R E S E A R C H
Open Access
Some generalized difference statistically
convergent sequence spaces in
-normed
space
Metin Ba¸sarır
1, ¸Sükran Konca
1*and Emrah Evren Kara
2*Correspondence:
1Department of Mathematics,
Sakarya University, Sakarya, 54187, Turkey
Full list of author information is available at the end of the article
Abstract
In this paper, we define a new generalized difference matrixBn
(m)and introduce some Bn
(m)-difference statistically convergent sequence spaces in a real linear 2-normed space. We also investigate some topological properties of these spaces.
MSC: Primary 40A05; secondary 46A45; 46E30
Keywords: statistical convergence; generalized difference sequence space; 2-norm; paranorm; completeness; solidity
1 Introduction
We shall writewfor the set of all real sequencesx= (xk) = (xk)∞k=. Letc,c,c,c,l∞,m
andmdenote the sets of all convergent, null, statistically convergent, statistically null,
bounded, bounded statistically convergent and bounded statistically null sequences, re-spectively. The difference sequence spacesl∞(),c() andc() were first defined by
Kızmaz in []. The idea of difference sequences is generalized by Et and Çolak [] as
Zn=x= (xk)∈w:
nxk
∈Z (n∈N)
forZ=l∞,c,c, wherenxk=n–xk–n–xk+andxk=xkfor allk∈N, the difference operator is equivalent to the following binomial representation:
nxk= n
v=
(–)v
n v
xk+v.
Et and Başarır [] generalized these spaces to E(n), where E =l
∞(p),c(p),c(p) are
Maddox’s sequence spaces. Tripathy and Esi [], who studied the spacesl∞(m),c(m) andc(m), gave a new type of generalization of the difference sequence spaces, where mx= (mxk) = (xk–xk+m). Tripathyet al.[] generalized this notion as follows:
Znm=x= (xk)∈w:
nmxk
∈Z (n,m∈N),
wherenmx= (nmxk) = (nm–xk–mn–xk+m) andmxk=xkfor allk∈N, which is equiva-lent to the following binomial representation:
nmxk= n
v=
(–)v
n v
xk+mv.
The difference sequence spaces have been studied by several authors, [, –].
The concept of -normed spaces has been initially introduced by Gähler in the s [], as an interesting non-linear generalization of a normed linear space, which has been subsequently studied by many authors [–]. Since then, a lot of activities have been started to study summability, sequence spaces and related topics on -normed spaces [– ]. Recently, some difference sequence spaces have been introduced in -normed spaces by several authors [, , ].
Dutta [] introduced the sequence spacesc(·,·,(nm),p), c(·,·,n(m),p),l∞(·,·,
n(m),p), m(·,·,n(m),p) andm(·,·,n(m),p), wherem,n∈Nandn(m)x= (n(m)xk) = (n(m–)xk–n(m–)xk–m), and(m)xk=xkfor allk∈N, which is equivalent to the following binomial representation:
n(m)xk= n
v=
(–)v
n v
xk–mv. (.)
In [], Başar and Altay introduced the generalized difference matrixB(r,s) = (bnk(r,s)) which is a generalization of
()-difference operator as follows:
bnk(r,s) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
r (k=n),
s (k=n– ),
(≤k<n– ) or (k>n)
for allk,n∈N,r,s∈R–{}. Recently, Başarır and Kayıkçı [] have defined the general-ized difference matrixBnof ordern, which reduced the difference operatorn
() in case
r= ,s= – and the binomial representation of this operator is
Bnxk= n
v=
n v
rn–vsvxk–v, (.)
wherer,s∈R–{}andn∈N.
Thus, for any sequence spaceZ, the spaceZ(Bn) is more general and more comprehen-sive than the corresponding consequences of the spaceZ(n()). For details, one may refer to [, , –].
The idea of statistical convergence was given by Zygmund [] in . The concept of statistical convergence was introduced by Fast [] and Schoenberg [], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [] and generalized to the concept of -normed space by Gürdal and Pehlivan []. The idea is based on the notion of natural density of subsets ofN, the set of positive integers, which is defined as follows: the natural density of a subsetEofNis denoted by
δ(E) = lim
n→∞
n{k∈E:k≤n},
2 Definitions and preliminaries
A sequence spaceE is said to be solid (or normal) if (xk)∈E implies (αkxk)∈E for all sequences of scalars (αk) with|αk| ≤ for allk∈N.
A linear topological spaceXover the real fieldRis said to be a paranormed space if there is a sub-additive functiong:X→Rsuch thatg(θ) = ,g(x) =g(–x),g(x+y)≤g(x) +g(y) and scalar multiplication is continuous,i.e.|λn–λ| → andg(xn–x)→ imply that
g(λnxn–λx)→ for allλ’s inRand allx’s inX, whereθ is the zero vector in the linear spaceX.
The following inequality will be used throughout the paper:
Letp= (pk) be a positive sequence of real numbers withinfkpk=h,supkpk =H and
D=max{, H–}. Then for alla
k,bk∈Cfor allk∈N, we have |ak+bk|pk≤D
|ak|pk+|bk|pk
and|λ|pk ≤max{|λ|h,|λ|H}forλ∈C.
A -norm on a vector spaceXofddimension, whered≥, is a function·,·:X×X→
R, which satisfies the following conditions:
() x,x= if and only ifx,xare linearly dependent,
() x,x=x,x,
() αx,x=|α|x,xfor anyα∈R,
() x+x,x ≤ x,x+x,x.
The pair (X,·,·) is then called a -normed space. For example, standard and Euclidean -norms onRare respectively given by
x,xS=
x,x x,x
x,x x,x
and
x,xE=abs
x x
x x
, xi= (xi,xi)∈R(i= , ), (.)
where ·,·stands for the inner product onX[].
Now we will give the following known example for -normed spaces.
Example . Consider the spaceZforl∞,candc. Let us define:
x,y=sup
i∈N
sup
j∈N
|xiyj–xjyi|,
wherex= (x,x, . . .) andy= (y,y, . . .)∈Z. Then·,·is a -norm onZ.
A sequence (xk) in a -normed space (X,·,·) is said to be convergent to someL∈Xin the -norm if
lim
A sequence (xk) in a -normed space (X,·,·) is said to be Cauchy sequence with respect to the -norm if
lim
k,l→∞xk–xl,z= for everyz∈X[].
If every Cauchy sequence inXconverges to someL∈X, thenXis said to be complete with respect to the -norm. Any complete -normed space is said to be a -Banach space [].
Let recall that a sequence (xk) is said to be statistically convergent toLif for everyε> the set{k∈N:xk–L,z ≥ε}has natural density zero for each nonzerozinX, in other words (xk) statistically converges toLin -normed space (X,·,·) if
lim
k→∞
kk∈N:xk–L,z ≥ε= ,
for each nonzerozinX. ForL= , we say this is statistically null [].
Firstly, we give the following lemma, which we need to establish our main results.
Lemma .[] Every closed linear subspace F of an arbitrary linear normed space E,
different from E,is a nowhere dense set in E.
Throughout the paperw(X),c(X),c(X),c(X),c(X),l∞(X),m(X) andm(X) denote the
spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null Xvalued sequence spaces, where (X,·,·) is a real -normed space. Byθ= (θ,θ,θ, . . .), we mean the zero element ofX.
3 Main results
In this section, we define the generalized difference matrix Bn
(m) and introduce
differ-ence sequdiffer-ence spaces c(Bn(m),p,·,·),c(Bn(m),p,·,·), m(Bn(m),p,·,·),m(Bn(m),p,·,·),
c(Bn(m),p,·,·), c(Bn(m),p,·,·), l∞(Bn(m),p,·,·), W(Bn(m),p,·,·), which are defined on
a real linear -normed space. We investigate some topological properties of the spaces
c(Bn(m),p,·,·), c(Bn(m),p,·,·), m(Bn(m),p,·,·) and m(Bn(m),p,·,·) including
linear-ity, existence of paranorm and solidity. Further, we show that the sequence spaces
m(Bn
(m),p,·,·) and m(Bn(m),p,·,·) are complete paranormed spaces when the base
space is a -Banach space. Moreover, we give some inclusion relations. By the notationxk
stat
→, we will mean thatxkis statistically convergent to zero, through-out the paper. Letm,nbe non-negative integers and p= (pk) be a sequence of strictly positive real numbers. Then we define new sequence spaces as follows:
cBn(m),p,·,·=x= (xk)∈w(X) :Bn(m)xk–L,zpk
stat
→,
for every nonzeroz∈Xand someL∈X,
c
Bn(m),p,·,·=x= (xk)∈w(X) :B(nm)xk,zpk
stat
→, for every nonzeroz∈X,
l∞Bn(m),p,·,·=
x= (xk)∈w(X) :sup k≥
Bn
(m)xk,z pk
<∞,
for every nonzeroz∈X
cBn(m),p,·,·=
x= (xk) : lim k→∞
Bn(m)xk–L,z pk
= ,
for every nonzeroz∈Xand someL∈X
,
c
Bn(m),p,·,·=
x= (xk) : lim k→∞B
n
(m)xk,z pk
= , for every nonzeroz∈X
,
WBn(m),p,·,·=
x= (xk)∈w(X) :lim j→∞
j
j
k= Bn
(m)xk–L,z pk
= ,
for every nonzeroz∈Xand someL∈X
,
mBn(m),p,·,·=cBn(m),p,·,·∩l∞Bn(m),p,·,·
and
m
Bn(m),p,·,·=c
Bn(m),p,·,·∩l∞Bn(m),p,·,·
,
whereBn
(m)x=Bn(m)xk=rBn(m–)xk+sBn(m–)xk–mandB(m)xk=xkfor allk∈N, which is equivalent to the binomial representation as follows:
Bn(m)xk= n
v=
n v
rn–vsvxk–mv.
In this representation, we obtain the matrixBn
()defined in [] forn> and in [] for
n= .
() If we taken= then the above sequence spaces are reduced toc(p,·,·),
c(p,·,·),l∞(p,·,·),c(p,·,·),c(p,·,·),W(p,·,·),m(p,·,·)and
m(p,·,·), respectively.
() If we taker= ,s= –, then the sequence spacesc(Bn(m),p,·,·),c(Bn(m),p,·,·),
l∞(Bn(m),p,·,·),W(Bn(m),p,·,·),m(Bn(m),p,·,·),m(Bn(m),p,·,·)are reduced to
c(n(m),p,·,·),c(n(m),p,·,·),l∞(n(m),p,·,·),W(n(m),p,·,·),
m(n(m),p,·,·)andm(n(m),p,·,·), respectively, which are studied in [].
() By takingpk= for allk∈N, then these sequence spaces are denoted by
c(Bn(m),·,·),c(Bn(m),·,·),l∞(Bn(m),·,·),c(B
n
(m),·,·),c(B
n
(m),·,·),
W(Bn
(m),·,·),m(Bn(m),·,·)andm(Bn(m),·,·), respectively.
() If we replace the base spaceX, which is a real linear-normed space byC, complete normed linear space, and takem= and taker= ,s= –, then the above sequence spaces are denoted byc(n(),p),c(n(),p),l∞(n(),p),c(n(),p),c(n(),p),
W(n(),p),m(n(),p)andm(n(),p), respectively.
() If we taker= ,s= –,pk= for allk∈N, then these sequence spaces are denoted byc(n(m),·,·),c(n(m),·,·),l∞(n(m),·,·),c(n(m),·,·),c(n(m),·,·),
W(n(m),·,·),m(n(m),·,·)andm((nm),p,·,·), respectively.
() If we replace the base spaceX, which is a real linear-normed space byC, we obtain the spacesc(Bn(m),p),c(B(nm),p),l∞(Bn(m),p),c(Bn(m),p),c(B(nm),p),W(Bn(m),p),
m(Bn(m),p)andm(Bn(m),p), respectively.
() Moreover, if we takeX=C,n= andpk= for allk∈N, we get the spacesc,c,l∞,
Theorem . Let p= (pk)be a sequence of strictly positive real numbers.Then the sequence
spaces Z(Bn
(m),p,·,·)are linear spaces where Z=c,c,l∞,W,m,m.
Proof The proof of the theorem can be obtained by similar techniques in [].
Theorem . For any two sequences p= (pk)and t= (tk)of positive real numbers and for
any two-norms·,·and·,· on X we have Z(Bn(m),p,·,·)∩Z(Bn(m),p,·,·)=∅,
where Z=c,c,m,m.
Proof The proof follows from the fact that the zero element belongs to each of the
se-quence spaces involved in the intersection.
Theorem . Let (X,·,·) be a -Banach space. Then the spaces m(Bn
(m),p,·,·),
m(Bn(m),p,·,·)are complete paranormed sequence spaces,paranormed by
g(x) = sup
k∈N,z∈X
Bn(m)xk,z pk
M, (.)
where M=max{,H}and H=supkpk,h=infkpk.
Proof We will prove the theorem for the sequence spacem(Bn(m),p,·,·). It can be proved
for the spacem(Bn
(m),p,·,·) similarly.
Clearlyg(–x) =g(x) andg(θ) = . From the following inequality, we have
g(x+y) = sup
k∈N,z∈X
Bn
(m)(xk+yk),z pk
M
≤ sup
k∈N,z∈X
Bn
(m)xk,z pk
M+ sup
k∈N,z∈X
Bn
(m)yk,z pk M.
This implies thatg(x+y)≤g(x) +g(y).
To prove the continuity of scalar multiplication, assume that (xn) be any sequence of the points inm(Bn(m),p,·,·) such thatg(xn–x)→ and (λn) be any sequence of scalars such thatλn→λ. Since the inequality
gxn≤g(x) +gxn–x
holds by subadditivity ofg, (g(xn)) is bounded. Thus, we have
gλnxn–λx
= sup
k∈N,z∈X
Bn(m)λnxnk–λxk,z pk
M
≤max|λn–λ|h,|λn–λ|H
M sup
k∈N,z∈X
Bn(m)xk,z pk
M
+max|λ|h,|λ|H
M sup
k∈N,z∈X
Bn
(m)
xn k–x
,zpkM
=max|λn–λ|h,|λn–λ|H
Mgxn +max|λ|h,|λ|H
which tends to zero asn→ ∞. Hence,gis a paranorm on the sequence spacem(Bn(m),p,
·,·).
To prove thatm(Bn(m),p,·,·) is complete, assume that (xi) is a Cauchy sequence in
m(Bn(m),p,·,·). Then for a givenε( <ε< ), there exists a positive integerNsuch that
g(xi–xj) <ε, for alli,j≥N
. This implies that
sup
k∈N,z∈X
Bn
(m)xik–Bn(m)x
j k,z
pk M<ε,
for alli,j≥N. It follows that for every nonzeroz∈X, Bn(m)xik–B(nm)xjk,z<ε,
for each k ≥ and for all i,j≥N. Hence (Bn(m)xik) is a Cauchy sequence inX for all
k∈N. SinceXis a -Banach space, (Bn
(m)xik) is convergent inXfor allk∈N, so we write (Bn(m)xik)→(Bn(m)xk) asi→ ∞. Now we have for alli,j≥N,
sup
k∈N,z∈X
Bn
(m)
xik–xjk,zpkM<ε
⇒ lim
j→∞
sup
k∈N,z∈X
Bn
(m)
xik–xjk,zpkM<ε
⇒ sup
k∈N,z∈X
Bn
(m)
xik–xk
,zpkM<ε,
for alli≥N. It follows that (xi–x)∈m(Bn(m),p,·,·). Since (xi)∈m(Bn(m),p,·,·) and
m(Bn(m),p,·,·) is a linear space, so we havex=xi– (xi–x)∈m(B(nm),p,·,·). This
com-pletes the proof.
Theorem .
() IfZ⊂Z,thenZ(Bn(m),p,·,·)⊂Z(Bn(m),p,·,·)and the inclusion is strict,where
Z,Z=c,c,l∞.
() Ifn<n,thenZ(Bn(m),p,·,·)⊂Z(Bn(m),p,·,·)and the inclusion is strict,where
Z=c,c,l∞.
Proof The parts of proof Z(Bn(m),p,·,·) ⊂ Z(Bn(m),p,·,·) and Z(Bn(m),p,·,·) ⊂
Z(Bn(m),p,·,·) are easy. To show the inclusions are strict, choose Z =c, Z =c,
x= (xk) = (k,k) and consider the -norm as defined in (.), letpk = for allk∈N,
m= ,n= ,r= ,s= –, thenx∈c(B
(),·,·) butx∈/c(B(),·,·). If we chooseZ=c,
x= (xk) = (k,k) andpk= for allk∈N,m= ,n= ,r= ,s= –, thenx∈c(B(),·,·) butx∈/c(B
(),·,·). These complete the proofs of parts () and () of the theorem,
respec-tively.
Theorem .
() c(Bn(m),·,·)⊂c(Bn(m),·,·)and the inclusion is strict. () c(·,·)⊂c(Bn
(m),·,·)and the inclusion is strict.
Proof
() It is clear thatc(Bn(m),·,·)⊂c(Bn(m),·,·). To show that the inclusion is strict, choose the sequencex= (xk)such that,
Bn(m)xk=
⎧ ⎨ ⎩
(,√k), k=n,
(, ), k=n, (.)
wheren∈N–{}, and consider the-norm as defined in(.). Then we obtain
Bn
(m)xk∈c(·,·), butBn(m)xk∈/c(·,·). That is,xk∈c(Bn(m),p,·,·), but
xk∈/c(Bn(m),p,·,·).
() It is easy to see thatc(·,·)⊂c(Bn
(m),·,·). To show that the inclusion is strict, let
us takex= (xk) = (k,k)and consider the-norm as defined in(.),m= ,n= ,
r= ,s= –, thenx∈c(B
(),·,·)butx∈/c(·,·).
() Since the sequencex=θbelongs to each of the sequence spaces, the overlapping part of the proof is obvious. For the other part of the proof, consider the sequence defined by(.)and the-norm as defined in(.). Thenx∈c(Bn(m),·,·), but
x∈/l∞(Bn(m),·,·). Conversely if we choose(Bn(m)xk) = (, , , , . . .)wherek= (k,k) for allk= , , thenBn(m)xk∈l∞(·,·)butBn(m)xk∈/c(·,·). That is,
x∈l∞(Bn(m),·,·)butx∈/c(Bn(m),·,·).
Theorem . The space Z(Bn
(m),p,·,·)is not solid in general,where Z=c,c,m,m.
Proof To show that the space is not solid in general, consider the following examples.
Example . Letm= ,n= ,r= ,s= – and consider the -normed space as defined
in Example .. Letpk= for allk∈N. Consider the sequence (xk), where xk= (xik) is defined by (xi
k) = (k,k,k, . . .) for each fixedk∈N. Thenxk∈Z(B(),p,·,·) forZ=c,m. Letαk= (–)k, then (αkxk) /∈Z(B(),p,·,·) forZ=c,m. ThusZ(B(),p,·,·) forZ=c,m is not solid in general.
Example . Let m= , n= ,r= ,s= – and consider the -normed space as
de-fined in Example .. Letpk= for all oddkandpk= for all evenk. Consider the se-quence (xk), wherexk= (xik) is defined by (xik) = (, , , . . .) for each fixedk∈N. Thenxk∈
Z(B(),p,·,·) forZ=c,m. Letαk= (–)k, then (αkxk) /∈Z(B(),p,·,·) forZ=c,m.
ThusZ(B(),p,·,·) forZ=c,mis not solid in general.
Theorem . The spaces m(Bn(m),p,·,·)and m(Bn(m),p,·,·)are nowhere dense subsets
of l∞(Bn(m),p,·,·).
Proof From Theorem ., it follows thatm(Bn(m),p,·,·) andm(Bn(m),p,·,·) are closed
subspaces ofl∞(Bn(m),p,·,·). Since the inclusion relations
m
Bn(m),p,·,·⊂l∞Bn(m),p,·,·
, mBn(m),p,·,·⊂l∞Bn(m),p,·,·
are strict, the spacesm(Bn(m),p,·,·) andm(Bn(m),p,·,·) are nowhere dense subsets of
Theorem . Let p= (pk)be a sequence of non-negative bounded real numbers such that
infkpk> .Then
WBn(m),p,·,·∩l∞Bn(m),p,·,·
⊂mBn(m),p,·,·.
Proof Let (xk)∈W(Bn(m),p,·,·)∩l∞(Bn(m),p,·,·). Then for a givenε> , we have
j
j
k=
Bn(m)xk–L,z pk ≥
j
j
k= Bn(m)xk–L,zpk≥ε
Bn(m)xk–L,z pk
≥ε
jk≤j:B
n
(m)xk–L,zpk≥ε.
If we take the limit for j→ ∞, it follows that (xk)∈c(Bn(m),p,·,·) from the inequality
above. Since (xk)∈l∞(B(nm),p,·,·), we have the result.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in the preparation of this article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Sakarya University, Sakarya, 54187, Turkey.2Department of Mathematics, Bilecik ¸Seyh
Edebali University, Bilecik, 11200, Turkey.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the anonymous reviewers for their comments and suggestions to improve the quality of the paper.
Received: 12 December 2012 Accepted: 2 April 2013 Published: 16 April 2013 References
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