or complex numbers. A real valued function ⋅...⋅
on 𝑛satises the following four conditions:
(1) 1,2,...,𝑛=0if and only if 1,2,...,and
𝑛are linearly dependent in ;
(2) 1,2,...,𝑛is invariant under permutation;
(3) 1,2,...,𝑛=||1,2,...,𝑛for any ∈;
(4) +,2,...,𝑛≤,2,...,𝑛+1,2,...,𝑛,
which is called an -norm on and the
pair (;⋅...⋅)is called an -normed space over
the eld . For example, we may take =R𝑛being
equipped with the -norm 1,2,...,𝑛𝐸=the
volume of the -dimensional parallelepiped spanned
by the vectors 1,2,...,and 𝑛which may be given
explicitlybytheformula
1,2,...,𝑛𝐸=det 𝑖𝑗
=abs 11 12 ⋅⋅⋅1𝑛
21 22 ⋅⋅⋅2𝑛
.
.
..
.
.d.
.
.
𝑛1 𝑛2 ⋅⋅⋅𝑛𝑛
, (11)
where 𝑖=(𝑖1,𝑖2,...,𝑖𝑛)for each ∈N.
Let (,⋅...⋅)be an -normed space of dimension ≥
≥2and {1,2,3,...,𝑛}a linearly independent set in .
en, the function ⋅...⋅∞on 𝑛−1 dened by
1,2,...,𝑛∞=max
1≤𝑖≤𝑛 1,2,...,𝑛−1,𝑖(12)
Abstract and Applied Analysis 3
denes an ( − 1)-norm on with respect to
1,2,3,...,and 𝑛andthisisknownasthederived
(−1)-norm. e standard ()-norm on ,arealinner
product space of dimension ≥,isasfollows:
1,2,...,𝑛𝑠
=abs1,11,2⋅⋅⋅1,𝑛
2,12,2⋅⋅⋅2,𝑛
.
.
..
.
.d.
.
.
𝑛,1𝑛,2⋅⋅⋅𝑛,𝑛1/2,(13)
where ⋅,⋅denotes the inner product on .Ifwetake=
R𝑛,then 1,2,...,𝑛𝐸=1,2,...,𝑛𝑠.(14)
For =1,this-norm is the usual norm 1=
1,1.
Denition 1. Asequence(𝑘)in an -normed space is said
to be convergent to ∈if
lim
𝑘→∞1,2,...,𝑛−1,𝑘−𝑛=0,
∀1,2,...,𝑛−1 ∈. (15)
Denition 2. Asequence (𝑘)in an -normed space is called
Cauchy (with respect to -norm) if
lim
𝑘,𝑗→ ∞1,2,...,𝑛−1,𝑘−𝑗𝑛=0,
∀1,2,...,𝑛−1 ∈. (16)
If every Cauchy sequence in converges to an ∈,
then is said to be complete (with respect to the -norm).
Acomplete -normed space is called an -Banach space.
Denition 3. Asequence(𝑘)in an -normed space (,⋅
...⋅)is said to be -convergent to 0∈with respect to -
norm, if, for each >0,theset
∈N:𝑘−0,1,2,...,𝑛−1
≥,for every 1,2,...,𝑛−1∈. (17)
Denition 4. Asequence(𝑘)in an -normed space (,⋅
...⋅)is said to be -Cauchy if, for each >0, there exists a
positive integer =()such that the set
∈N:𝑘−𝑚,1,2,...,𝑛−1
≥,for every 1,2,...,𝑛−1∈. (18)
Let =(𝑘)be a sequence; then ()denotes the set
of all permutations of the elements of (𝑘);thatis,()=
(𝜋(𝑛)):is a permutation of N.
Denition 5. Asequencespaceis said to be symmetric
if ()⊂for all ∈.
Denition 6. Asequencespaceis said to be normal (or
solid) if (𝑘𝑘)∈,whenever (𝑘)∈and for all sequences
(𝑘)of scalars with |𝑘|≤1for all ∈N.
Denition 7. Asequencespaceis said to be a sequence
algebra if ,∈;then ⋅=(𝑘𝑘)∈.
Lemma 8. Every -normed space is an (−)-normed space
for all =1,2,3,...,−1.Inparticular,every -normed space
is a normed space.
Lemma 9. On a standard -normed space ,thederived (−
1)-norm ⋅...⋅∞dened with respect to the orthogonal
set {1,2,...,𝑛}is equivalent to the standard ( − 1)-
norm ⋅...⋅𝑠.Tobeprecise,onehas
1,2,...,𝑛−1∞≤⋅...⋅𝑠≤1,2,...,𝑛−1∞,(19)
for all 1,2,...,𝑛−1 ∈,where1,2,...,𝑛−1∞=
max1≤𝑖≤𝑛{1,2,...,𝑛−1,𝑖𝑆}.
For any bounded sequence (𝑛)of positive numbers, one
has the following well known inequality: if 0≤𝑘≤sup𝑘𝑘=
and =max(1,2𝐺−1),then |𝑛+𝑛|𝑝𝑛≤(|𝑛|𝑝𝑛+|𝑛|𝑝𝑛),
for all and 𝑘,𝑘∈C.
3. Main Results
In this section, we dene some new ideal convergent
sequence spaces and investigate their linear topological
structures.Wendoutsomerelationsrelatedtothese
sequence spaces. Let be an admissible ideal of N,
M=(
𝑗)a Musielak-Orlicz function, and 𝑠
𝑚the
forward generalized dierence operator on the class of all
sequences (𝑗)satisfying the property Λand an -normed
space (,⋅...⋅). Further, let =(𝑘)be any bounded
sequence of positive real numbers; we will dene the follow-
ing sequence spaces:
,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
=
∈(−):∀>0
×
∈N:∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()−
,
1,2,...,𝑛−1𝑝𝑗≥
∈,
4Abstract and Applied Analysis
for some >0,∈and each 1,2,...,𝑛−1 ∈
,
,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
0
=
∈(−):∀>0
×
∈N:∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
,
1,2,...,𝑛−1𝑝𝑗≥
∈,
for some >0,and each 1,2,...,𝑛−1 ∈
,
,M,𝑠
𝑚,Λ,,⋅...⋅∞
=
∈(−):
∃>0st.sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
,
1,2,...,𝑛−1𝑝𝑗
<∞,
for some >0and each 1,2,...,𝑛−1 ∈
,
,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
∞
=
∈(−):
∃>0,s.t.
∈N:∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
,
1,2,...,𝑛−1𝑝𝑗≥
∈,
for some >0,
and each 1,2,...,𝑛−1 ∈
.(20)
Let us consider a few special cases of the aforementioned
sets.
(1) If 𝑘()=(),forall∈Nthen the previous
classes of sequences are denoted by [,,𝑠
𝑚,Λ,,⋅...⋅
]𝐼,[,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,[,,𝑠
𝑚,Λ,,⋅...⋅]∞,
and [,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞,respectively.
(2) If 𝑘=1for all ∈Nthen the previous classes
of sequences are denoted by [,M,𝑠
𝑚,Λ,⋅...⋅]I,
[,M,𝑠
𝑚,Λ,⋅...⋅]𝐼
0,[,M,𝑠
𝑚,Λ,⋅...⋅]∞,and
[,M,𝑠
𝑚,Λ,⋅...⋅]𝐼
∞,respectively.
(3) If 𝑘() = ,forall∈Nand ∈[0,∞[,
then the previous classes of sequences are denoted by
[,𝑠
𝑚,Λ,,⋅...⋅]𝐼,[,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,
[,𝑠
𝑚,Λ,,⋅...⋅]∞,and[,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞,
respectively.
(4) If we take 𝑘()= (),forall∈Nand =
(𝑘𝑗)as
𝑘𝑗 =
1, ≥,
0, otherwise,(21)
then we denote the previous classes of sequences
by [,,𝑠
𝑚,Λ,,⋅...⋅]𝐼,[,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,
[,,𝑠
𝑚,,⋅...⋅]∞,and [,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞,
respectively.
(5) If we take 𝑘()=()and =(𝑘𝑗 )as
𝑘𝑗 =
1
𝑘,∈
𝑘=−𝑘+1,,
0, otherwise,(22)
where (𝑘)is a nondecreasing sequence of positive numbers
tending to ∞,1=1,and𝑘+1 ≤𝑘+1,thenwedenote
the previous classes of sequences by [Φ,,𝑠
𝑚,Λ,,⋅...⋅
]𝐼,[Φ,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,[Φ,,𝑠
𝑚,Λ,,⋅...⋅]∞,
and [Φ,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞.
(6) If =(
𝑘𝑗)as in (22), then we denote the previ-
ous classes of sequences by [Φ,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼,
[Φ,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,[Φ,M,𝑠
𝑚,Λ,,⋅...⋅]∞,
and [Φ,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞.
And if 𝑗=for all ∈N, then the previous classes
of sequences are denoted by [Φ,M,𝑠
𝑚,,,⋅...⋅]𝐼,
[Φ,M,𝑠
𝑚,,,⋅...⋅]𝐼
0,[Φ,M,𝑠
𝑚,,,⋅...⋅
]∞,and[Φ,M,𝑠
𝑚,,,⋅...⋅]𝐼
∞and they are a
Abstract and Applied Analysis 5
generalization of the sequence spaces dened by Bakery et al.
[22].
(7) By a lacunary =(𝑟),=0,1,2,...,where0=0,
we will mean an increasing sequence of nonnegative integers
with 𝑟−𝑟−1 →∞as →∞. e interval determined
by will be denoted by 𝑟=]𝑟−1,𝑟]and 𝑟=𝑟−𝑟−1 and
let =(𝑘𝑗 )as
𝑘𝑗 =
1
𝑟,∈
𝑟=𝑟−1,𝑟,
0, otherwise.(23)
en we denote the previous classes of sequences
by [,,𝑠
𝑚,Λ,,⋅...⋅]𝐼,[,,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,
[,,𝑠
𝑚,Λ,,⋅...⋅]∞,and [,,𝑠
𝑚,Λ,,⋅...⋅
]𝐼
∞,respectively.
(8) If 𝑘() = (),forall∈N,=,
and 𝑗=, then the previous classes of sequences are
denoted by [,𝑠
𝑚,,,⋅...⋅]𝐼,[,𝑠
𝑚,,,⋅...⋅
]𝐼
0,[,𝑠
𝑚,,,⋅...⋅]∞,and [,𝑠
𝑚,,,⋅...⋅]𝐼
∞.
(9) If =1, then the previous classes of sequences are
denoted by [,M,𝑚,Λ,,⋅...⋅]𝐼,[,𝑚,,,⋅
...⋅]𝐼
0,[,𝑚,,,⋅...⋅]∞,and[,𝑚,,,⋅
...⋅]𝐼
∞.
(10) If =1, then the previous classes of sequences are
denoted by [,M,𝑠,Λ,,⋅...⋅]𝐼,[,𝑠,,,⋅...⋅]𝐼
0,
[,𝑠,,,⋅...⋅]∞,and[,𝑠,,,⋅...⋅]𝐼
∞.
eorem 10. e spaces [,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼,
[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0and [,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞
are linear spaces.
Proof. Wewillprovetheassertionfor[,M,𝑠
𝑚,Λ,,⋅
...⋅]𝐼
0; the others can be proved similarly. Assume that =
(𝑘),=(𝑘)∈[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,and,∈C.
en, there exist 1and 2such that the sets
∈N:∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗
≥2
∈,
(24)
∈N:
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚(Λ𝑗()
2,1,2,...,𝑛−1𝑝𝑗
≥2
∈. (25)
Since (,⋅...⋅)is an -norm, 𝑠
𝑚and Λ𝑗are linear,
and the Orlicz function 𝑗is convex for all ∈N,the
following inequality holds:
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗+
||1𝐹 +2,
1,2,...,𝑛−1𝑝𝑗
≤∞
𝑗=1𝑘𝑗 ||1
||1+2
×𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
+∞
𝑗=1𝑘𝑗 2
||1+2
×𝑗𝑠
𝑚Λ𝑗
2,1,2,...,𝑛−1𝑝𝑗
≤∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
+∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗
2,
1,2,...,𝑛−1𝑝𝑗,(26)
where =max{||1/(||1+||2),||2/(||1+||2)}.
On the other hand from the above inequality we get
∈N:∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗+
||1+2,
6Abstract and Applied Analysis
1,2,...,𝑛−1𝑝𝑗≥
⊆
∈N:
×∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗≥2
∪
∈N:
×∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗
2,1,2,...,𝑛−1𝑝𝑗
≥2
.(27)
Sincethetwosetsontherighthandsidebelongto ,this
completes the proof.
eorem 11. e spaces [,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼,
[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,and[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞
are paranormed spaces (not totally paranormed) with respect to
the paranorm Δdened by
Δ()
=𝑚𝑠
𝑗=1 𝑗,1,2,...,𝑛−1
+inf
𝑝𝑘/𝐻 :
sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗
1/𝐻
≤1, for some >0,
and each 1,2,...,𝑛−1 ∈
,(28)
where =max{1,sup𝑘𝑘}.
Proof. Clearly Δ(−) = Δ()and Δ() = 0.Let=
(𝑘)and =(𝑘)∈[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0.en,
for >0we set
1=
:sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
,
1,2,...,𝑛−1𝑝𝑗
1/𝐻 ≤1,
for each 1,2,...,𝑛−1 ∈
,
2
=
:sup
𝑘
∞
𝑗=1𝑘𝑗
×
𝑗𝑠
𝑚Λ𝑗
,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻
≤1, for each 1,2,...,𝑛−1 ∈
.(29)
Let 1∈1,2∈2,and =1+2;thenwehave
sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗+
,
1,2,...,𝑛−1𝑝𝑗
1/𝐻
Abstract and Applied Analysis 7
≤1
1+2
×sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗
1/𝐻
+2
1+2
×sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗
2,
1,2,...,𝑛−1𝑝𝑗
1/𝐻 ≤1,
Δ+
=𝑚𝑠
𝑗=1 𝑗+𝑗,1,2,...,𝑛−1
+inf 1+2𝑝𝑘/𝐻 :1∈1,2∈2
≤𝑚𝑠
𝑗=1 𝑗,1,2,...,𝑛−1
+inf 1𝑝𝑘/𝐻 :1∈1
+𝑚𝑠
𝑗=1 𝑗,1,2,...,𝑛−1
+inf 2𝑝𝑘/𝐻 :2∈2
=Δ()+Δ. (30)
Let 𝑡→where 𝑡,∈C,andletΔ(𝑡−)→0
as →∞.WehavetoshowthatΔ(𝑡𝑡−)→0as
→∞.Weset
3
=
𝑡:sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
𝑡,
1,2,...,𝑛−1𝑝𝑗
1/𝐻 ≤1,
for each 1,2,...,𝑛−1 ∈
,
4=
1
𝑡:sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗
1
𝑡,
1,2,...,𝑛−1𝑝𝑗
1/𝐻 ≤1,
for each 1,2,...,𝑛−1 ∈
.(31)
If 𝑡∈3and 1
𝑡∈4, then by using non-decreasing and
convexity of the Orlicz function 𝑗for all ∈Nwe get
sup
𝑘
∞
𝑗=1𝑘𝑗
×
𝑗𝑠
𝑚𝑡𝑡
𝑗−𝑗
|𝑡−|𝑡+||1
𝑡,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻
≤sup
𝑘
∞
𝑗=1𝑘𝑗
×
𝑗𝑠
𝑚𝑡𝑡
𝑗−𝑡
𝑗
|𝑡−|𝑡+||1
𝑡,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻
+sup
𝑘
∞
𝑗=1𝑘𝑗
8Abstract and Applied Analysis
×
𝑗𝑠
𝑚𝑡
𝑗−𝑗
|𝑡−|𝑡+||1
𝑡,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻
≤𝑡−𝑡
|𝑡−|𝑡+||1
𝑡
×sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚𝑡
𝑗
𝑡,1,2,...,𝑛−1𝑝𝑗
1/𝐻
+||1
𝑡
|𝑡−|𝑡+||1
𝑡
×sup
𝑘
∞
𝑗=1𝑘𝑗
𝑗𝑠
𝑚𝑡
𝑗−𝑗
1
𝑡,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻.(32)
From the previous inequality, it follows that
sup
𝑘
∞
𝑗=1𝑘𝑗
×
𝑗𝑠
𝑚𝑡𝑡
𝑗−𝑗
|𝑡−|𝑡+||1
𝑡,
1,2,...,𝑛−1
𝑝𝑗
1/𝐻 ≤1,
(33)
and consequently
Δ𝑡𝑡−
=𝑚𝑠
𝑗=1 𝑡𝑡
𝑗−𝑗,1,2,...,𝑛−1
+inf 𝑡−𝑡+||1
𝑡𝑝𝑘/𝐻 :𝑡∈3,1
𝑡∈4
≤𝑡−𝑚𝑠
𝑗=1 𝑡
𝑗,1,2,...,𝑛−1
+𝑡−𝑝𝑘/𝐻 inf 𝑡𝑝𝑘/𝐻 :𝑡∈3
+||𝑚𝑠
𝑗=1 𝑡𝑡
𝑗−𝑗,1,2,...,𝑛−1
+||𝑝𝑘/𝐻 inf 1
𝑡𝑝𝑘/𝐻 :1
𝑡∈4
≤max 𝑡−,𝑡−𝑝𝑘/𝐻Δ𝑡
+max ||,||𝑝𝑘/𝐻Δ𝑡−. (34)
Note that Δ(𝑡)≤Δ()+Δ(𝑡−),forall∈N.
Hence,byourassumption,therighthandof (34)tendsto0
as →∞, and the result follows. is completes the proof
of the theorem.
eorem 12. Let M=(𝑗),M=(
𝑗),andM =(
𝑗)
be the Musielak-Orlicz functions. en, the following hold:
(a) [,M,𝑠
𝑚,Λ,, ⋅ ... ⋅ ]𝐼
0⊆
[,M.M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,provided=(𝑘)
such that 0=inf 𝑘>0,
(b) [,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0⊆[,M+
M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0.
Proof. (a) Let >0be given. Choose 1>0such
that sup𝑘(∑∞
𝑗=1 𝑘𝑗)max{𝐺
1,𝐺0
1}<.Usingthecontinuityof
the Orlicz function ,choose 0<<1such that 0<<
implies that ()<1.
Let =(𝑘)be any element in [,M,𝑠
𝑚,Λ,,⋅...⋅
]𝐼
0and put
𝛿=
∈N:
∞
𝑗=1𝑘𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
≥𝐺
.(35)
en, by the denition of ideal convergent, we have the
set 𝛿∈.If ∉𝛿,thenwehave
Abstract and Applied Analysis 9
∞
𝑗=1𝑘𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
<𝐺
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
<𝐺⇒
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1<.
(36)
Using the continuity of the Orlicz function 𝑗for
all and the relation (36), we have
𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1<1.(37)
Consequently, we get
∞
𝑗=1𝑘𝑗𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
<sup
𝑘∞
𝑗=1𝑘𝑗max 𝐺
1,𝐺0
1<
⇒∞
𝑗=1𝑘𝑗𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
<. (38)
is shows that
∈N:
∞
𝑗=1𝑘𝑗 𝑗
𝑗𝑠
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗
≥
⊆𝛿∈.
(39)
is proves the assertion.
(b) Let =(𝑘)be any element in [,M,Λ,,⋅...⋅
]𝐼
0. en, by the following inequality, the results follow:
∞
𝑗=1𝑘𝑗
𝑗+
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
≤∞
𝑗=1𝑘𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗
+∞
𝑗=1𝑘𝑗
𝑗𝑠
𝑚Λ𝑗()
1,1,2,...,𝑛−1𝑝𝑗.
(40)
eorem 13. e inclusions [,M,𝑠−1
𝑚,Λ,⋅...⋅]⊆
[,M,𝑠
𝑚,Λ,⋅...⋅]are strict for ,≥1in general
where =𝐼,𝐼
0,and𝐼
∞.
Proof. We will give the proof for [,M,𝑠−1
𝑚,Λ,⋅...⋅]𝐼
0⊆
[,M,𝑠
𝑚,Λ,⋅...⋅]𝐼
0only. e others can be proved by
similar arguments. Let =(𝑘)∈[,M,𝑠−1
𝑚,Λ,⋅...⋅]𝐼
0.
en let >0be given; there exist >0such that
∈N:∞
𝑗=1𝑘𝑗𝑗𝑠−1
𝑚Λ𝑗()
,1,2,...,𝑛−1
≥2
∈. (41)
Since 𝑗for all ∈Nis non-decreasing and convex, it
follows that
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
2 ,1,2,...,𝑛−1
=∞
𝑗=1𝑘𝑗
×𝑗𝑠−1
𝑚Λ𝑗+1 ()−𝑠−1
𝑚Λ𝑗()
2 ,
1,2,...,𝑛−1
≤12∞
𝑗=1𝑘𝑗𝑗𝑠−1
𝑚Λ𝑗+1 ()
,1,2,...,𝑛−1
+12∞
𝑗=1𝑘𝑗𝑗𝑠−1
𝑚Λ𝑗()
,1,2,...,𝑛−1,(42)
10 Abstract and Applied Analysis
andthenwehave
∈N:∞
𝑗=1𝑘𝑗
×𝑗𝑠
𝑚Λ𝑗()
2 ,
1,2,...,𝑛−1≥
⊆
∈N:12
×∞
𝑗=1𝑘𝑗𝑗𝑠−1
𝑚𝑗+1
,
1,2,...,𝑛−1≥2
∪
∈N:12
×∞
𝑗=1𝑘𝑗𝑗𝑠−1
𝑚Λ𝑗()
,
1,2,...,𝑛−1≥2
.(43)
Let 𝑘()=()=for all ∈[0,∞[,∈Nand
𝑘=for all ∈N. Consider a sequence =(𝑘)=(𝑠).
en, ∈[,M,𝑠
𝑚,Λ,⋅...⋅]𝐼
0but does not belong to
[,M,𝑠−1
𝑚,Λ,⋅...⋅]𝐼
0,for==1.isshowsthat
the inclusion is strict.
eorem 14. Let 0<𝑘≤𝑘for all ∈N;then
,M,𝑠
𝑚,Λ,,⋅...⋅∞
⊆,M,𝑠
𝑚,Λ,,⋅...⋅∞.(44)
Proof. Let =(𝑗)∈[,M,𝑠
𝑚,Λ,,⋅...⋅]∞;then
there exists some >0such that
sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1𝑝𝑗<∞.
(45)
is implies that
𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1<1, (46)
for a suciently large value of .Since𝑗for all ∈Nis
non-decreasing, we get
sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1𝑞𝑗
≤sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1𝑝𝑗<∞.
(47)
us, ∈[,M,𝑠
𝑚,Λ,,⋅...⋅]∞.iscompletes
the proof of the theorem.
eorem 15. (i) If 0<inf 𝑘≤𝑘<1,then
,M,𝑠
𝑚,Λ,,⋅...⋅∞
⊆,M,𝑠
𝑚,Λ,⋅...⋅∞.(48)
(ii) If 1<𝑘≤sup𝑘𝑘<∞,then[,M,𝑠
𝑚,Λ,⋅...⋅
]∞⊆[,M,𝑠
𝑚,Λ,,⋅...⋅]∞.
Proof. (i) Let =(𝑗)∈[,M,𝑠
𝑚,Λ,,⋅...⋅]∞;since
0<inf𝑘𝑘≤𝑘<1,thenwehave
sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1
≤sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1𝑝𝑗<∞,
(49)
and hence ∈[,M,𝑠
𝑚,Λ,⋅...⋅]∞.
(ii) Let 1<
𝑘≤sup𝑘𝑘<∞and =(
𝑗)∈
[,M,𝑠
𝑚,Λ,⋅...⋅]∞. en for each 0<<1there
exists a positive integer 0such that
sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1
≤<1, (50)
for all ≥0. is implies that
sup
𝑘
∞
𝑗=1𝑘𝑗𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1𝑝𝑗
≤sup
𝑘
∞
𝑗=1𝑘𝑗 𝑗𝑠
𝑚Λ𝑗()
,1,2,...,𝑛−1<∞.
(51)
us ∈[,M,𝑠
𝑚,Λ,,⋅...⋅]∞and this completes
the proof.
eorem 16. For any sequence of the Orlicz functions
M=(
𝑗)which satises the 2-condition, we have
[,𝑠
𝑚,Λ,,⋅...⋅]𝐼⊂[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼.
Abstract and Applied Analysis 11
Proof. Let =(𝑗)∈[,𝑠
𝑚,Λ,,⋅...⋅]𝐼and >0be
given. en, there exist >0such that the set
∈N:
∞
𝑗=1𝑘𝑗 𝑠
𝑚Λ𝑗()−
,
1,2,...,𝑛−1𝑝𝑗≥
∈,for some .
(52)
By taking 𝑗=(𝑠
𝑚(Λ𝑗())−)/,1,2,...,𝑛−1,let
>0and choose wit 0<<1such that 𝑗()<for all
∈N;for0≤≤, consider that
∞
𝑗=1𝑗𝑗𝑝𝑗
=∞
𝑗=1,𝑦𝑗≤𝛿𝑗𝑗𝑝𝑗
+∞
𝑗=1,𝑦𝑗>𝛿𝑗𝑗𝑝𝑗,
(53)
since 𝑗is continuous for all ∈N.
∑𝑗∈𝐼𝑘,𝑦𝑗≤𝛿[𝑗(𝑗)]𝑝𝑗<and for 𝑗>, we use the fact
that 𝑗<𝑗/<1+𝑗/.SinceM=(𝑗)is non-decreasing
and convex, it follows that
𝑗𝑗<𝑗1+𝑗
<12𝑗(2)+12𝑗2𝑗
. (54)
Since M=(𝑗)satises the 2-condition, then
𝑗𝑗<𝑗
2𝑗(2)+𝑗
2𝑗(2)=𝑗
𝑗(2).(55)
Hence
∞
𝑗=1,𝑦𝑗>𝛿𝑗𝑗𝑝𝑗
<max 1,sup
𝑗−1𝑗(2)𝑝𝑗
×∞
𝑗=1,𝑦𝑗>𝛿𝑗𝑝𝑗,
(56)
andthenwehave
∞
𝑗=1𝑗𝑗𝑝𝑗
<+max 1,sup
𝑗−1𝑗(2)𝑝𝑗∞
𝑗=1,𝑦𝑗>𝛿𝑗𝑝𝑗.(57)
isprovesthat[,𝑠
𝑚,Λ,, ⋅... ⋅]𝐼⊂
[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼.
eorem 17. Let 0<𝑛≤𝑛<1and (𝑛/𝑛)be bounded;
then ,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
⊆,M,𝑠
𝑚,Λ,,⋅...⋅𝐼.(58)
Proof. Let =(𝑗)∈[,M,Λ,,⋅...⋅]∞and we put
𝑗=𝑗𝑠
𝑚Λ𝑗()−
,1,2,...,𝑛−1𝑞𝑗,
𝑗=𝑗
𝑗∀∈N.(59)
en 0<𝑗≤1,forall∈N.Letitbesuchthat0<
≤𝑗for all ∈N. Dene the sequences (𝑗)and (𝑗)as
follows: for 𝑗≥1,let 𝑗=𝑗and 𝑗=0;for 𝑗<1,let 𝑗=
0and 𝑗=𝑗.enclearly,forall∈Nwe have 𝑗=𝑗+
𝑗,𝛽𝑗
𝑗=𝛽𝑗
𝑗+𝛽𝑗
𝑗,𝛽𝑗
𝑗≤𝑗≤𝑗,and 𝛽𝑗
𝑗≤𝛽
𝑗. erefore, we
have
∞
𝑗=1𝑘𝑗𝛽𝑗
𝑗≤∞
𝑗=1𝑘𝑗𝑗≤
∞
𝑗=1𝑘𝑗𝑗
𝛽.(60)
Hence ∈[,M,𝑠
𝑚,Λ,,⋅...⋅]∞.
eorem 18. For any two sequences =(
𝑘)and =
(𝑘)of positive real numbers and for any two n-
norms ⋅...⋅1and ⋅...⋅2on ,thefollowingholds:
,M,𝑠
𝑚,Λ,,⋅...⋅1
∩,M,𝑠
𝑚,Λ,,⋅...⋅2 =, (61)
where =𝐼,𝐼
0,𝐼
∞,and∞.
Proof. e proof of the theorem is obvious, because the zero
element belongs to each of the sequence spaces involved in
the intersection.
eorem 19. e sequence spaces [,M,𝑠
𝑚,Λ,,⋅...⋅
]𝐼,[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
0,[,M,𝑠
𝑚,Λ,,⋅...⋅
]∞,and[,M,𝑠
𝑚,Λ,,⋅...⋅]𝐼
∞are neither solid nor
symmetric nor sequence algebras for ,≥1in general.
Proof. e proof is obtained by using the same techniques of
Et [23] and eorems 15,17,and18.
Note 1. It is clear from denitions that
,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
0
⊆,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
⊆,M,𝑠
𝑚,Λ,,⋅...⋅𝐼
∞.(62)
12 Abstract and Applied Analysis
eorem 20. e spaces [,M,𝑠
𝑚,Λ,,⋅...⋅]and
[,M,,⋅...⋅]are equivalent as topological spaces, where
=𝐼,𝐼
0,𝐼
∞,and∞.
Proof. Consider the mapping
:,M,𝑠
𝑚,Λ,,⋅...⋅→,M,,⋅...⋅,
(63)
dened by () = (𝑠
𝑚(Λ𝑗)) for each ∈
[,M,𝑠
𝑚,Λ,,⋅...⋅].en,clearlyis a linear
homeomorphism and the proof follows.
Remark 21. If we replace the dierence operator 𝑠
𝑚by (𝑠)
𝑚,
then for each >0we get the following sequence spaces:
,M,(𝑠)
𝑚,Λ,,⋅...⋅𝐼
=
∈(−):
∈N:
−1
𝑘
𝑗∈𝐼𝑘𝑗(𝑠)
𝑚𝑗−
,
1,2,...,𝑛−1𝑝𝑗
≥
∈,
for some >0,∈and each
1,2,...,𝑛−1 ∈
,
,M,(𝑠)
𝑚,Λ,,⋅...⋅𝐼
0
=
∈(−):
∈N:
−1
𝑘
𝑗∈𝐼𝑘𝑗(𝑠)
𝑚𝑗
,
1,2,...,𝑛−1𝑝𝑗
≥
∈,
for some >0and each 1,2,...,𝑛−1 ∈
,
,M,(𝑠)
𝑚,Λ,,⋅...⋅∞
=
∈(−):
sup
𝑘−1
𝑘
𝑗∈𝐼𝑘𝑗(𝑠)
𝑚𝑗
,1,2,...,𝑛−1𝑝𝑗
<∞,
for some >0and each 1,2,...,𝑛−1 ∈
,
,M,(𝑠)
𝑚,Λ,,⋅...⋅𝐼
∞
=
∈(−):∃>0,
s.t.
∈N:−1
𝑘
𝑗∈𝐼𝑘𝑗(𝑠)
𝑚𝑗
,
1,2,...,𝑛−1
𝑝𝑗
≥
∈,
for some >0and each 1,2,...,𝑛−1 ∈
.
(64)
Corollary 22. e sequence spaces [,M,𝑠
𝑚,Λ,,⋅
...⋅],where=
𝐼,𝐼
0,𝐼
∞,and∞, are para-
normed spaces (not totally paranormed) with respect to the
paranorm Δdened by
Δ()=𝑚𝑠
𝑗=1 𝑗,1,2,...,𝑛−1
+inf
𝑝𝑘/𝐻 :
Abstract and Applied Analysis 13
sup
𝑘
∞
𝑗=1𝑘𝑗
×𝑗(𝑠)
𝑚Λ𝑗()
1,
1,2,...,𝑛−1𝑝𝑗
1/𝐻
≤1, for some >0,
and each 1,2,...,𝑛−1 ∈
,(65)
where =max{1,sup𝑘𝑘}and =𝐼,𝐼
0,𝐼
∞,and ∞.
Also it is clear that the paranorms Δand Δare equivalent.
We state the following theorem in view of Lemma 9.
eorem 23. Let be a standard -normed space and
{1,2,...,𝑛}an orthogonal set in .en,thefollowinghold:
(a) [,M,(𝑠)
𝑚,Λ,,⋅...⋅∞]𝐼=[,M,(𝑠)
𝑚,Λ,
,⋅...⋅𝑛−1]𝐼,
(b) [,M,(𝑠)
𝑚,Λ,,⋅...⋅∞]𝐼
0=[,M,(𝑠)
𝑚,Λ,
,⋅...⋅𝑛−1]𝐼
0,
(c) [,M,(𝑠)
𝑚,Λ,,⋅...⋅∞]∞=[,M,(𝑠)
𝑚,Λ,
,⋅...⋅𝑛−1]∞,
(d) [,M,(𝑠)
𝑚,Λ,,⋅...⋅∞]𝐼
∞=[,M,(𝑠)
𝑚,Λ,
,⋅...⋅𝑛−1]𝐼
∞,
where ⋅...⋅∞is the derived (−1)-norm dened with
respect to the set {1,2,...,𝑛}and ⋅...⋅𝑛−1 is the
standard (−1)-norm on .
Acknowledgment
e author is most grateful to the editor and anonymous ref-
eree for careful reading of the paper and valuable suggestions
which helped in improving an earlier version of it.
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