Electronic Journal of Mathematical Analysis and Applications
Vol. 4(2) July 2016, pp. 150-159.
ISSN: 2090-729(online)
http://fcag-egypt.com/Journals/EJMAA/
————————————————————————————————
LACUNARY I-CONVERGENT AND LACUNARY I-BOUNDED
SEQUENCE SPACES DEFINED BY AN ORLICZ FUNCTION
EMRAH EVREN KARA, MERVE İLKHAN
Abstract. A lacunary sequence is an increasing sequence θ = (kr ) such that
kr − kr−1 → ∞ as r → ∞. In this paper, we define the spaces of lacunary
ideal convergent and lacunary ideal bounded sequences with respect to an
Orlicz function. We establish some inclusion relations of the resulting sequence
spaces.
1. Preliminaries, background and notation
Let ω denote the space of all real or complex valued sequences and N, C stand
for the set of natural numbers, complex numbers.
Let X 6= ∅. A class I ⊆ 2X is called an ideal if I is additive (i.e., A, B ∈ I ⇒
A ∪ B ∈ I) and hereditary (i.e., A ∈ I, B ⊆ A ⇒ B ∈ I) ([29]). An ideal is called
non-trivial if X ∈
/ I. A non-trivial ideal I is said to be admissible if I contains
every finite subset of X.
The notion of ideal convergence was first introduced by Kostyrko et al [25] in
the following way. Let I be a non-trivial ideal in N. A sequence x = (xn )∞
n=1
of real numbers is said to be I-convergent to l ∈ R if for every ε > 0 the set
A(ε) = {n ∈ N : |xn − l| ≥ ε} belongs to I. Also, Kostyrko et al [27] studied the
concept of I-convergence in metric spaces. Later the idea of I-convergence was
extended to an arbitrary topological space by Lahiri and Das [31]. Note that if I
is the ideal of all finite subsets of N, then the ideal convergence coincides with the
usual convergence.
A sequence x = (xn ) is said to be I-bounded if there exists an K > 0 such that
{n ∈ N : |xn | > K} ∈ I.
For more details about I-convergence, we refer to [6, 7, 15, 16, 17, 26, 30, 36, 41].
An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous,
nondecreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) → ∞
as x → ∞. Since M is a convex function and M (0) = 0, M (αx) ≤ αM (x) for all
α ∈ (0, 1).
M is said to satisfy ∆2 -condition for all x ∈ [0, ∞) if there exists a constant
K > 0 such that M (Lx) ≤ KLM (x), where L > 1 (see [28]).
2010 Mathematics Subject Classification. 40A05, 40A35, 46A45.
Key words and phrases. Lacunary sequence; ideal convergence; Orlicz functions.
Submitted Jan. 12, 2016.
150
LACUNARY I-CONVERGENT SEQUENCE SPACES
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151
Throughout this paper, p = (pi ) will be a sequence of positive real numbers such
that 0 < h = inf pi ≤ pi ≤ H = sup pi < ∞. Also, the inequalities for every
i = 1, 2, ...
|ai + bi |pi ≤ D {|ai |pi + |bi |pi }
(1)
and
|a|pi ≤ max{1, |a|H }
will be used, where a, ai , bi ∈ C and D = max 1, 2H−1 .
Let A = (aij ) be an infinite matrix of real or complex numbers aij , where i, j ∈ N.
∞
P
We write Ax = (Ai (x)) if Ai (x) =
aij xj converges for each i ∈ N.
j=1
Lindenstrauss and Tzafriri [32] used the idea of Orlicz function to construct the
following sequence space:
)
(
∞
X
|xi |
< ∞, for some ρ > 0 .
`M = x ∈ ω :
M
ρ
i=1
The space `M becomes a Banach space with the norm
)
(
∞
X
|xn |
≤1 ,
M
kxk = inf ρ > 0 :
ρ
n=1
which is called an Orlicz sequence space.
By using Orlicz function, the following sequence spaces were defined in [34]:
(
)
pi
∞ X
|xi |
`M (p) = x ∈ ω :
M
< ∞ for some ρ > 0 ,
ρ
i=1
(
)
pi
n |xi − l|
1X
M
W (M, p) = x ∈ ω :
→ 0 as n → ∞ for some ρ > 0 and l > 0 ,
n i=1
ρ
(
)
pi
n |xi |
1X
M
→ 0 as n → ∞ for some ρ > 0
W0 (M, p) = x ∈ ω :
n i=1
ρ
and
(
W∞ (M, p) =
)
pi
n |xi |
1X
x ∈ ω : sup
M
< ∞ for some ρ > 0 ,
ρ
n n
i=1
where p = (pi ) is any sequence of positive real numbers.
Later on Orlicz sequence spaces were investigated by Esi [8], Bhardwaj and Singh
[2], Esi and Et [10], Et [12], Tripathy et al [42], Bektaş and Altın [1], Şahiner and
Gürdal [40] and many others.
Recently, the authors introduced new spaces by using ideal convergence, Orlicz
function and an infinite matrix.
For example, Hazarika et al [19] introduced paranorm ideal convergent sequence
spaces by using Zweir transform and Orlicz function as follows:
pn
|(Z p x)n − L|
I
≥ ε ∈ I for some L ∈ C
Z (M, p) = x ∈ ω : n ∈ N : M
ρ
and
Z0I (M, p)
=
x∈ω:
pn
|(Z p x)n |
n∈N: M
≥ε ∈I ,
ρ
152
E. E. KARA, M. İLKHAN
EJMAA-2016/4(2)
where the Zweir matrix Z p = (znk ) defined by

(n = k)
 p,
1 − p, (n − 1 = k); (n, k ∈ N)
znk =

0,
otherwise.
Ideal convergent sequence spaces combined by a sequence of Orlicz functions
(Mi ) and the matrix Λ
p
|Λi (x) − L| i
= 0, for some L and ρ > 0 ,
cI (M, Λ, p) = x ∈ ω : I − lim Mi
i
ρ
cI0 (M, Λ, p)
x ∈ ω : I − lim Mi
=
i
|Λi (x)|
ρ
pi
= 0, for some ρ > 0
and
`∞ (M, Λ, p) =
x ∈ ω : sup Mi
i
|Λi (x)|
ρ
pi
< ∞, for some ρ > 0
Pi
were defined by Mursaleen and Sharma [33], where Λi (x) = λ1i m=1 (λm −λm−1 )xm
and (λm ) is a strictly increasing sequence of positive real numbers tending to infinity.
Kara and İlkhan [21] defined the following spaces:
pn
|An (x) − L|
= 0, for some L and ρ > 0 ,
cI (M, A, p) = x ∈ ω : I − lim M
n
ρ
cI0 (M, A, p)
=
pn
|An (x)|
x ∈ ω : I − lim M
= 0, for some ρ > 0 ,
n
ρ
and
`∞ (M, A, p) =
pn
|An (x)|
< ∞, for some ρ > 0 .
x ∈ ω : sup M
ρ
n
In the literature, there are more papers related to sequence spaces defined by
using ideal convergence, an Orlicz function and an infinite matrix. For some of
these papers, one can see [18, 20, 35, 37, 43].
An increasing integer sequence θ = (kr ) is called a lacunary sequence if k0 = 0
and hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ is denoted by
Ir = (kr−1 , kr ] and the ratio kr /kr−1 is written as qr = kr /kr−1 .
Freedman et al [13] defined the space of lacunary strongly convergent sequences
Nθ in the following way:
)
(
1 X
Nθ = x ∈ ω : lim
|xi − l| = 0, for some l
r→∞ hr
i∈Ir
for any lacunary sequence θ = (kr ).
The sequence spaces defined by combining a lacunary sequence, an Orlicz function and an infinite matrix were investigated by many authors. We refer to Bhardwaj and Singh [3], Bilgin [4, 5], Savaş and Rhoades [38], Karakaya [22], Güngör et
al [14] for some of related papers.
LACUNARY I-CONVERGENT SEQUENCE SPACES
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153
In [44], the authors defined the notions of lacunary I-convergence and lacunary
I-null sequences. If for every ε > 0,
)
(
)
(
1 X
1 X
r∈N:
|xi − l| ≥ ε ∈ I and r ∈ N :
|xi | ≥ ε ∈ I,
hr
hr
i∈Ir
i∈Ir
then the sequence x = (xi ) is said to be lacunary I- convergent to l and lacunary
I-null, respectively.
It can be seen in [9, 11, 23, 24, 39, 45], the authors used the concept of ideal
convergence and lacunary sequence to define different types of sequence spaces.
In this paper, we define some new sequence spaces using the concept of ideal
convergence, lacunary sequence, Orlicz function and A-transform as follows:
(
)
X |Ai (x)| pi
1
I−Nθ0 (A, M, p) = x ∈ ω : I − lim
M
= 0, for some ρ > 0 ,
r hr
ρ
i∈Ir
)
pi
|Ai (x) − L|
1 X
= 0, for some L and ρ > 0 ,
M
x ∈ ω : I − lim
r hr
ρ
(
I−Nθ (A, M, p) =
i∈Ir
(
I−Nθ∞ (A, M, p)
=
x∈ω:
)
pi !
1 X
|Ai (x)|
is I − bounded for some ρ > 0 .
M
hr
ρ
i∈Ir
In the case, M (x) = x for all x ∈ [0, ∞), we write I − Nθ0 (A, p), I − Nθ (A, p)
and I − Nθ∞ (A, p) instead of the spaces I − Nθ0 (A, M, p), I − Nθ (A, M, p) and
I − Nθ∞ (A, M, p), respectively.
In the case, pi = 1 for all i ∈ N, we write I − Nθ0 (A, M ), I − Nθ (A, M ) and
I − Nθ∞ (A, M ) instead of the spaces I − Nθ0 (A, M, p), I − Nθ (A, M, p) and I −
Nθ∞ (A, M, p), respectively.
If A = I, pi = 1 for all i ∈ N and M (x) = x for all x ∈ [0, ∞), the spaces
I − Nθ0 (A, M, p) and I − Nθ (A, M, p) reduce to the spaces of lacunary I-null and
lacunary I-convergent sequences defined by Tripathy et al [44].
The main purpose of this paper is to introduce the spaces I − Nθ0 (A, M, p),
I − Nθ (A, M, p) and I − Nθ∞ (A, M, p) and give some inclusion theorems.
2. Main results
In this section, we investigate some inclusion relations related to the spaces
I − Nθ0 (A, M, p), I − Nθ (A, M, p) and I − Nθ∞ (A, M, p). Firstly, we prove that
these spaces are linear over the set of complex or real numbers.
Theorem 1 I − Nθ0 (A, M, p), I − Nθ (A, M, p) and I − Nθ∞ (A, M, p) are linear
spaces.
Proof.
Let x, y ∈ I − Nθ0 (A, M, p) and α, β be scalars. Then there exist
ρ1 , ρ2 > 0 such that for every ε > 0
(
)
pi
1 X
|Ai (x)|
ε
M
≥
∈ I,
A1 = r ∈ N :
hr
ρ1
2D
i∈Ir
(
)
pi
ε
1 X
|Ai (y)|
A2 = r ∈ N :
M
≥
∈ I.
hr
ρ2
2D
i∈Ir
154
E. E. KARA, M. İLKHAN
EJMAA-2016/4(2)
To prove that αx + βy ∈ I − Nθ0 (A, M, p), let define ρ = max{2|α|ρ1 , 2|β|ρ2 }.
Suppose that r ∈
/ A1 ∪ A2 . By using inequalitty (1), we have
(
pi
pi
pi )
1 X
|Ai (αx + βy)|
1 X
|Ai (x)|
1 X
|Ai (y)|
M
≤D
M
+
M
hr
ρ
hr
ρ1
hr
ρ2
i∈Ir
i∈Ir
i∈Ir
o
n ε
ε
=ε
<D
+
2D 2D
since
M is an Orlicz
and i
A is a linear
transformation. Hence r ∈
/ A0 =
n
h function
o
pi
P
|Ai (αx+βy)|
1
r ∈ N : hr i∈Ir M
≥ ε . This implies that A0 ⊂ A1 ∪ A2 . By
ρ
additivity and heritability of I, we have A0 ∈ I. Consequently, I − Nθ0 (A, M, p) is
a linear space.
In a similar way, one can prove that I − Nθ (A, M, p) and I − Nθ∞ (A, M, p) are
linear spaces.
Now, we give some inclusion relations.
Theorem 2 The following inclusion relations hold:
I − Nθ0 (A, M, p) ⊂ I − Nθ (A, M, p) ⊂ I − Nθ∞ (A, M, p).
Proof.
Clearly, the first inclusion is true. To prove that the inclusion I −
Nθ (A, M, p) ⊂ I − Nθ∞ (A, M, p) holds, let x ∈ I − Nθ (A, M, p). Then there exists
ρ1 > 0 such that for every ε > 0
(
)
pi
1 X
|Ai (x) − L|
A0 = r ∈ N :
M
≥ ε ∈ I.
hr
ρ1
i∈Ir
Let define ρ = 2ρ1 . Since M is non-decreasing and convex, we have
|Ai (x)|
|Ai (x) − L|
|L|
M
≤M
+M
.
ρ
ρ1
ρ1
Suppose that r ∈
/ A0 . Hence by the last inequality and (1), the following inequality
is obtained:
(
pi
pi
pi )
1 X
|Ai (x)|
|Ai (x) − L|
1 X
|L|
1 X
≤D
M
M
+
M
hr
ρ
hr
ρ1
hr
ρ1
i∈Ir
i∈Ir
i∈Ir
(
)
pi
1 X
|L|
<D ε+
M
.
hr
ρ1
i∈Ir
h i h ipi
H
|L|
Because of the fact that M ρ1
≤ max 1, M |L|
, we have
ρ1
n
Put K = D ε +
1
hr
pi
1 X
|L|
< ∞.
M
hr
ρ1
i∈Ir
h ipi o
n
P
|L|
. It follows that r ∈ N:
i∈Ir M
ρ1
1
hr
P
i∈Ir
h
M
|Ai (x)|
ρ
I which means x ∈ I − Nθ∞ (A, M, p). This completes the proof.
Theorem 3 Let M and M 0 be Orlicz functions which satisfies ∆2 -condition.
Then the following inclusion relations hold:
(1) I − Nθ0 (A, M, p) ⊆ I − Nθ0 (A, M 0 ◦ M, p), I − Nθ (A, M, p) ⊆ I − Nθ (A, M 0 ◦
M, p) and I − Nθ∞ (A, M, p) ⊆ I − Nθ∞ (A, M 0 ◦ M, p).
ipi
o
>K ∈
LACUNARY I-CONVERGENT SEQUENCE SPACES
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155
(2) I−Nθ0 (A, M, p)∩ I−Nθ0 (A, M 0 , p) ⊆ I−Nθ0 (A, M +M 0 , p), I−Nθ (A, M, p)∩
I − Nθ (A, M 0 , p) ⊆ I − Nθ (A, M + M 0 , p) and I − Nθ∞ (A, M, p) ∩ I −
Nθ∞ (A, M 0 , p) ⊆ I − Nθ∞ (A, M + M 0 , p).
Proof.
We prove only the third inclusions since the others can be proved
similarly.
(1) Let x ∈ I − Nθ∞ (A, M, p). Then there exists K1 > 0 such that
(
)
pi
1 X
|Ai (x)|
A0 = r ∈ N :
M
> K1 ∈ I
hr
ρ
i∈Ir
for a ρ > 0. Since M 0 is nondecreasing and convex, and satisfies ∆2 -condition, we
obtain the inequality
( pi
H )
pi
X
X
|A
(x)|
Ai (x)
1
1
1
i
0
0
M M
M
≤ max 1, K M (2)
hr
ρ
δ
hr
ρ
A (x)
A (x)
i∈Ir , M
i
ρ
>δ
i
ρ
i∈Ir , M
>δ
(2)
for K ≥ 1. By continuity of M 0 , we have
pi
X
1
|Ai (x)|
1
M0 M
≤
hr
ρ
h
r
A (x)
i∈Ir , M
i
ρ
≤δ
X
i∈Ir , M
ε pi ≤
Ai (x)
ρ
1
hr
≤δ
X
i∈Ir , M
max{εh , εH }.
Ai (x)
ρ
≤δ
(3)
Suppose that r ∈
/ A0 . Then by using the inequalities (2) and (3), we have
pi
pi
X
1X 0
1
|Ai (x)|
1
|Ai (x)|
=
M0 M
+
M M
hr
ρ
hr
ρ
h
r
A (x)
i∈Ir
i∈Ir , M
i
ρ
>δ
X
i∈Ir , M
Ai (x)
ρ
pi
|Ai (x)|
M0 M
ρ
≤δ
( H )
1 0
≤ max 1, K M (2)
K1 + max{εh , εH } = K2 .
δ
Hence r ∈
/ B =
n
r∈N:
1
hr
P
h
i∈Ir
ipi
o
M 0 M |Aiρ(x)|
> K2 and so B ⊂ A0
which implies B ∈ I. We conclude that x ∈ I − Nθ∞ (A, M 0 ◦ M, p).
(2) Let x ∈ I − Nθ∞ (A, M, p) ∩ I − Nθ∞ (A, M 0 , p). There exist K1 > 0 and
K2 > 0 such that
(
)
pi
1 X
|Ai (x)|
A1 = r ∈ N :
M
> K1 ∈ I
hr
ρ
i∈Ir
and
(
A2 =
)
pi
1 X
|Ai (x)|
r∈N:
M
> K2 ∈ I
hr
ρ
i∈Ir
for some ρ > 0. Let r ∈
/ A1 ∪ A2 . Then we have
(
pi
pi )
X |Ai (x)| pi
X
1 X
|A
(x)|
1
1
|A
(x)|
i
i
(M + M 0 )
≤D
M
+
M0
hr
ρ
hr
ρ
hr
ρ
i∈Ir
i∈Ir
< D {K1 + K2 } = K.
i∈Ir
156
E. E. KARA, M. İLKHAN
EJMAA-2016/4(2)
h
ipi
o
n
P
>K .
Thus r is not contained in B = r ∈ N : h1r i∈Ir (M + M 0 ) |Aiρ(x)|
We have A1 ∪ A2 ∈ I and B ⊂ A1 ∪ A2 which imply B ∈ I. This means x ∈
I − Nθ∞ (A, M + M 0 , p) and completes the proof.
Corollary 1 Let M be an Orlicz function which satisfies ∆2 -condition. Then
the inclusions I − Nθ0 (A, p) ⊂ I − Nθ0 (A, M, p), I − Nθ (A, p) ⊂ I − Nθ (A, M, p)
and I − Nθ∞ (A, p) ⊂ I − Nθ∞ (A, M, p) hold.
Proof. The proof follows from the first part of Theorem 3 by using M (x) = x
and M 0 (x) = M (x) for all x ∈ [0, ∞).
qi
pi
Theorem 4 Let 0 < pi ≤ qi and
hold:
be bounded. Then the following inclusions
I − Nθ0 (A, M, q) ⊂ I − Nθ0 (A, M, p) and I − Nθ (A, M, q) ⊂ I − Nθ (A, M, p).
Proof.
We prove only the first inclusion and the other inclusion can be
0
carried
out
by
h iqi using a similar technique. Let x ∈ I − Nθ (A, M, q). Write ti =
M |Aiρ(x)|
and λi = pqii , so that 0 < λ ≤ λi ≤ 1. By using Hölder inequality,
we obtain
X
X
1 X
1
1
(ti )λi +
(ti )λi
(ti )λi =
hr
hr
hr
i∈Ir , ti ≥1
i∈Ir
1
≤
hr
1
=
hr
X
i∈Ir , ti ≥1
X
i∈Ir , ti ≥1
i∈Ir , ti <1
1
ti +
hr
X
i∈Ir , ti <1
X
ti +
i∈Ir , ti <1

≤
1
hr
X
i∈Ir , ti ≥1
(ti )λ
ti + 
1
ti
hr
"
X
i∈Ir , ti <1
X
i∈Ir , ti ≥1
1
ti + 
hr
1
ti
hr
1
hr
1−λ
λ # λ1
λ 
"
X
 
i∈Ir , ti <1
1
hr
1
1−λ # 1−λ
1−λ

λ

1
≤
hr
λ X
ti  .
i∈Ir , ti <1
Hence for every ε > 0 we have
) 
(
pi

|Ai (x)|
1
1 X
M
≥ε ⊂ r∈N:
r∈N:

hr
ρ
hr
i∈Ir


1
∪ r∈N:

hr
i∈Ir , ti <1
n
This last inclusion implies that r ∈ N :
|Ai (x)|
ρ
1
hr
P
i∈Ir
h
M
X
i∈Ir , ti ≥1
X

qi
|Ai (x)|
ε
M
≥
ρ
2
ipi
M
|Ai (x)|
ρ
qi
o
≥ ε ∈ I and so
x ∈ I − Nθ0 (A, M, p).
Corollary 2
(1) If 0 < inf pi ≤ 1, then the inclusions I − Nθ0 (A, M ) ⊂ I − Nθ0 (A, M, p) and
I − Nθ (A, M ) ⊂ I − Nθ (A, M, p) hold.
(2) If 1 ≤ pi ≤ sup pi < ∞, then the inclusions I −Nθ0 (A, M, p) ⊂ I −Nθ0 (A, M )
and I − Nθ (A, M, p) ⊂ I − Nθ (A, M ) hold.
≥

ε λ1 
2

.
LACUNARY I-CONVERGENT SEQUENCE SPACES
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157
Proof. The proof follows from Theorem 4 taking tk = 1 for all k and changing
the roles of pk and tk only for the second part of the corollary.
Theorem 5 If lim pi > 0 and x → L(I − Nθ (A, M, p)), then L is unique.
i→∞
Proof.
Let lim pi = p0 > 0 and assume that x → L(I − Nθ (A, M, p)) and
i→∞
x → L0 (I − Nθ (A, M, p)) for L 6= L0 . Then there exist ρ1 , ρ2 > 0 such that
(
)
pi
ε
1 X
|Ai (x) − L|
≥
r∈N:
M
∈I
hr
ρ1
2D
i∈Ir
and
(
)
pi
1 X
|Ai (x) − L0 |
ε
r∈N:
∈ I.
M
≥
hr
ρ2
2D
i∈Ir
Then we have
(
pi
pi
pi )
1 X
|L − L0 |
|Ai (x) − L|
|Ai (x) − L0 |
1 X
1 X
M
M
M
,
≤D
+
hr
ρ
hr
ρ1
hr
ρ2
i∈Ir
i∈Ir
i∈Ir
where ρ = max{2ρ1 , 2ρ2 }.
Hence it is obtained that
(
)
pi
1 X
|L − L0 |
r∈N:
M
≥ε ∈I
hr
ρ
i∈Ir
Also we have
pi
p0
|L − L0 |
|L − L0 |
M
→ M
ρ
ρ
0
i
h p
0
|
= 0. This implies that L = L0 .
as i → ∞ and so M |L−L
ρ
Acknowledgements
The authors would like to thank the reviewers for their very helpful comments and
valuable suggestions.
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E. E. Kara
DEPARTMENT OF MATHEMATICS, DÜZCE UNIVERSITY, DÜZCE, TURKEY
E-mail address: [email protected]
M. İlkhan
DEPARTMENT OF MATHEMATICS, DÜZCE UNIVERSITY, DÜZCE, TURKEY
E-mail address: [email protected]