R E S E A R C H
Open Access
The binomial sequence spaces which
include the spaces
p
and
∞
and geometric
properties
Mustafa Cemil Bi¸sgin
**Correspondence:
[email protected] Department of Mathematics, Faculty of Arts and Sciences, Recep Tayyip Erdo ˘gan University, Zihni Derin Campus, Rize, 53100, Turkey
Abstract
In this work, we introduce the binomial sequence spacesbrp,sandbr∞,s which include the spaces
pand
∞, in turn. Moreover, we show that the spacesbpr,sandbr∞,s are BK-spaces and prove that these spaces are linearly isomorphic to the spaces
pand
∞, respectively. Furthermore, we speak of some inclusion relations and give the Schauder basis of the spacebr,s
p. Lastly, we determine the
α-,
β
-, andγ
-duals of thosespaces and give some geometric properties of the spacebr,s p.
MSC: 40C05; 40H05; 46B45
Keywords: matrix transformations; matrix domain; Schauder basis;
α
-,β- and
γ
-duals; matrix classes1 The basic information and notations
The set of all real (or complex) valued sequences is symbolized bywwhich becomes a
vector space under point-wise addition and scalar multiplication. Any vector subspace of
wis called a sequence space. The spaces of all bounded, null, convergent, and absolutely
p-summable sequences are denoted by∞,c,c, andp, respectively, where ≤p<∞.
A Banach sequence space is called aBK-space provided each of the mapspn:X−→C
defined bypn=xnis continuous for alln∈N[]. By considering the notion ofBK-space,
one can say that the sequence spaces∞,c, andcareBK-spaces according to their usual
sup-normdefined byx∞=supk∈N|xk|andp is aBK-space according to itsp-norm
defined by
xp=
∞
k=
|xk|p
p
,
where ≤p<∞.
For an arbitrary infinite matrixA= (ank) of real (or complex) entries andx= (xk)∈w,
theA-transform ofxis defined by
(Ax)n=
∞
k=
ankxk (.)
and is supposed to be convergent for alln∈N[]. In terms of the ease of use, we prefer
that the summation without limits runs from to∞.
Given two sequence spacesXandY, and an infinite matrixA= (ank), the sequence space
XAis defined by
XA=
x= (xk)∈w:Ax∈X
(.)
which is called the domain of an infinite matrixA. Also, by (X:Y), we denote the class of
all matrices such thatX⊂YA. Ifank = fork>nandann= for alln,k∈N, an infinite
matrixA= (ank) is called a triangle. Also, a triangle matrixAuniquely has an inverseA–
which is a triangle matrix.
Let the summation matrixS= (snk) be defined as follows:
snk=
, ≤k≤n,
, k>n
for allk,n∈N. Then the spaces of all bounded and convergent series are defined by means
of the summation matrix such thatbs= (∞)Sandcs=cS, respectively.
The theory of matrix transformation was set in motion by the theory of summability
which was developed by Cesàro, Norlund, Riesz,etc.By taking into account this theory,
many authors have constructed new sequence spaces. For example, (∞)NqandcNqin [],
XpandX∞ in [],arpandar∞in []. Furthermore, many authors have used especially the
Euler matrix for defining new sequence spaces. These areer
andercin [],erp ander∞ in
[] and [],er
(),erc() ander∞() in [],er((m)),erc((m)) ander∞((m)) in [],er(B(m)),
erc(B(m)), ander∞(B(m)) in [],er(,p),erc(,p), ander∞(,p) in [],er(u,p) anderc(u,p)
in [].
In this work, we introduce the binomial sequence spacesbrp,sandbr∞,swhich include the
spacesp and∞, in turn. Moreover, we show that the spacesbpr,sandbr∞,sareBK-spaces
and prove that these spaces are linearly isomorphic to the spacespand∞, respectively.
Furthermore, we speak of some inclusion relations and give the Schauder basis of the space br,s
p . Lastly, we determine theα-,β-, andγ-duals of those spaces and give some
geometric properties of the spacebr,s
p .
2 The binomial sequence spaces which include the spaces
pand
∞
In this part, we define the binomial sequence spacesbr,s
p andbr∞,swhich include the spaces
p and∞, respectively. Furthermore, we show that those spaces areBK-spaces and are
linearly isomorphic to the spacespand∞. Also, we show that the binomial sequence
spacebr,s
p is not a Hilbert space except the casep= , where ≤p<∞.
Letr,sbe nonzero real numbers withr+s= . Then the binomial matrix Br,s= (br,s
nk) is
defined as follows:
brnk,s=
(s+r)n
n
k sn–krk, ≤k≤n,
, k>n
for allk,n∈N. Forsr> , one can easily check that the following properties hold for the
(ii) limn→∞brnk,s= (eachk∈N),
(iii) limn→∞kb r,s nk= .
Thus, the binomial matrix is regular wheneversr> . Here and in the following, unless
stated otherwise, we suppose thatsr> .
By taking into account the binomial matrixBr,s= (br,s
nk), the binomial sequence spaces
br,s
p andbr∞,sare defined by
brp,s=
x= (xk)∈w:
n
(s+r)n
n
k=
n k
sn–krkxk
p
<∞
, ≤p<∞,
and
br,s
∞=
x= (xk)∈w:sup n∈N
(s+r)n
n
k=
n k
sn–krkx k
<∞
.
By considering the notation of (.), the binomial sequence spacesbrp,sandbr∞,scan be
redefined by the matrix domain ofBr,s= (brnk,s) as follows:
brp,s= (p)Br,s and br∞,s= (∞)Br,s. (.)
Let us define a sequencey= (yk) as follows:
Br,sx k=yk=
(s+r)k
k
j=
k j
sk–jrjxj (.)
for allk∈N. This sequence will be frequently used as theBr,s-transform ofx.
We would like to touch on a point, if we takes+r= , we obtain the Euler matrixEr=
(er
nk). So, the binomial matrixBr,s= (b
r,s
nk) generalizes the Euler matrix.
Now, we want to continue with the following theorem which is needed in the next.
Theorem . The binomial sequence spaces br,s
p and br∞,sare BK -spaces according to their
norms defined by
xbr,s p =B
r,sx
p=
∞
n=
Br,sx np
p
and
xbr,s ∞=B
r,sx
∞=sup
n∈N
Br,sx n,
where≤p<∞.
Proof We know that the sequence spacesp and∞ areBK-spaces with theirp-norm
andsup-norm, respectively, where ≤p<∞. Furthermore, (.) holds and the binomial
matrixBr,s= (brnk,s) is a triangle matrix. By taking into account these three facts and
The-orem .. of Wilansky [], we conclude that the binomial sequence spacesbr,s
p andbr∞,s
Theorem . The binomial sequence spaces br,s
p and br∞,sare linearly isomorphic to the
sequence spacespand∞,in turn,where≤p<∞.
Proof To refrain from the usage of similar statements, we prove the theorem for only the
sequence spacebr,s
p , where ≤p<∞. For the proof of the theorem, we need to show the
existence of a linear bijection between the spacesbr,s
p andp. LetLbe a transformation
such thatL:br,s
p −→p,L(x) =Br,sx. By the definition of the binomial sequence spacebrp,s,
we conclude that, for allx∈br,s
p ,L(x) =Br,sx∈p. Furthermore, it is obvious thatLis a
linear transformation andx= wheneverL(x) = . Therefore,Lis injective.
For giveny= (yk)∈p, let us define a sequencex= (xk) such that
xk=
rk k
j=
k j
(–s)k–j(s+r)jyj
for allk∈N. Then we get
xbr,s p =B
r,sx
p
=
∞
n=
Br,sx np
p
=
∞
n=
(s+r)n
n
k=
n k
sn–krkxk
pp
=
∞
n=
(s+r)n
n
k=
n k
sn–k
k
j=
k j
(–s)k–j(s+r)jyj
p
p
=
∞
n=
|yn|p
p
=yp
=L(x)p<∞.
Hence, we conclude thatLis norm preserving andx∈brp,s, namelyLis surjective. As a
consequence,Lis a linear bijection. This means that the spacesbr,s
p andp are linearly
isomorphic, that is,brp,s∼=p, where ≤p<∞. This completes the proof of the theorem.
Theorem . The binomial sequence space br,s
p is not a Hilbert space except the case p= ,
where≤p<∞.
Proof Letp= . Remembering Theorem ., one can say thatbr,sis aBK-space according
to its-norm defined by
xbr,s
=B
r,sx
= ∞
n=
Br,sx n
Moreover, this norm can be generated by an inner product such that
xbr,s=
Br,sx,Br,sx
.
Therefore,br,sis a Hilbert space.
Now, we assume that ≤p<∞andp= . We define two sequencesy= (yk) andz= (zk)
as follows:
yk=
–s+k(r+s)
r
–s
r
k–
and zk= –
s+k(r+s)
r
–s
r
k–
for allk∈N. Then we obtain
y+zbr,s
p +y–z
brp,s= =
p+= y
brp,s+z
brp,s .
Thus, the norm of the binomial sequence space br,s
p does not satisfy the parallelogram
equality. As a consequence, the norm cannot be generated by an inner product, that is,
the binomial sequence spacebr,s
p is not a Hilbert space wheneverp= . This completes the
proof of the theorem.
3 The inclusion relations and Schauder basis
In this part, we speak of some inclusion relations and give the Schauder basis for the
bi-nomial sequence spacebr,s
p , where ≤p<∞.
Theorem . The inclusions er
p⊂brp,sand er∞⊂b∞r,sstrictly hold,where erpand er∞are the
Euler sequence spaces which include the spacespand∞,respectively.
Proof If r+s= , one can easily see that Er =Br,s. Therefore, the inclusioner
∞ ⊂br∞,s
holds. Suppose that <r< ands= . Let us now consider a sequencex= (xk) such that
xk= (–r)kfor allk∈N. Then it is clear thatx= (xk) = ((–r)k) /∈∞,Erx= ((– –r)k) /∈∞
andBr,sx= (( +r)
k)∈
∞. As a result of this,x= (xk)∈b∞r,s\er∞. This shows that the
in-clusioner
∞⊂br∞,sis strictly. We can prove the other part of the theorem by using a similar
technique. This completes the proof of the theorem.
Theorem . The inclusionp⊂bpr,sis strict,where≤p<∞.
Proof First we assume that <p<∞. From the definition of the spacep, we write
k
|xk|p<∞
for allx= (xk)∈p. For given an arbitrary sequencex= (xk)∈p, by taking into account
the equality (.) and the Hölder inequality, we obtain
Br,sx k
p
=
(s+r)k
k
j=
k j
sk–jrjxj
p
≤
|s+r|k
pk
j=
k j
|s|k–j|r|j
p–
×
k
j=
k j
|s|k–j|r|j|xj|p
= |s+r|k
k
j=
k j
|s|k–j|r|j|xj|p
=
k
j=
k j
s+srkrsj|xj|p,
where ≤p<∞. And
k
Br,sx kp≤
k k
j=
k j
s+srkrsj|xj|p
=
j
|xj|p
∞
k=j
k j
s+srkrsj
=s+r
s
j
|xj|p.
If we consider the comparison test, we conclude thatBr,sx∈
p, namelyx∈brp,s. As a
con-sequencep⊂brp,s, where <p<∞.
Now, we keep in view the sequencev= (vk) defined byvk= (–)kfor allk∈N. Then it is
clear thatv= (vk) /∈pandBr,sv= ((ss–+rr)k)∈p, namelyv= (vk)∈brp,s. Because ofv= (vk)∈
brp,s\p, the inclusionp⊂brp,sis strict. In case ofp= , the theorem can be proved by using
a similar method. This completes the proof of the theorem.
Theorem . The spaces br,s
p and∞overlap but these spaces do not include each other,
where≤p<∞.
Proof It is obvious thatv= ((–)k)∈
∞andv= ((–)k)∈bpr,s. So, the spacesbrp,sand∞
overlap, where ≤p<∞. Here, we consider the sequencese= (, , , . . .) andu= (uk)
defined by uk= (–sr)k for all k∈N, where |sr|> . Then we conclude thate∈∞ but
Br,se=e∈/
p, that is,e∈/ bpr,sandu∈/∞ butBr,su= (, , , . . .)∈p, namelyu∈brp,s. As
a consequence,e∈∞\brp,sandu∈bpr,s\∞. On account of this,brp,sand∞ do not
in-clude each other, where ≤p<∞. This completes the proof of the theorem.
Theorem . The inclusions∞⊂br∞,sand brp,s⊂br∞,sare strict,where≤p<∞.
Proof The inequality
xbr,s ∞=sup
k∈N
(s+r)k
k
j=
k j
sk–jrjxj
≤ x∞
holds for allx∈∞. In this way, the inclusion∞⊂br,s
∞holds. Now, we consider the
se-quencev= (vk) defined byvk= (–s+rr)kfor allk∈N. Then we conclude thatv= (vk) /∈∞
butBr,sv= ((–r+rs)k)∈∞, namelyv= (vk)∈br∞,s. Therefore, the inclusion∞⊂br∞,sstrictly holds.
For givenx= (xk)∈bpr,s, where ≤p<∞, by taking into account Theorem . and the
br,s
∞holds. Also, it is clear thate∈br∞,s\bpr,s. Hence, the inclusionbrp,s⊂br∞,sis strict. This
completes the proof of the theorem.
Now, let us continue with the definition of the Schauder basis of a normed space. Let
(X, · X) be a normed sequence space andd= (dk) be a sequence inX. If for everyx∈X,
there exists a unique sequence of scalarsλ= (λk) such that
lim
n→∞
x–
n
k= λkdk
X
=
thend= (dk) is called a Schauder basis forX[].
Theorem . Letμk={Br,sx}k be given for all k∈N.We define the sequence g(k)(r,s) =
{gn(k)(r,s)}n∈Nof the elements of the binomial sequence space brp,sas follows:
gn(k)(r,s) =
, ≤n<k,
rn
n
k (–s)n–k(s+r)k, n≥k
for all fixed k∈N.Then the sequence{g(k)(r,s)}
k∈N is a Schauder basis for the binomial
sequence space br,s
p ,and every x∈brp,shas a unique representation of the form
x=
k
μkg(k)(r,s),
where≤p<∞.
Proof Letx= (xk)∈bpr,sbe given, where ≤p<∞. For all non-negative integerm, we
define
x[m]=
m
k=
μkg(k)(r,s).
Then, if we apply the binomial matrixBr,s= (br,s
nk) tox[m], we write
Br,sx[m]=
m
k=
μkBr,sg(k)(r,s) = m
k=
Br,sx ke(k)
and
Br,sx–x[m] n=
, ≤n≤m,
(Br,sx)n, n>m
for allm,n∈N.
For any given> , there exists a non-negative integermsuch that
∞
n=m+
Br,sx np≤
for allm≥m. Thus,
x–x[m]br,s p =
∞
n=m+
Br,sx np
p
≤
∞
n=m+
Br,sx np
p
≤ <
for allm≥m. This shows us that
x=
k
μkg(k)(r,s).
Lastly, we should show the uniqueness of this representation. For this purpose, assume that
x=
k
λkg(k)(r,s).
Since the linear transformationLdefined frombrp,stop in the proof of Theorem . is
continuous, we have
Br,sx n=
k
λk
Br,sg(k)(r,s)n=
k
λke(nk)=λn
for everyn∈N, which contradicts the fact that (Br,sx)
n=μnfor everyn∈N. Therefore,
everyx∈brp,shas a unique representation. This completes the proof of the theorem.
From Theorem ., we know thatbr,s
p is a Banach space, where ≤p<∞. If we consider
this fact and Theorem ., we can give the next corollary.
Corollary . The binomial sequence space brp,sis separable,where≤p<∞.
4 The
α
- ,β
-, andγ
-dualsIn this part, we determine theα-,β-, andγ-duals of the binomial sequence spacesbr,s
p and
br,s
∞, where ≤p<∞.
Now, we start with a definition. The multiplier space of the sequence spacesXandYis
denoted byM(X,Y) and defined by
M(X,Y) =y= (yk)∈w:xy= (xkyk)∈Yfor allx= (xk)∈X
.
By taking into account the definition of a multiplier space, theα-,β -,and γ-duals of a
sequence spaceXare defined by
Xα=M(X,), Xβ=M(X,cs) and Xγ=M(X,bs),
For use in the next lemma, we now give some properties:
sup
n∈N
k
|ank|q<∞, (.)
sup
n,k∈N
|ank|<∞, (.)
lim
n→∞ank=ak for eachk∈N, (.)
sup
K∈F
k
n∈K
ank
q<∞, (.)
lim
n→∞
k
|ank|=
k
lim
n→∞ank
, (.)
sup
k∈N
n
|ank|<∞, (.)
whereFis the collection of all finite subsets ofN,
p+
q = and <p≤ ∞.
Lemma .(see []) Let A= (ank)be an infinite matrix,then the following hold:
(i) A= (ank)∈(:)⇔(.)holds,
(ii) A= (ank)∈(:c)⇔(.)and(.)hold,
(iii) A= (ank)∈(:∞)⇔(.)holds,
(iv) A= (ank)∈(p:)⇔(.)holds withp+q= and <p≤ ∞,
(v) A= (ank)∈(p:c)⇔(.)and(.)hold with p+q= and <p<∞,
(vi) A= (ank)∈(p:∞)⇔(.)holds withp+
q= and <p<∞,
(vii) A= (ank)∈(∞:c)⇔(.)and(.)hold,
(viii) A= (ank)∈(∞:∞)⇔(.)holds withq= .
Theorem . Let vr,sand vr,sbe defined as follows:
vr,s=
a= (ak)∈w:sup K∈F
k
n∈K
n k
(–s)n–kr–n(r+s)kan
q<∞
and
vr,s=
a= (ak)∈w:sup k∈N
n
nk(–s)n–kr–n(r+s)kan
<∞
.
Then{br,s}α=vr,s
and{brp,s}α=vr,s,where <p≤ ∞.
Proof Leta= (an)∈wbe given. Remembering the sequencex= (xn), which is defined in
the proof of Theorem ., we have
anxn= n
k=
n k
(–s)n–kr–n(r+s)kanyk=
Hr,sy n
for all n∈N. Then, by considering the equality above, we deduce that ax= (anxn)∈
(yk)∈ ory= (yk)∈p, respectively, where <p≤ ∞. This shows us thata= (an)∈
{br,s}α or a= (a
n)∈ {brp,s}α if and only ifHr,s∈( :) orHr,s∈(p :), respectively,
where <p≤ ∞. If we combine these two facts and Lemma .(i) and (iv), we
ob-tain
a= (an)∈
br,sα⇔sup
k∈N
n
nk(–s)n–kr–n(r+s)kan
<∞
or
a= (an)∈
brp,sα⇔sup
K∈F
k
n∈K
n k
(–s)n–kr–n(r+s)kan
q<∞,
respectively, where <p≤ ∞. Therefore,{br,s}α=vr,s
and{brp,s}α=v r,s
, where <p≤ ∞.
This completes the proof of the theorem.
Theorem . Let vr,s,vr,s,vr,s,vr,s,and vr,sbe defined as follows:
vr,s=
a= (ak)∈w:
∞
j=k
j k
(–s)j–kr–j(r+s)kajexists for each k∈N
,
vr,s=
a= (ak)∈w: sup n,k∈N
n
j=k
j k
(–s)j–kr–j(r+s)kaj
<∞
,
vr,s=
a= (ak)∈w: lim n→∞ k n
j=k
j k
(–s)j–kr–j(r+s)kaj
= k ∞
j=k
j k
(–s)j–kr–j(r+s)kaj
,
vr,s=
a= (ak)∈w:sup n∈N n k= n
j=k
j k
(–s)j–kr–j(r+s)kaj
q <∞
, <q<∞,
vr,s=
a= (ak)∈w:sup n∈N n k= n
j=k
j k
(–s)j–kr–j(r+s)kaj
<∞
.
Then the following equalities hold: (I) {br,s}β=vr,s
∩vr,s,
(II) {br,s
p }β=vr,s∩vr,s,where <p<∞,
(III) {br,s
∞}β=vr,s∩vr,s,
(IV) {br,s}γ =vr,s
,
(V) {br,s
p }γ=vr,s,where <p<∞,
(VI) {br,s
∞}γ =vr,s.
Proof To avoid the repetition of similar statements, we give the proof of the theorem for
only the sequence spacebr,s
Leta= (ak)∈wbe given. By considering the sequencex= (xk), which is used in the
proof of Theorem ., we obtain
n
k=
akxk= n
k=
rk k
j=
k j
(–s)k–j(r+s)jyj
ak
=
n
k= n
j=k
j k
(–s)j–kr–j(r+s)kaj
yk
=Gr,sy n
for alln∈N, where the matrixGr,s= (gr,s
nk) is defined by
gnkr,s=
n
j=k
j
k (–s)j–kr–j(r+s)kaj, ≤k≤n,
, k>n
for allk,n∈N. Then:
(II)ax= (akxk)∈cswheneverx= (xk)∈bpr,sif and only ifGr,sy∈cwhenevery= (yk)∈p,
where <p<∞. This fact shows thata= (ak)∈ {bpr,s}β if and only ifGr,s∈(p:c), where
<p<∞. By combining this result and Lemma .(v), we deduce that
sup
n∈N
n
k=
n
j=k
j k
(–s)j–kr–j(r+s)kaj
q
<∞ (.)
and
∞
j=k
j k
(–s)j–kr–j(r+s)ka
j exists for eachk∈N,
where <p<∞and p + q = . As a result of this, we obtain{br,s
p }β=v r,s
∩v
r,s
, where
<p<∞.
(V) By following a similar way,ax= (akxk)∈bswheneverx= (xk)∈brp,sif and only if
Gr,sy∈
∞whenevery= (yk)∈p, where <p<∞. This says us thata= (ak)∈ {brp,s}γ if
and only ifGr,s∈(p:∞), where <p<∞. By using this result and Lemma .(vi), we
conclude that (.) holds, where <p<∞and p +q = . As a consequence of this, we
obtain{br,s
p }γ =vr,s, where <p<∞. This completes the proof of the theorem.
5 Geometric properties of the binomial sequence spacebr,s p
In this part, we give some geometric properties of the binomial sequence spacebr,s
p . Let us
start with some notions.
Let (X, · X) be a Banach space. ThenXis said to have the Banach-Saks property, if
every bounded sequenceu= (un) contains a subsequencev= (vn) such that the Cesàro
meansn+nk=vkare norm convergent [].
Xis said to have the weak Banach-Saks property, if every weakly null sequenceu= (un)
contains a subsequencev= (vn) such that the Cesàro meansn+
n
k=vkare norm
Xis said to have Banach-Saks typep, if every weakly null sequenceu= (un) has a
subse-quencev= (vn) such that, for someM> ,
n
k=
vk
X
≤M(n+ )p
for alln∈N, where <p<∞[].
LetCbe a weakly compact convex subset ofX. ThenXis said to have the weak fixed
point property, if every self mappingT:C−→Cthat providesTx–Ty ≤ x–yfor all
x,y∈Chas a fixed point [].
LetXbe a normed linear space andS(X) be a unit sphere ofX. Then the Gurarii modulus
of convexity is defined as follows:
βX() =inf
– inf
≤λ≤
λx+ ( –λ)y:x,y∈S(X),x–y=
,
where ≤≤ [].
Theorem .(see []) A Banach space X has the weak fixed point property,if X provides the condition
R(X) =sup
lim inf
n→∞ xn+x
< ,
where the supremum is taken over all weakly null sequences(xn)of the unit ball and all
points x of the unit ball.
Theorem . The binomial sequence space br,s
p is of the Banach-Saks type p.
Proof Let (un) be a weakly null sequence in theB(bpr,s) unit ball ofbrp,s. We suppose that
(n) is a sequence of positive numbers provided
n≤. Constructv=u= andv=
un=u. Then we can find anm∈Nsuch that
∞
i=m+
v(i)e(i)
brp,s <.
By virtue ofun
w
−→ implyingun−→ coordinatewise, we can find ann∈Nsuch that
m
i=
un(i)e(i)
brp,s <,
asn≥n. Constructv=un. Then we can find anm>msuch that
∞
i=m+
v(i)e(i)
brp,s <.
If we usexn−→ coordinatewise one more time, we can find ann>nsuch that
m
i=
un(i)e(i)
brp,s <,
By continuing this method, we can constitute two increasing sequences (mk) and (nk) such that mk i=
un(i)e(i)
brp,s <k
for alln≥nk+and
∞
i=mk+ v(i)e(i)
brp,s <k,
wherevk=unk. Thus
n k= vk
brp,s = n k= mk–
i=
vk(i)e(i)+ mk
i=mk–+
vk(i)e(i)+
∞
i=mk+ vk(i)e(i)
brp,s
≤ n k= mk
i=mk–+
vk(i)e(i)
brp,s + n k= k and n k= mk
i=mk–+
vk(i)e(i)
p
brp,s = n k= mk
i=mk–+
(s+r)i
i j= i j
si–jrjvk(j)
p ≤ n k= ∞ i= (s+r)i
i j= i j
si–jrjvk(j)
p
≤n+ .
Thus we obtain
n k= vk
brp,s
≤(n+ )p+ ≤(n+ )p.
As a consequence, the binomial sequence spacebr,s
p is of the Banach-Saks typep. This
completes the proof of the theorem.
We know from Theorem . thatbrp,s is linearly isomorphic top. So, it is clear that
R(br,s
p ) =R(p) =
p.
By combining this fact and Theorem ., we can give the next theorem.
Theorem . The binomial sequence space brp,shas the weak fixed point property,where
<p<∞.
Theorem . The inequalityβbr,s
p()≤ – [ – (
)p]
pholds,where≤≤.
Proof Let ≤≤ be given. By assuming the inverse of the binomial matrixBr,sisD, we
construct two sequencesuandvas follows:
u= D –
pp
v=
D
–
pp
,D
–
, , , . . .
.
Then we obtain
Br,sup=ubrp,s= and B
r,sv
p=vbrp,s= .
This shows thatu,v∈S(br,s
p ) andBr,su–Br,svp=u–vbrp,s=.
For ≤λ≤, we have
λu+ ( –λ)vpbr,s p =λB
r,su+ ( –λ)Br,svp
p
= –
p
+|λ– |
p
and
inf
≤λ≤λu+ ( –λ)v
p brp,s= –
p
. (.)
Thus, we obtain
βbr,s
p()≤ –
–
pp
.
This completes the proof of the theorem.
By using the equality (.), we find two more results.
Corollary . Sinceβbr,s
p() = ,the binomial sequence space b
r,s
p is strictly convex.
Corollary . Since <βbr,s
p()≤,for <≤,the binomial sequence space b
r,s p is
uni-formly convex.
6 Conclusion
By taking into account the binomial matrixBr,s= (br,s
nk), we conclude thatBr,s= (b
r,s nk)
re-duces in the caser+s= toEr= (er
nk) which is called the Euler matrix of orderr. Therefore,
our results obtained from the matrix domain of the binomial matrixBr,s= (brnk,s) are more
general and more extensive than the results on the matrix domain of the Euler matrix of
orderr. Furthermore, the binomial matrixBr,s= (br,s
nk) is not a special case of the weighed
mean matrices. Thus, this paper has filled up a gap in the existent literature.
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
I would like to express my thanks to the anonymous reviewers for their valuable comments.
References
1. Choudhary, B, Nanda, S: Functional Analysis with Applications. Wiley, New Delhi (1989)
2. Wilansky, A: Summability Through Functional Analysis. North-Holland Mathematics Studies, vol. 85. Elsevier, Amsterdam (1984)
3. Wang, C-S: On Nörlund sequence spaces. Tamkang J. Math.9, 269-274 (1978)
4. Ng, P-N, Lee, P-Y: Cesàro sequence spaces of non-absolute type. Comment. Math. Prace Mat.20(2), 429-433 (1978) 5. Aydın, C, Ba¸sar, F: Some new sequence spaces which include the spacespand∞. Demonstr. Math.38(3), 641-656
(2005)
6. Altay, B, Ba¸sar, F: Some Euler sequence spaces of non-absolute type. Ukr. Math. J.57(1), 1-17 (2005)
7. Altay, B, Ba¸sar, F, Mursaleen, M: On the Euler sequence spaces which include the spacespand∞. I. Inf. Sci.176(10),
1450-1462 (2006)
8. Mursaleen, M, Ba¸sar, F, Altay, B: On the Euler sequence spaces which include the spacespand∞. II. Nonlinear Anal.
65(3), 707-717 (2006)
9. Altay, B, Polat, H: On some new Euler difference sequence spaces. Southeast Asian Bull. Math.30(2), 209-220 (2006) 10. Polat, H, Ba¸sar, F: Some Euler spaces of difference sequences of orderm. Acta Math. Sci. Ser. B Engl. Ed.27(2), 254-266
(2007)
11. Kara, EE, Ba¸sarır, M: On compact operators and some EulerB(m)-difference sequence spaces. J. Math. Anal. Appl. 379(2), 499-511 (2011)
12. Karakaya, V, Polat, H: Some new paranormed sequence spaces defined by Euler difference operators. Acta Sci. Math. 76, 87-100 (2010)
13. Demiriz, S, Çakan, C: On some new paranormed Euler sequence spaces and Euler core. Acta Math. Sin. Engl. Ser. 26(7), 1207-1222 (2010)
14. Stieglitz, M, Tietz, H: Matrix transformationen von folgenräumen eine ergebnisübersicht. Math. Z.154, 1-16 (1977) 15. Beauzamy, B: Banach-Saks properties and spreading models. Math. Scand.44, 357-384 (1997)
16. Knaust, H: Orlicz sequence spaces of Banach-Saks type. Arch. Math.59, 562-565 (1992)
17. Garcia-Falset, J: Stability and fixed points for nonexpansive mappings. Houst. J. Math.20, 495-505 (1994) 18. Sánchez, L, Ullán, A: Some properties of Gurarii’s modulus of continuity. Arch. Math.71, 399-406 (1998) 19. Garcia-Falset, J: The fixed point property in Banach spaces with NUS-property. J. Math. Anal. Appl.215(2), 532-542