R E S E A R C H
Open Access
On some
A
I
-convergent difference sequence
spaces of fuzzy numbers defined by the
sequence of Orlicz functions
Ekrem Savas
**Correspondence:
[email protected] Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey
Abstract
In this paper, using the difference operator of orderm, the sequences of Orlicz functions, and an infinite matrix, we introduce and examine some classes of sequences of fuzzy numbers defined byI-convergence. We study some basic topological and algebraic properties of these spaces. In addition, we shall establish inclusion theorems between these sequence spaces.
MSC: 40A05; 40G15; 46A45
Keywords: ideal;I-convergent; infinite matrix; Orlicz function; fuzzy number; difference space
1 Introduction
The notion of ideal convergence was introduced first by Kostyrkoet al.[] as a generaliza-tion of statistical convergence [, ], which was further studied in topological spaces []. More applications of ideals can be seen in [–].
The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh []. Subsequently, several authors have discussed various aspects of the theory and applica-tions of fuzzy sets such as fuzzy topological spaces, similarity relaapplica-tions and fuzzy ordering, fuzzy measures of fuzzy events, and fuzzy mathematical programming. In particular, the concept of fuzzy topology has very important applications in quantum particle physics, especially in connection with both string andε∞theory, which were given and studied by El Naschie []. The theory of sequences of fuzzy numbers was first introduced by Matloka []. Matloka introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties, and showed that every convergent sequence of fuzzy numbers is bounded. In [], Nanda studied sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Different classes of sequences of fuzzy real numbers have been discussed by Nuray and Savas [], Altinok, Colak, and Et [], Savas [–], Savas and Mursaleen [], and many others.
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence spacelMcontains a subspace
isomorphic tolp(≤p<∞). The Orlicz sequence spaces are the special cases of Orlicz
spaces studied in []. Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Although the Orlicz sequence spaces are the generalization
oflpspaces, thelp-spaces find themselves enveloped in Orlicz spaces []. Recently, Savas
[] generalizedc() andl∞() for a single sequence of fuzzy numbers by using the Orlicz
function and also established some inclusion theorems.
In the later stage, different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [], Savas [–], and many others.
Throughout the article,wF denotes the class of all fuzzy real-valued sequence spaces.
Also,NandRdenote the set of positive integers and the set of real numbers, respectively. The operator n:w
F →wF is defined by (X)k =Xk; (X)k=Xk =Xk–Xk+;
(nX)
k=nXk=n–Xk–n–Xk+(n≥) for alln∈N. The generalized difference has
the following binomial expression forn≥:
nxk= n
ν=
n
ν
(–)νxk+ν. ()
In this paper, we study some new sequence spaces of fuzzy numbers usingI-convergence, the sequence of Orlicz functions, an infinite matrix, and the difference operator. We es-tablish the inclusion relation between the sequence spaceswI(F)[A,M,m,p],wI(F)[A,M,
m,p]
,wF[A,M,m,p]∞, andwI(F)[A,M,m,p]∞, wherep= (pk) denotes the sequence
of positive real numbers for allk∈NandM= (Mk) is a sequence of Orlicz functions. In
addition, we study some algebraic and topological properties of these new spaces.
2 Definitions and notations
Before continuing with this paper, we present some definitions and preliminaries which we shall use throughout this paper.
LetXandYbe two nonempty subsets of the spacewof complex sequences. LetA= (ank)
(n,k= , , . . .) be an infinite matrix of complex numbers. We writeAx= (An(x)) ifAn(x) =
kankxkconverges for eachn. (Throughout,
kdenotes summation overkfromk= to
k=∞). Ifx= (xk)∈X⇒Ax= (An(x))∈Y, we say thatAdefines a (matrix) transformation
fromXtoYand we denote it byA:X→Y.
LetXbe a nonempty set, then a family of setsI⊂X (the class of all subsets ofX) is
called anidealif and only if for eachA,B∈I, we haveA∪B∈I, and for eachA∈Iand eachB⊂A, we haveB∈I. A nonempty family of setsF⊂Xis afilteron X if and only if
∈/F, for eachA,B∈F, we haveA∩B∈F, and for eachA∈Fand eachA⊂B, we have
B∈F. An idealIis callednon-trivialideal ifI=andX∈/I. Clearly,I⊂Xis a non-trivial
ideal if and only ifF=F(I) ={X–A:A∈I}is a filter onX. A non-trivial idealI⊂Xis
calledadmissibleif and only if{{x}:x∈X} ⊂I. A non-trivial idealIis maximal if there cannot exist any non-trivial idealJ=IcontainingIas a subset. Further details on ideals of Xcan be found in Kostyrkoet al.[].
LetDdenote the set of all closed and bounded intervalsX= [x,x] on the real lineR.
ForX,Y∈D, we defineX≤Yif and only ifx≤yandx≤y,
d(X,Y) =max|x–y|,|x–y|
, whereX= [x,x] andY= [y,y].
twith its grade of membershipX(t). A fuzzy numberXisconvexifX(t)≥X(s)∧X(r) =
min{X(s),X(r)}, wheres<t<r. If there existst∈Rsuch thatX(t) = , then the fuzzy
numberXis callednormal. A fuzzy numberXis said to be upper semicontinuous if for each> ,X–([,a+)) for alla∈[, ] is open in the usual topology inR. LetR(J) denote
the set of all fuzzy numbers which are upper semicontinuous and have compact support,
i.e., ifX∈R(J), then for anyα∈[, ], [X]αis compact, where
[X]α=t∈R:X(t)≥α, ifα∈[, ], [X]= closure oft∈R:X(t) >α, ifα= .
The setRof real numbers can be embedded inR(J) if we definer∈R(J) by
r(t) =
⎧ ⎨ ⎩
, ift=r; , ift=r.
The additive identity and multiplicative identity ofR(J) are defined by and respec-tively.
The arithmetic operations onR(J) are defined as follows: (X⊕Y)(t) =supX(s)∧Y(t–s), t∈R,
(XY)(t) =supX(s)∧Y(s–t), t∈R, (X⊗Y)(t) =sup
X(s)∧Y
t s
, t∈R,
X Y
(t) =supX(st)∧Y(s), t∈R.
LetX,Y∈R(J) and theα-level sets be [X]α= [xα
,xα], [Y]α= [yα,yα],α∈[, ]. Then the
above operations can be defined in terms ofα-level sets as follows: [X⊕Y]α=xα +yα,xα+yα,
[XY]α=xα –yα,xα–yα, [X⊗Y]α=min
i∈{,}x
α iyαi,max
i∈{,}x
α iyαi
,
X–α=xα –,xα –, xαi > , for each <α≤. Forr∈RandX∈R(J), the productrXis defined as follows:
rX(t) =
⎧ ⎨ ⎩
X(r–t), ifr= ;
, ifr= .
The absolute value|X|ofX∈R(J) is defined by
|X|(t) =
⎧ ⎨ ⎩
Define a mappingd¯:R(J)×R(J)→R+∪ {}by ¯
d(X,Y) = sup ≤α≤
d[X]α, [Y]α .
A metric d¯ onR(J) is said to be a translation invariant ifd¯(X+Z,Y+Z) =d¯(X,Y) for
X,Y,Z∈R(J).
Proposition . Ifd is a translation invariant metric on¯ R(J),then
(i) d¯(X+Z, )≤ ¯d(X, ) +d¯(Y, ), (ii) d¯(λX, )≤ |λ|¯d(X, ),|λ|> .
The proof is easy and so it is omitted.
A sequenceX= (Xk) of fuzzy numbers is said toconvergeto a fuzzy numberXif for
every> , there exists a positive integernsuch thatd¯(Xk,X) <for alln≥n.
A sequence X= (Xk) of fuzzy numbers is said to bebounded if the set{Xk:k∈N}
of fuzzy numbers is bounded. A sequence X= (Xk) of fuzzy numbers is said to be I
-convergent to a fuzzy numberXif for each> such that A=k∈N:d¯(Xk,X)≥
∈I.
The fuzzy numberXis calledI-limit of the sequence (Xk) of fuzzy numbers, and we write
I-limXk=X.
A sequenceX= (Xk) of fuzzy numbers is said to beI-bounded if there existsM> such
that
k∈N:d¯(Xk,) >¯ M
∈I.
Example . If we takeI=If={A⊆N:Ais a finite subset}, thenIfis a non-trivial
admis-sible ideal ofN, and the corresponding convergence coincides with the usual convergence.
Example . If we takeI=Iδ={A⊆N:δ(A) = }, whereδ(A) denotes the asymptotic
density of the setA, thenIδ is a non-trivial admissible ideal ofN, and the corresponding
convergence coincides with the statistical convergence.
Lemma .(Kostyrko, Salat, and Wilczynski [], Lemma .) If I⊂N
is a maximal ideal,
then for each A⊂N,we have either A∈I orN–A∈I.
Recall in [] that the Orlicz functionM: [,∞)→[,∞) is a continuous, convex, non-decreasing function such thatM() = andM(x) > forx> , andM(x)→ ∞asx→ ∞. If convexity of the Orlicz function is replaced byM(x+y)≤M(x) +M(y), then this function is called the modulus function and characterized by Ruckle []. An Orlicz functionMis said to satisfy the-condition for all values ofu, if there existsK> such thatM(u)≤ KM(u),u≥.
The following well-known inequality will be used throughout the article. Letp= (pk) be
any sequence of positive real numbers with ≤pk≤supkpk=G,H=max{, G–}, then
|ak+bk|pk≤H
|ak|pk+|bk|pk
3 Some new sequence spaces of fuzzy numbers
In this section, using the sequence of Orlicz functions, an infinite matrix, the difference operatorm, andI-convergence, we introduce the following new sequence spaces and examine some properties of the resulting sequence spaces. LetIbe an admissible ideal ofN, and letp= (pk) be a sequence of positive real numbers for allk∈NandA= (ank)
be an infinite matrix. LetM= (Mk) be a sequence of Orlicz functions andX= (Xk) be a
sequence of fuzzy numbers. We define the following new sequence spaces:
wI(F)A,M,m,p
=
(Xk)∈wF:∀ε> ,
n∈N:
∞
k= ank
Mk
¯
d(mX k,X)
ρ
pk ≥ε
∈I,
for someρ> andX∈R(J)
,
wI(F)A,M,m,p
=
(Xk)∈wF:∀ε> ,
n∈N:
∞
k= ank
Mk
¯
d(mX k,)¯ ρ
pk ≥ε
∈I,
for someρ>
,
wFA,M,m,p∞
=
(Xk)∈wF:sup n
∞
k= ank
Mk
¯
d(mX k,)¯ ρ
pk
<∞, for someρ>
,
and
wI(F)A,M,m,p∞
=
(Xk)∈wF:∃K> s.t.
n∈N:
∞
k= ank
Mk
¯
d(mXk,)¯ ρ
pk ≥K
∈I,
for someρ>
.
Let us consider a few special cases of the above sets.
(i) Ifm= , then the above classes of sequences are denoted bywI(F)[A,M,p], wI(F)[A,M,p]
,wF[A,M,p]∞, andwI(F)[A,M,p]∞, respectively.
(ii) IfMk(x) =xfor allk∈N, then the above classes of sequences are denoted by
wI(F)[A,m,p],wI(F)[A,m,p]
,wF[A,m,p]∞, andwI(F)[A,m,p]∞, respectively.
(iii) Ifp= (pk) = (, , , . . .), then we denote the above spaces bywI(F)[A,M,m],
wI(F)[A,M,m]
,wF[A,M,m]∞, andwI(F)[A,M,m]∞.
(iv) If we takeA= (C, ),i.e., the Cesàro matrix, then the above classes of sequences are denoted bywI(F)[M,m,p],wI(F)[M,m,p]
,wF[M,m,p]∞, and wI(F)[M,m,p]
(v) If we takeA= (ank)is a de la Valée Poussin mean,i.e.,
ank=
⎧ ⎨ ⎩
λn, ifk∈In= [n–λn+ ,n],
, otherwise,
where(λn)is a non-decreasing sequence of positive numbers tending to∞and λn+≤λn+ ,λ= , then the above classes of sequences are denoted by wIλ(F)[A,M,m,p],wIλ(F)[M,m,p],wλF[M,m,p]∞, andw
I(F)
λ [M,m,p]∞, respectively (see []).
(vi) By a lacunaryθ= (kr),r= , , , . . .wherek= , we shall mean an increasing
sequence of non-negative integers withkr–kr–asr→ ∞. The intervals
determined byθwill be denoted byIr= (kr–,kr]andhr=kr–kr–. As a final
illustration, let
ank=
⎧ ⎨ ⎩
hr, ifkr–<k≤kr,
, otherwise.
Then we denote the above classes of sequences bywIθ(F)[M,m,p],
wIθ(F)[M,m,p],wFθ[M,m,p]∞, andw I(F)
θ [M,m,p]∞, respectively. (vii) IfI=If, then we obtain
wFA,M,m,p
=
(Xk)∈wF: lim n→∞
∞
k= ank
Mk
¯
d(mXk,X)
ρ
pk
= ,
for someρ> andX∈R(J)
,
wFA,M,m,p
=
(Xk)∈wF: lim n→∞
∞
k= ank
Mk
¯
d(mXk,)¯ ρ
pk
= ,for someρ>
,
wFA,M,m,p∞
=
(Xk)∈wF:sup n
∞
k= ank
Mk
¯
d(mX k,)¯ ρ
pk
<∞,for someρ>
.
IfX= (Xk)∈wF[A,M,m,p], then we say thatX= (Xk)is stronglyA-convergent with respect to the sequence of Orlicz functionsM.
(viii) IfI=Iδis an admissible ideal ofN, then we obtain
wI(F)A,M,m,p
=
(Xk)∈wF:∀ε> ,
n∈N:
∞
k= ank
Mk
¯
d(mX k,X)
ρ
pk ≥ε
∈Iδ,
for someρ> andX∈R(J)
wI(F)A,M,m,p
=
(Xk)∈wF:∀ε> ,
n∈N:
∞
k= ank
Mk
¯
d(mXk,)¯ ρ
pk ≥ε
∈Iδ,
for someρ>
,
and
wI(F)A,M,m,p∞
=
(Xk)∈wF:∃K>
s.t.
n∈N:
∞
k= ank
Mk
¯
d(mXk,)¯ ρ
pk ≥K
∈Iδ,for someρ>
.
4 Main results
In this section, we examine the basic topological and algebraic properties of the new se-quence spaces and obtain the inclusion relation related to these spaces.
Theorem . Let(pk)be a bounded sequence.Then the sequence spaces wI(F)[A,M,m,p],
wI(F)[A,M,m,p]
,and wI(F)[A,M,m,p]∞are linear spaces.
Proof We will prove the result for the spacewIθ(F)[M,m,p]only and others can be proved
in a similar way.
LetX= (Xk) andY= (Yk) be two elements inwθI(F)[M,m,p]. Then there existρ>
andρ> such that
Aε =
r∈N:
∞
k= ank
Mk
¯
d(mX k,)¯ ρ
pk ≥ε
∈I
and
Bε =
r∈N:
∞
k= ank
Mk
¯
d(mY k,)¯ ρ
pk ≥ε
∈I.
Letα,β be two scalars. By the continuity of the functionM= (Mk), the following
in-equality holds:
∞
k= ank
Mk
¯
d(m(αXk+βYk,))¯
|α|ρ+|β|ρ
pk
≤D ∞
k= ank
|α|
|α|ρ+|β|ρ Mk
¯
d(mXk,)¯ ρ
pk
+D ∞
k= ank
|β|
|α|ρ+|β|ρ Mk
¯
d(mYk,)¯ ρ
≤DK ∞ k= ank Mk ¯
d(mXk,)¯ ρ pk +DK ∞ k= ank Mk ¯
d(mYk,)¯ ρ
pk
,
whereK=max{, (|α|ρ|α|
+|β|ρ)
G, ( |α|
|β|ρ+|β|ρ) G}.
From the above relation, we obtain the following:
n∈N:
∞ k= ank Mk ¯
d(m(αX
k+βYk,))¯
|α|ρ+|β|ρ
pk ≥ε
⊆
n∈N:DK ∞ k= ank Mk ¯
d(mXk,)¯ ρ pk ≥ ε ∪
n∈N:DK ∞ k= ank Mk ¯
d(mYk,)¯ ρ
pk ≥ε
∈I.
This completes the proof.
Theorem . wI(F)[A,M,m,p],wI(F)[A,M,m,p],and wI(F)[A,M,m,p]∞ are linear topological spaces with the paranorm gdefined by
g(X) =inf
ρpnH :
∞ k= ank Mk ¯
d(mX k,)¯ ρ
pk/H ≤,
for someρ> ,n= , , , . . .
,
where H=max{,supkpk}.
Proof Clearly,g(–X) =g(X) andg(θ) = . LetX= (Xk) andY= (Yk) be two elements
inwI(F)[A,M,m,p]
∞. Then for everyρ> , we write
A=
ρ> :
∞ k= ank Mk ¯
d(mX k,)¯ ρ pk H ≤ and
A=
ρ> :
∞ k= ank Mk ¯
d(mXk,)¯ ρ pk H ≤ .
Letρ∈Aandρ∈A. Ifρ=ρ+ρ, then we get the following:
∞ k= ank Mk ¯
d(mXk,)¯ ρ
≤ ρ
ρ+ρ
∞ k= ank Mk ¯
d(mXk,)¯ ρ
+ ρ
ρ+ρ
∞ k= ank Mk ¯
d(mXk,)¯ ρ
Hence, we obtain
∞
k= ank
Mk
¯
d(mXk,)¯ ρ
pk ≤
and
g(x+y) =inf(ρ+ρ) pn
H :ρ∈A,ρ∈A
≤inf(ρ) pn
H :ρ
∈A
+inf(ρ) pn
H :ρ
∈A
=g(x) +g(y). Letum
k →t, whereumk,u∈C, and letg(Xkm–Xk)→ asm→ ∞. To prove thatg(umkXmk –
uXk)→ asm→ ∞, letuk→u, whereuk,u∈Candg(Xkm–Xk)→ asm→ ∞.
We have
A=
ρk> :
∞
k= ank
Mk
¯
d(mXk,)¯ ρk
pk ≤
and
A=
ρk> :
∞
k= ank
Mk
¯
d(mX k,)¯ ρk
pk ≤
.
Ifρk∈Aandρk∈Aand by continuity of the functionM=Mk, we have that
Mk
¯
d(m(umXm
k –uX),)¯
|um–u|ρ k+|u|ρk
≤Mk
¯
d(m(umXkm–uXk),)¯
|um–u|ρ k+|u|ρk
+Mk
¯
d(m(uX
k–uX),)¯
|um–u|ρ k+|u|ρk
≤ |um–u|ρk
|um–u|ρ k+|u|ρk
Mk
¯
d(mXm k,)¯ ρk
+ |u|ρ
k
|um–u|ρ k+|u|ρk
Mk
¯
d(m(Xm
k –Xk),)¯ ρk
.
From the above inequality, it follows that
∞
k= ank
Mk
¯
d(m(umXkm–uX),)¯
|um–u|ρ k+|u|ρk
pk ≤,
and consequently
gumkxk–uxk =infumk –uρk+|u|ρk
pn
G :ρ
k∈A,ρk∈A
≤umk –upnG inf(ρ
k)
pn
G :ρk∈A+|u|pnG infρ
k
pn
G :ρ
k∈A
≤max|u|,|u|pnGgxm
k –xk .
Hence, by our assumption, the right-hand side tends to asm→ ∞. This completes the proof of the theorem.
Theorem . Let I be an admissible ideal andM= (Mk)be a sequence of Orlicz functions.
Then the following hold:
wI(F)[A,M,m–,p]
⊂wI(F)[A,M,m,p];wI(F)[A,M,m–,p]⊂wI(F)[A,M,m,p]; wI(F)[A,M,m–,p]∞⊂wI(F)[A,M,m,p]∞form≥and the inclusions are strict.
In general,for all i= , , , . . . ,m– ,the following hold:
wI(F)[A,M,i,p]
⊂wI(F)[A,M,m,p];wI(F)[A,M,i,p]⊂wI(F)[A,M,m,p]; wI(F)[A,M,i,p]
∞⊂wI(F)[A,M,m,p]∞and the inclusions are strict.
Proof LetX= (Xk) be an element inwI(F)[A,M,m–,p]∞. Then there existsK> , and
for givenε> ,ρ> , we have
n∈N:
∞ k= ank Mk ¯
d(m–X
k,)¯ ρ
pk ≥K
∈I.
SinceM= (Mk) is non-decreasing and convex, it follows that
∞ k= ank Mk ¯
d(mX k,)¯
ρ pk ≤ ∞ k= ank Mk ¯
d(m–X
k+–m–Xk,)¯
ρ pk ≤D ∞ k= ank Mk
¯
d(m–X
k+,)¯
ρ pk +D ∞ k= ank Mk
¯
d(m–X
k,)¯ ρ pk ≤D ∞ k= ank Mk ¯
d(m–X
k+,)¯
ρ pk +D ∞ k= ank Mk ¯
d(m–X
k,)¯ ρ
pk
.
Hence, we have
n∈N:
∞ k= ank Mk ¯
d(mX k,)¯
ρ pk ≥K ⊆
n∈N:D ∞ k= ank Mk ¯
d(m–Xk+,)¯
ρ pk ≥K ∪
n∈N:D ∞ k= ank Mk ¯
d(m–Xk,)¯ ρ
pk ≥K
∈I.
Example . LetMk(x) =x,pk= for all k∈NandA= (C, ), i.e., the Cesàro matrix,
m= . Consider the sequenceX= (Xk) of fuzzy numbers as follows:
Xk(t) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–kt–– , fork– ≤t≤;
–kt+– , for <t≤k+ ;
, otherwise.
Forα∈(, ],α-level sets ofXk,Xk,Xk, andXkare
[Xk]α=
( –α)k– , ( –α)k+ , [Xk]α=
( –α)–k– k– , ( –α)–k– k+ ,
Xk
α
=( –α)(k+ ), ( –α)(k+ ),
Xk
α
=–( –α), ( –α),
respectively. It is easy to see that the sequence [X
k]αis notI-bounded although [Xk]α
isI-bounded.
Theorem . (a)Let <infpk≤pk≤.Then
wI(F)[A,M,m,p]⊆wI(F)[A,M,m];wI(F)[A,M,m,p]⊆wI(F)[A,M,m].
(b)Let≤pk≤suppk<∞.Then
wI(F)[A,M,m]⊆wI(F)[A,M,m,p];wI(F)[A,M,m]
⊆wI(F)[A,M,m,p].
Proof (a) LetX= (Xk) be an element inwI(F)[A,M,m,p]. Since <infpk≤pk≤, we
have
∞
k= ank
Mk
¯
d(mX k,X)
ρ
≤ ∞
k= ank
Mk
¯
d(mX k,X)
ρ
pk
.
Therefore,
n∈N:
∞
k= ank
Mk
¯
d(mXk,X)
ρ
≥ε
⊆
n∈N:
∞
k= ank
Mk
¯
d(mXk,X)
ρ
pk ≥ε
∈I.
The other part can be proved in a similar way.
(b) LetX= (Xk) be an element inwI(F)[A,M,m,p]. Since ≤pk≤suppk<∞, then for
each <ε< , there exists a positive integernsuch that
∞
k= ank
Mk
¯
d(mX k,X)
ρ
≤ε< for alln≥n.
This implies that
∞
k= ank
Mk
¯
d(mXk,X)
ρ
pk ≤
∞
k= ank
Mk
¯
d(mXk,X)
ρ
Therefore, we have
n∈N:
∞
k= ank
Mk
¯
d(mX k,X)
ρ
pk ≥ε
⊆
n∈N:
∞
k= ank
Mk
¯
d(mXk,X)
ρ
≥ε
∈I.
The other part can be proved in a similar way.
The following corollary follows immediately from the above theorem.
Corollary . Let A= (C, ),i.e.,the Cesàro matrix,andM= (Mk)be a sequence of Orlicz
functions.
(a)Let <infpk≤pk≤.Then
wI(F)[M,m,p]⊆wI(F)[M,m];wI(F)[M,m,p]
⊆wI(F)[M,m].
(b)Let≤pk≤suppk<∞.Then
wI(F)[M,m]⊆wI(F)[M,m,p];wI(F)[M,m]
⊆wI(F)[M,m,p].
Competing interests
The author declares that they have no competing interests.
Received: 9 October 2011 Accepted: 9 October 2012 Published: 6 November 2012 References
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