1
Gen. Math. Notes, Vol. 9, No. 1, March 2012, pp.44-51
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
Lacunary Strongly Almost Generalized
Convergence with Respect to Orlicz Function
Ayhan Esi
Adiyaman University, Science and Art Faculty,
Department of Mathematics, 02040, Adiyaman, Turkey
E-mail: aesi23@hotmail.com, aesi@adiyaman.edu.tr
(Received: 24-1-12/Accepted: 28-2-12)
Abstract
Kizmaz [5] defined the concept of difference sequence spaces. Later some
authors introduced and studied some generalizations of this idea. In this paper,
we study some properties of [b
c, M ]θ (∆m ) -convergence which was defined by
Esi [1] .
Keywords: Lacunary sequence, difference sequence, Orlicz function, strongly
almost convergence.
1
Definitions and Notations
Let l∞ , c and c0 be the sequence spaces of bounded, convergent and null sequences x = (xi ) , respectively. A sequence x = (xi ) ∈ l∞ is said to be almost
convergent [8] if all Banach limits of x = (xi ) coincide. In [8] ,it was shown
that
(
)
n
X
b
c = x = (xi ) : lim
xi+s exists, uniformly in s .
n→∞
i=1
In [9, 10], Maddox defined a sequence x = (xi ) strongly almost convergent to
a number L, if
n
X
lim
|xi+s − L| = 0, uniformly in s.
n→∞
i=1
45
Lacunary Strongly Almost Generalized...
By a lacunary sequence θ = (kr ), r = 0, 1, 2, ...,where k0 = 0, we shall
mean increasing sequence of non-negative integers hr = kr − kr−1 → ∞ as
r → ∞.The intervals determined by θ are denoted by Ir = (kr−1 , kr ] and the
kr
will be denoted by qr . The space of lacunary strongly convergent
ratio kr−1
sequence Nθ was defined by Freedman et al. [3] as follows:
(
)
1 X
Nθ = x = (xi ) : lim
|xi − L| = 0 for some L .
r→∞ hr
i∈I
r
In [5] , Kizmaz defined the sequence spaces Z (∆) = {x = (xi ) : (∆xi ) ∈ Z}
for Z = l∞ , c and c0 ,where ∆x = (∆xi ) = (xi − xi−1 ) . After, Et and Colak
[2] defined generalized the difference sequence spaces as follows: Z (∆m ) =
{x = (xi ) : (∆m xi ) ∈ Z} for Z = l∞ , c and c0 ,where m ∈ N, ∆0 x = xi , ∆x =
(xi − xi−1 ) , ∆m xi = (∆m xi ) = (∆m−1 xi − ∆m−1 xi+1 ) and so that
m
X
v m
m
∆ xi =
(−1)
xi+v .
v
v=0
An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous,
non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) →
∞ as x → ∞.
An Orlicz function M is said to be satisfy ∆2 −condition for all values of t,
if there exists a constant T > 0 such that M (2t) ≤ T M (t) , (t ≥ 0) .
Remark 1.The ∆2 −condition is equivalent to the satisfaction of the inequality M (Lt) ≤ T LM (t) for all values of t and for L > 1.This inequality was
used in some published articles [12] ,[13] and many others. But this is not true,
which is shown by the simplest example such as M (t) = t2 . Then the Orlicz
function M satisfies ∆2 −condition with T = 4, but for M (Lt) = L2 t2 > 4Lt2
when L ≥ 5.
Remark 2. An Orlicz function M satisfies the inequality M (λt) ≤ λM (t)
for all λ with 0 < λ < 1.
Lindenstrauss and Tzafriri [7] used the idea of Orlicz functions to construct
Orlicz sequence spaces
)
(
X |xi | < ∞, for some ρ > 0 .
lM = x = (xi ) :
M
ρ
i
The sequence space lM with the norm
(
kxk = inf
ρ>0:
X
i
M
|xi |
ρ
)
≤1
46
Ayhan Esi
becomes a Banach space with is called an Orlicz Sequence Space. The space
lM is closely related to the space lp , which is an Orlicz Sequence Space with
M (x) = xp for 1 ≤ p < ∞.
Let M be an Orlicz function. Göngör and Et [4] defined the following
sequence spaces:
(
[b
c, M ] (∆m ) =
m
)
P
−L|
=
0,
x = (xi ) : limn→∞ n1 ni=1 M |∆ xi+s
ρ
,
uniformly in s, for some ρ > 0 and L > 0
(
)
m
n
X
1
|∆ xi+s |
[b
c, M ]0 (∆m ) = x = (xi ) : lim
M
= 0, uniformly in s, for some ρ > 0 ,
n→∞ n
ρ
i=1
)
(
m
n
X
|∆
x
|
1
i+s
M
< ∞, for some ρ > 0 .
[b
c, M ]∞ (∆m ) = x = (xi ) : sup
ρ
n,s n
i=1
Let M be an Orlicz function. We defined in [1] new generalized difference
sequence spaces as follows:
(
[b
c, M ]θ (∆m ) =
limr→∞ h1r
P
|∆m xi+s −L|
ρ
x = (xi ) :
= 0,
i∈Ir M
uniformly in s, for some ρ > 0 and L > 0
)
,
(
)
X |∆m xi+s | 1
= 0, uniformly in s, for some ρ > 0 ,
(∆m ) = x = (xi ) : lim
M
r→∞ hr
ρ
i∈Ir
(
)
m
X
1
|∆
x
|
i+s
[b
c, M ]θ∞ (∆m ) = x = (xi ) : sup
M
< ∞ for some ρ > 0 .
ρ
r,s hr
i∈I
[b
c, M ]θ0
r
If x = (xi ) ∈ [b
c, M ]θ (∆m ), we say that x = (xi ) is lacunary strongly almost
generalized ∆m −convergence to the number L with respect to Orlicz function
M.In this case we write [b
c, M ]θ (∆m ) − lim x = L.When M (x) = x, then we
write [b
c]θ (∆m ) , [b
c]θ0 (∆m ) and [b
c]θ∞ (∆m ) for the spaces [b
c, M ]θ (∆m ) , [b
c, M ]θ0 (∆m )
and [b
c, M ]θ∞ (∆m ) .
The purpose of this paper is to examine some properties of these new
sequence spaces as a concept of lacunary almost generalized ∆m −convergence
using Orlicz function which also generalize the well known Orlicz sequence
space lM , strongly summable sequence spaces [C, 1] , [C, 1]0 and [C, 1]∞ .
47
Lacunary Strongly Almost Generalized...
2
Main Results
In this section we prove some results involving the sequence spaces [b
c, M ]θ (∆m ) , [b
c, M ]θ0 (∆m )
and [b
c, M ]θ∞ (∆m ) .
Theorem 2.1. The spaces [b
c, M ]θ (∆m ) , [b
c, M ]θ0 (∆m ) and [b
c, M ]θ∞ (∆m )
are linear spaces over the complex field C.
Proof. Let x = (xi ) , y = (yi ) ∈ [b
c, M ]θ0 (∆m ) and α, β ∈ C . Then there
exist positive numbers ρ1 and ρ2 such that
m
1 X
|∆ xi+s |
lim
M
= 0, uniformly in s
r→∞ hr
ρ1
i∈I
r
and
m
|∆ yi+s |
1 X
M
lim
= 0, uniformly in s.
r→∞ hr
ρ
2
i∈I
r
Let ρ3 = max (2 |α| ρ1 , 2 |β| ρ2 ) .Since M is non-decreasing convex function,
we have
m
m
m
|∆ (αxi+s + βyi+s )|
|∆ (αxi+s )|
1 X
1 X
|∆ (βyi+s )|
M
M
≤
+
hr i∈I
ρ3
hr i∈I
ρ3
ρ3
r
r
m
m
|∆ xi+s |
|∆ yi+s |
1 X
1 X
M
M
≤
+
.
hr i∈I
ρ1
hr i∈I
ρ2
r
[b
c, M ]θ0
r
m
(∆ ) .
Therefore αx + βy ∈
The proof for other two cases are routine work in view of above proof.
Theorem 2.2. For any Orlicz function M , [b
c, M ]θ∞ (∆m ) is a semi-normed
linear space, semi-normed by
(
)
m
m
X
1 X
|∆ xi+s |
h∆m (x) =
|xi |+inf ρ > 0 : sup
M
≤ 1, r = 1, 2, ...s = 1, 2, ... .
ρ
r,s hr
i=1
i∈I
r
−
Proof. Clearly; h∆m(x) = h∆m (−x) , x = 0 implies ∆m xi+s = 0 for
all
−
−
−
i, s ∈ N and as such M 0 = 0,where 0 = (0, 0, ...).Therefore h∆m 0 =0.
Next, let ρ1 and ρ2 be such that
m
|∆ xi+s |
1 X
sup
M
≤1
ρ
r,s hr
1
i∈I
r
48
Ayhan Esi
and
m
1 X
|∆ yi+s |
sup
M
≤ 1.
ρ2
r,s hr
i∈I
r
Let ρ = ρ1 + ρ2 .Then, we have
m
1 X
|∆ (xi+s + yi+s )|
sup
M
ρ
r,s hr
i∈I
r
m
m
1 X
|∆ xi+s |
ρ2
1 X
|∆ yi+s |
ρ1
sup
M
+
sup
M
≤ 1.
≤
ρ1 + ρ2 r,s hr i∈I
ρ1
ρ1 + ρ2 r,s hr i∈I
ρ2
r
r
0
Since the ρ s non-negative, so we have
h∆m (x + y) =
m
X
(
|xi + yi |+inf
i=1
m
1 X
|∆ (xi+s + yi+s )|
ρ > 0 : sup
M
≤ 1, r = 1, 2, ...s = 1, 2,
ρ
r,s hr
i∈I
r
)
m
|∆ xi+s |
1 X
M
≤ 1, r = 1, 2, ...s = 1, 2, ...
≤
|xi |+inf ρ1 > 0 : sup
ρ
r,s hr
1
i=1
i∈Ir
)
(
m
m
X
|∆ yi+s |
1 X
M
≤ 1, r = 1, 2, ...s = 1, 2, ... .
+
|yi |+inf ρ2 > 0 : sup
ρ2
r,s hr
i=1
i∈I
m
X
(
r
So, h∆m (x + y) ≤ h∆m (x) + h∆m (y) . Finally for λ ∈C, without loss of generality λ 6= 0, then
)
(
m
m
X
1 X
|∆ λxi+s |
≤ 1, r = 1, 2, ...s = 1, 2, ...
h∆m (λx) =
|λxi |+inf ρ > 0 : sup
M
ρ
r,s hr
i=1
i∈I
r
= |λ|
m
X
i=1
(
|xi |+inf
)
m
1 X
|∆ xi+s |
ρ
|λ| r > 0 : sup
M
≤ 1, r = 1, 2, ...s = 1, 2, ... , where r =
r
λ
r,s hr
i∈I
r
= |λ| h∆m (λx) .
This completes the proof.
Theorem 2.3. If θ = (kr ) be a lacunary sequence with lim inf qr > 1,then
[b
c, M ] (∆m ) ⊂ [b
c, M ]θ (∆m )
49
Lacunary Strongly Almost Generalized...
where
(
[b
c, M ] (∆m ) =
m
)
P
−L|
x = (xi ) : limn→∞ n1 ni=1 M |∆ xi+s
=
0,
ρ
.
uniformly in s, for some ρ > 0 and L > 0
Proof. Let lim inf qr > 1. Then there exists δ > 0 such that qr > 1 + δ and
hence
hr
kr−1
1
δ
=1−
>1−
=
.
kr
kr
1+δ
1+δ
Therefore,
m
m
m
kr
1 X
|∆ xi+s |
|∆ xi+s |
|∆ xi+s |
1 X
δ 1 X
M
M
M
≥
≥
kr i=1
ρ
kr i∈I
ρ
1 + δ hr i∈I
ρ
r
r
and if x = (xi ) ∈ [b
c, M ] (∆m ) ,then it follows that x = (xi ) ∈ [b
c, M ]θ (∆m ) .
Theorem 2.4. If θ = (kr ) be a lacunary sequence with lim sup qr <
∞,then
[b
c, M ]θ (∆m ) ⊂ [b
c, M ] (∆m ) .
Proof. Let x = (xi ) ∈ [b
c, M ]θ (∆m ) .Then for ε > 0, there exists j0 such
that for every j ≥ j0 and for all s ∈ N
m
1 X
|∆ xi+s |
ajs =
M
<ε
hj i∈I
ρ
j
that is, we can find some positive constant T, such that
ajs < T
(1)
for all j and s. Given lim sup qr < ∞ implies that there exists some positive
number K such that
qr < K
(2)
for all r ≥ 1. Therefore, for kr−1 < n ≤ kr , we have by (1) and (2)
n
1X
M
n i=1
≤
r
1 XX
kr−1
j=1 i∈Ij
M
|∆m xi+s − L|
ρ
≤
kr
1 X
kr−1
i=1
M
|∆m xi+s − L|
ρ
" j
#
r
0
X
X
X |∆m xi+s − L| 1
|∆m xi+s − L|
=
+
M
ρ
kr−1 j=1 j=j +1 i∈I
ρ
0
j
1
kr − kj0
≤
sup aps kj0 + ε
kr−1 1≤p≤j0
kr−1
50
Ayhan Esi
≤T
kj0
+ εK.
kr−1
Since kr−1 → ∞ as r → ∞,we get x = (xi ) ∈ [b
c, M ] (∆m ) .This completes the
proof.
Theorem 2.5. Let θ = (kr ) be a lacunary sequence with 1¡lim inf qr ≤
lim sup qr < ∞, then
[b
c, M ]θ (∆m ) = [b
c, M ] (∆m ) .
Proof. It follows from Theorem 2.3. and Theorem 2.4..
Theorem 2.6. Let x = (xi ) ∈ [b
c, M ] (∆m ) ∩ [b
c, M ]θ (∆m ) . Then
[b
c, M ]θ (∆m ) − lim x = [b
c, M ] (∆m ) − lim x
and [b
c, M ]θ (∆m ) − lim x is unique for any lacunary sequence θ = (kr ) .
Proof. Let x = (xi ) ∈ [b
c, M ] (∆m ) ∩ [b
c, M ]θ (∆m ) and [b
c, M ]θ (∆m ) −
lim x = Lo , [b
c, M ] (∆m ) − lim x = L.Suppose that L 6= L0 .We can see that
m
m
|∆ xi+s − L|
|∆ xi+s − Lo |
1 X
1 X
|L − Lo |
M
M
≤
+
M
ρ
hr i∈I
ρ
hr i∈I
ρ
r
r
≤ lim sup
r
s
1 X
M
hr i∈I
|∆m xi+s − L|
ρ
+ 0.
r
Hence, there exists ro , such that for r > ro
m
1 X
|∆ xi+s − L|
1
|L − Lo |
M
> M
.
hr i∈I
ρ
2
ρ
r
Since [b
c, M ] (∆m ) − lim x = L, it follows that
hr
|L − Lo |
|L − Lo |
M
≥ lim inf M
≥0
0 ≥ lim sup
r
kr
ρ
ρ
r
and so limr qr = 1. Hence by Theorem 2.2., [b
c, M ]θ (∆m ) ⊂ [b
c, M ] (∆m ) and
θ
m
m
[b
c, M ] (∆ ) − lim x = Lo = [b
c, M ] (∆ ) − lim x = L. Further
m
m
n
n
1X
|L − Lo |
1X
|∆ xi+s − L|
|∆ xi+s − Lo |
M
+
M
≥M
≥0
n i=1
ρ
n i=1
ρ
ρ
|L−Lo |
and taking the limit on both sides as n → ∞, we have M
= 0, i.e., L =
ρ
Lo for any Orlicz function M. This completes the proof.
Lacunary Strongly Almost Generalized...
51
References
[1] A. Esi, Generalized difference sequence spaces defined by Orlicz functions,
General Mathematics, 17(2)(2009), 53-66.
[2] M. Et and R. Çolak, On some generalized difference sequence spaces,
Soochow J. Math., 21(4)(1995), 377-386.
[3] A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. London Math. Soc., 37(3) (1978), 508-520.
[4] M. Güngör and M. Et, Strongly almost summable sequences defined by
Orlicz functions, Indian J. Pure Appl. Math., 34(8) (2003), 1141-1151.
[5] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24(2) (1981),
169-176.
[6] M.A. Krasnoselskii and Y.B. Rutitsky, Convex functions and Orlicz
Spaces, Groningen, Netherland, (1961).
[7] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J.
Math., 10(1971), 345-355.
[8] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta
Mathematica, 80(1) (1948), 167-190.
[9] I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math.,
18(1967), 345-355.
[10] I.J. Maddox, A new type of convergence, Math. Proc. Camb. Phil. Soc.,
83(1978), 61-64.
[11] I.J. Maddox, On strong almost convergence, Math. Proc. Camb. Phil.
Soc., 85(1979), 345-350.
[12] S.D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(1994), 419-428.
[13] T.Bilgin, Some new difference sequence spaces defined by an Orlicz funtion, Filomat, 17(2003), 1-8.