Generalized C
λ
-Rate Sequence Spaces of Difference
Sequence Defined by a Modulus Function in a Locally
Convex Space
B. ¨
Ozaltın
,
˙I. Da˘gadur
∗Department of Mathematics, Faculty of Science and Literature, Mersin University, 33343, Mersin, Turkey
Copyright c⃝2015 Horizon Research Publishing All rights reserved.
Abstract
The idea of difference sequence spaces was introduced by Kızmaz[14]and this concept was generalized by Et and C¸ olak[6].Recently the difference sequence spaces have been studied in (see, [3],[7],[17],[18]). The purpose of this article is to introduce the sequence spacesCcλ0π(∆m, f, p, q), Ccλπ(∆m, f, p, q)andC(λℓ∞) (∆
m, f, p, q)using a
modulusfunctionfandmoregeneralCλ−methodinvievofArmitageandMaddox[2].Several
π
propertiesof these spaces, andsomeinclusionrelationshavebeenexamined.
Keywords
FK-spaces, Modulus Function, Rate Sequence Space, Difference Sequence,Cλ−Summability Method2000 Mathematics Subject Classification:
40C05, 40D25, 40G05, 42A05, 42A101
Introduction
The notion of a modulus was introduced by Nakano[19]. We recall that a modulusf is a function from[0,∞)to[0,∞)such that
i)f(x) = 0if and only ifx= 0,
ii)f(x+y)≤f(x) +f(y)forx, y≥0, iii)fis increasing,
iv)f is continuous from the right at0.
It follows thatf must be continuous everywhere on[0,∞). Maddox[15]and Ruckle[22]used a modulus function to construct some sequence spaces. Later on using a modulus different sequence spaces have been studied by Altın and Et[1], Et[5], Nuray and Savas[20], Tripathy and Chandra[26]and many others.
The notion of difference sequence spaces was introduced by Kızmaz[14]and the notion was generalized by Et and C¸ olak [6]. Recently the difference sequence spaces have been studied in ([3],[7],[10],[17],[18]).
Letw be the set of all sequences of real or complex numbers andℓ∞, candc0 be respectively the Banach spaces of
bounded, convergent and null sequencesx = (xk)with the usual norm∥x∥∞ = sup|xk|,wherek ∈ N = {1,2, . . .},
the set of positive integers. Also bybs, cs, ℓ1 andℓp; we denote the spaces ofall bounded, convergent, absolutely and
p−absolutelyconvergent series, respectively.
A sequence space E with a linear topology is called aK−space provided each of the maps pi : E → Cdefined by
pi(x) = xi is continuous for each i ∈ N, whereC denotes the complex field. AK−space E is called an F K−space
providedEis a complete linear metric space. AnF K−space whose topology is normable is called aBK−space.The basic properties ofF K−spaces may be found in([27],[28],[29],[30]).
Letπ= (πn)be a sequence of positive numbers i.e,πn >0,∀n∈NandXis anF K−space. We shall consider the sets
of sequencesx= (xn)
Xπ={x∈w:
( xn
πn
)
∈X}.
Let F be an infinite subset of NandF as the range of a strictly increasing sequence of positive integers, say F =
{λ(n)}∞n=1. The Ces´aro submethodCλis defined as
(Cλx)n =
1 λ(n)
λ∑(n)
k=1
xk, (n= 1,2, ...),
where{xk} is a sequence of a real or complex numbers. Therefore, the Cλ-method yields a subsequence of the Ces´aro
methodC1, and hence it is regular for anyλ. Cλis obtained by deleting a set of rows from Ces´aro matrix. Ifλ(n) =nis
taken, thenCλ=C1is obteined. On a range of sequences
lim
n (Cλx)n:= limn (C1x)n,
we will writeCλ∼C1.The basic properties ofCλ−method can be found in[2]and[21].
We need the following inequality throughout the paper. Let p = (pk)be a sequence of positive real numbers with
G= supkpk andD = max(1, 2G−1).Then, it is well known that for allak, bk ∈C,the field of complex numbers, for all
k∈N,
|ak+bk|
pk≤D(|a
k| pk+|b
k|
pk). (1)
Also for any complexµ,
µpk≤max(1, µG) (2)
see in[16].
LetXbe a sequence space. ThenXis called;
i)Solid (or normal) if(αkxk)∈Xwhenever(xk)∈X for all sequences(αk)of scalars with|αk| ≤1,
ii)Symmetric if(xk)∈Ximplies
( xπ(k)
)
∈X,whereπis a permutation ofN, iii)Sequence algebra ifXis closed under multiplication.
2
Main Results
In this section we give the main results of this paper.
Definition 2.1Letf be a modulus function,Xbe a locally convex Hausdorff topological linear space whose topology is determined by a setQof continuous seminormsqandp= (pk)be a sequence of positive real numbers. The symbolw(X)
denotes the space of all sequences defined overX.
Ccλ
0π(∆
m, f, p, q) =
x∈w(X) : λ(1n)
λ∑(n)
k=1
[ f
( q
( ∆m xk
πk
))]pk
−→0
asn−→ ∞
Ccλπ(∆m, f, p, q) =
x∈w(X) : λ(1n)
λ∑(n)
k=1
[ f
( q
( ∆m xk
πk−L
))]pk
−→0
asn−→ ∞, for someL
C(λℓ
∞)π(∆
m, f, p, q) =
x∈w(X) : supn
1 λ(n)
λ∑(n)
k=1
[ f
( q
( ∆mxk
πk
))]pk
<∞
,
where∆0x= xk
πk,∆
mx = (∆m−1xk
πk −∆
m−1xk+1
πk+1)and∆
mx
k =
∑m
v=0(−1)
v(m v
)xk+v
πk+v.Forf(x) =xwe shall write
Cλ c0π(∆
m, p, q), Cλ cπ(∆
m, p, q)andCλ
(ℓ∞)π(∆
mp, q)instead ofCλ c0π(∆
m, f, p, q), Cλ cπ(∆
m, f, p, q)andCλ
(ℓ∞)π(∆
m, f, p, q)
respecxtively.
The proof of each of the following results is fairly straightforward, so we choose to state these results without proof.
Theorem 2.1. Letp= (pk)be bounded, thenCcλπ(∆
m, f, p, q), Cλ c0π(∆
m, f, p, q)andCλ
(ℓ∞)π(∆
m, f, p, q)are
linear spaces.
Theorem 2.2.Cλ c0π(∆
m, f, p, q)is a paranormed space(not totally paranormed), paranormed by
g∆(x) = sup
n
1 λ(n)
λ∑(n)
k=1
[ f
( q
( ∆mxk
πk
))]pk
1
M
Theorem 2.3.Let f, f1, f2are modulus functions and
0< h= infpk≤pk≤sup k
pk =G <∞,
then (i)Cλ
c0π(∆
m, f
1, p, q)⊆Ccλ0π(∆
m, f◦f
1, p, q),
(ii)Ccλ0π(∆m, f1, p, q)∩Ccλ0π(∆
m, f
2, p, q)⊆Ccλ0π(∆
m, f
1+f2, p, q).
Proof . (i)Letx = (
xk
πk
)
∈ Cλ c0π(∆
m, f
1, p, q). Letε > 0and chooseδwith0 < δ < 1such that f(t) < εfor
0≤t≤δ. Writeyk =f1
( q
( ∆m xk
πk
))
and consider
λ∑(n)
k=1
[f (yk)]pk =
∑
1
[f (yk)]pk+
∑
2
[f (yk)]pk
where the first summation is overyk ≤δand the second overyk > δ. Sincef is continuous, we get
∑
1
[f(yk)]
pk < λ(n) max(εh, εG) (3)
and foryk> δwe use the fact that
yk<
yk
δ <1 + yk
δ. By the definition off we have foryk > δ,
f(yk)≤f(1)
[ 1 +
(yk
δ )]
≤2f(1)yk δ . Hence 1 λ(n) ∑ 2
[f(yk)]
pk ≤max
( 1,
( 2f(1)
δ
)G)
1 λ(n)
λ∑(n)
k=1
[yk]
pk−→0. (4)
By(3)and(4)we haveCcλ0π(∆
m, f
1, p, q)⊆Ccλ0π(∆
m, f◦f
1, p, q).
(ii)Let (
xk
πk
)
∈Cλ c0π(∆
m, f
1, p, q)∩Ccλ0π(∆
m, f
2, p, q). Then there existf1andf2such that
1 λ(n)
λ∑(n)
k=1 [ f1 ( q ( ∆mxk
πk
))]pk
−→0asn−→ ∞, (5)
1 λ(n)
λ∑(n)
k=1 [ f2 ( q ( ∆mxk
πk
))]pk
−→0asn−→ ∞. (6)
Then using(1)it can be shown that
1 λ(n)
λ∑(n)
k=1
[
(f1+f2)
( q
( ∆mxk
πk
))]pk
= 1
λ(n)
λ∑(n)
k=1 [ f1 ( q ( ∆mxk
πk
)) +f2
( q
( ∆mxk
πk
))]pk
≤ D 1
λ(n)
λ∑(n)
k=1 [ f1 ( q ( ∆mxk
πk
))]pk
+D 1 λ(n)
λ∑(n)
k=1 [ f2 ( q ( ∆mxk
πk
))]pk
−→ 0asn−→ ∞
By(5)and(6), thenλ(1n)
λ∑(n)
k=1
[
(f1+f2)
( q
( ∆m xk
πk
))]pk
−→0asn−→ ∞.Therefore (
xk
πk
)
∈Cλ c0π(∆
m, f
1+f2, p, q).
Hence
Ccλ0π(∆
m, f
1, p, q)∩Ccλ0π(∆
m, f
2, p, q)⊆Ccλ0π(∆
m, f
1+f2, p, q).
The proof of the following result is a routune work in view of the above Theorem.
Corallary 2.4.f, f1, f2 are modulus functions. Then
(i)Cλ cπ(∆
m, f
1, p, q)⊆Ccλπ(∆
m, f◦f
1, p, q),
(ii)Cλ cπ(∆
m, f
1, p, q)∩Ccλπ(∆
m, f
2, p, q)⊆Ccλπ(∆
m, f
(iii)Cλ
(ℓ∞)π(∆
m, f
1, p, q)⊆C(λℓ
∞)π(∆
m, f◦f
1, p, q),
(iv)Cλ
(ℓ∞)π(∆
m, f
1, p, q)∩C(λℓ∞)π(∆
m, f
2, p, q)⊆C(λℓ∞)π(∆
m, f
1+f2, p, q)
The following result is a routune work.
Proposition 2.5.Cλ cπ
(
∆m−1, f, p, q)⊆Cλ cπ(∆
m, f, p, q).
Theorem 2.6.Letm≥1, then the following inclusions are strict. (i)Ccλ0π(∆m−1, f, q)⊆Ccλ0π(∆m, f, q)
(ii)Cλ cπ
(
∆m−1, f, q)⊆Cλ cπ(∆
m, f, q)
(iii)C(λℓ
∞)π
(
∆m−1, f, q)⊆C(λℓ
∞)π(∆
m, f, q).
Proof . We prove the case(i)only. The other cases follow in a similar way. Let (
xk
πk
)
∈Cλ c0π
(
∆m−1, f, q). Then we
have
1 λ(n)
λ∑(n)
k=1
f (
q (
∆m−1xk πk
))
−→0as n−→ ∞.
By the definition off andq, we have
1 λ(n)
λ∑(n)
k=1
f (
q (
∆mxk πk
))
≤ 1
λ(n)
λ∑(n)
k=1
f (
q (
∆m−1xk πk −
∆m−1xk+1 πk+1
))
≤ 1
λ(n)
λ∑(n)
k=1
f (
q (
∆m−1xk πk
))
+ 1
λ(n)
λ∑(n)
k=1
f (
q (
∆m−1xk+1 πk+1
))
−→ 0as n−→ ∞.
This completes the proof. In generalCcλπ(∆i, f, q)⊆Ccλπ(∆m, f, q)for alli= 1,2,3, ..., m−1and the inclusion is strict. To show that the inclusion is strict, consider the following example.
Example 2.7. LetX =C, f(x) =x,andq(x) = |x|. Consider the sequences(xk) = (km+α)and(πk) =
( kα+1),
wherex= (
xk
πk
)
andm∈N, α∈R.Thenx∈Cλ c0π(∆
m, f, q)butx /∈Cλ c0π
(
∆m−1, f, q), since∆m xk
πk = 0,∆
m−1xk
πk =
(−1)m−1 (m−1)!for∀k∈N.
Theorem 2.8. For any two sequencep= (pk)andt = (tk)of positive real numbers and any two seminormsq1,q2we
have (i)Cλ
c0π(∆
m, f, p, q
1)∩Ccλ0π(∆
m, f, t, q
2)̸=∅
(ii)Cλ cπ(∆
m, f, p, q
1)∩Ccλπ(∆
m, f, t, q
2)̸=∅
(iii)Cλ
(ℓ∞)π(∆
m, f, p, q
1)∩C(λℓ∞)
π(∆
m, f, t, q
2)̸=∅.
Proof.Since the zero element belongs to each of the above classes of sequences, thus the intersection is non empty. The following result is a consequence of Theorem 2. 3(i)and Corollary 2. 4(i)and(iii).
Proposition 2.9.Letf be a modulus function. Then (i)Ccλ0π(∆
m, p, q)⊆Cλ c0π(∆
m, f, p, q),
(ii)Cλ cπ(∆
m, p, q)⊆Cλ cπ(∆
m, f, p, q),
(iii)Cλ
(ℓ∞)π(∆
m, p, q)⊆Cλ
(ℓ∞)π(∆
m, f, p, q).
Theorem 2.10.Let0< pk≤rkand
(
rk
pk
)
be bounded, then Ccλπ(∆
m, f, r, q)⊆Cλ cπ(∆
m, f, p, q).
Proof.Omitted.
Theorem.2.11. The sequence spaces Cλ c0π(∆
m, f, p, q),Cλ cπ(∆
m, f, p, q)andCλ
(ℓ∞)π(∆
m, f, p, q) are neither
solid nor symmetric, nor sequence algebras form≥1.
Proof. Letm= 1, pk = 1for allk∈N,f(x) =xandq(x) =|x|. If the sequences(xk) =
(
kn+1)and(πk) = (kn)
are taken, then the sequence (
xk
πk
)
belongs toCλ
(ℓ∞)π(∆)andC
λ
cπ(∆),wheren∈R. Letαk = (−1)
k
, then(αk x)does
not belong toCλ
(ℓ∞)π(∆)andC
λ
cπ(∆). HenceC
λ cπ(∆
m, f, p, q)andCλ
(ℓ∞)π(∆
m, f, p, q)are not solid. The other cases
3
Results Related to Statical Convergence
Fast[8]and (independently) Schoenberg[24]introduced the notion of statistical convergence. The idea depends on the density of subsets of the setNof natural numbers. The density ofEa subset ofNis defined byδ(E) = limn→∞ n1∑nk=1χE(k) provided the limit exists, whereχEis the characteristic function ofE. A sequencex= (xk)is called statistically convergent
to a numberL,if for everyε >0, δ{k∈N:|xk−L| ≥ε} = 0. Later on it was further investigated from the sequence
space point of view and linked with summability theory by Fridy[9]and Salat[23]and many others.
Definition 3.1. Letπ = (πk)be a sequence of pozitif numbers. A sequencex= (xk)is said to be∆mq −statistically
convergent toL∈Xif for allq∈Qand anyε >0, lim
n−→∞
1 λ(n)
{k≤λ(n) :q (
∆mxk πk −
L )
≥ε}= 0,
where the vertical bars indicate the number of elements in the closed set. In this case we write (
xk
πk
)
−→L (S(∆m q
)) .
Theorem 3.2.Letfbe a modulus function andsupkpk=G <∞.ThenCcλπ(∆
m, f, p, q)⊂S(∆m q
) .
Proof. Let x ∈ Cλ cπ(∆
m, f, p, q). Take q ∈ Q, ε > 0 and let ∑
1 denote the sum over k ≤ λ(n) with
q (
∆m xk
πk−L
)
>εand∑2denote the sum overk≤λ(n)withq (
∆m xk
πk −L
)
< ε.Then
1 λ(n)
λ∑(n)
k=1 [ f ( q ( ∆mxk
πk −
L
))]pk
= 1 λ(n) ( ∑ 1 [ f ( q ( ∆mxk
πk −
L
))]pk
+∑ 2 [ f ( q ( ∆mxk
πk −
L
))]pk)
≥ 1 λ(n) ∑ 1 [ f ( q ( ∆mxk
πk −
L
))]pk
≥ 1
λ(n) ∑
1
[f(ε)]pk
≥ 1 λ(n) ∑ 1 min (
[f(ε)]infpk, [f(ε)]G)
= 1
λ(n)
{k≤λ(n) : q (
∆mxk πk −
L )
≥ε}min (
[f(ε)]infpk, [f(ε)]G
) .
Hencex∈S(∆mq
) .
Theorem 3.3Letf be a bounded and0< h= infpk ≤pk ≤supkpk =G <∞. ThenS
( ∆m
q
)
⊂Cλ cπ(∆
m, f, p, q).
Proof. Suppose thatf is bounded. Letq∈Q , ε >0and let∑1and∑2was denoted in previous theorem. Sincef is bounded there exists an integerKsuch thatf(x)< K,for allx≥0.Then
1 λ(n)
λ∑(n)
k=1 [ f ( q ( ∆mxk
πk −
L
))]pk
≤ 1 λ(n) ∑ 1 [ f ( q ( ∆mxk
πk −
L
))]pk
+ 1 λ(n) ∑ 2 [ f ( q ( ∆mxk
πk −
L
))]pk
≤ 1
λ(n) ∑
1
max(Kh, KG)+ 1 λ(n)
∑
2
[f(ε)]pk
≤ max(Kh, KG) 1 λ(n)
{k≤λ(n) : q (
∆mxk πk
−L )
≥ε}
+ max (
f(ε)h, f(ε)G )
.
Hencex∈Ccλπ(∆m, f, p, q).
Theorem 3.4.S(∆m q
) =Cλ
cπ(∆
m, f, q)if and only iff is bounded.
Proof :Letf be bounded. By Theorem 3.2 and Theorem 3.3 we haveS(∆mq )=Ccλπ(∆m, f, q). Conversely, suppose thatS(∆m
q
) =Cλ
cπ(∆
m, f, q)andfis unbounded. Then there exists a positive sequence(t λ(n)
)
withf(tλ(n)
)
= (λ(n))2, n= 1,2,3, ....If we choose
∆mxk πk
= {
tλ(n), k= (λ(n)) 2
, n= 1,2,3, ...
then we have
1 λ(n)
{k≤λ(n) :q (
∆mxk πk
−L )
≥ε}≤ √
λ(n)
λ(n) −→0as n−→ ∞.
Hencexk −→0
( S(∆m
q
))
,butx /∈Cλ cπ(∆
m, f, q)forq=|x|andX =C.Indeed letq=|x|andX=C.Then
∆mxk πk
= (
∆mx1
π1, ∆
mx2
π2, ∆
mx3
π3, ∆
mx4
π4, ..., ∆
mx9
π9, ∆
mx10
π10, ..., ∆
mx16
π16, ... )
= (tλ(1), 0, 0, tλ(2), 0, ...,0, tλ(3), 0, ...,0, tλ(4), 0, ...,0, tλ(5), 0, ...
) .
Letsn =λ(1n)
∑n
k=1f(∆
m xk
πk
).Then
s(λ(n))2 =
1
(λ(n))2 (
(λ(1))2+ (λ(2))2+ (λ(3))2+...+ (λ(n))2 )
= λ(n) (λ(n) + 1) (2λ(n) + 1) 6 (λ(n))2
by(7).Now the subsequence (
s(λ(n))2
)
of (sλ(n)
)
is unbounded. Therefore (sλ(n)
) /
∈ C(λℓ
∞)π(∆
m, f, q)and hence
( sλ(n)
) /
∈Ccλπ(∆
m, f, q).Then contradicts toS(∆m q
)
=Ccλπ(∆
m, f, q).
Consequently, the results obtained in[1]are generalized by usingCλ−method and modulus function with more general and
weaker conditions. The analogous results in[31]may be obtained by this method.
Acknowledgements
The authors wish to thank the referee for his carefully reading of the manuscript and valuable suggestions.
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