Applied Mathematics, 2014, 5, 2602-2611 Published Online September 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.516248
Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space Mohammad Aiyub Department of Mathematics, University of Bahrain, Sakhir, Bahrain Email:
[email protected] Received 11 July 2014; revised 15 August 2014; accepted 25 August 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.
Keywords Invariant Means, Musielak-Orlicz Functions, 2-Norm Space, Lacunary Sequence
1. Introduction Let ω be the set of all sequences of real numbers ∞ , c and c0 be respectively the Banach spaces of bounded, convergent and null sequences x = ( xk ) with ( xk ) ∈ or the usual norm x = sup k xk , 1, 2,3, , the positive integers. where k ∈ = The idea of difference sequence spaces was first introduced by Kizmaz [1] and then the concept was generalized by Et and Çolak [2]. Later on Et and Esi [3] extended the difference sequence spaces to the sequence spaces:
( ) {
(
)
}
X ∆ vm = x = ( xk ) : ∆ vm x ∈ X ,
for X = ∞ , c and c0 , where v = ( vk ) be any fixed sequence of non zero complex numbers and ∆ vm xk = ∆ vm −1 xk − ∆ vm −1 xk +1 .
(
) (
)
The generalized difference operator has the following binomial representation, How to cite this paper: Aiyub, M. (2014) Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space. Applied Mathematics, 5, 2602-2611. http://dx.doi.org/10.4236/am.2014.516248
m i m ∆ vm xk = ∑ ( −1) vk + i xk + i , for all k ∈ . i =0 i
M. Aiyub
The sequence spaces ∆ vm ( ∞ ) , ∆ vm ( c ) and ∆ vm ( c0 ) are Banach spaces normed by m
∑ vi xi
= x∆
i =1
+ ∆ vm x . ∞
The concept of 2-normed space was initially introduces by Gahler [4] as an interesting linear generalization of normed linear space which was subsequently studied by many others [5] [6]). Recently a lot of activities have started to study summability, sequence spaces and related topics in these linear spaces [7] [8]). Let X be a real vector space of dimension d , where 2 ≤ d < ∞ . A 2-norme on X is a function .,. : X × X → which satisfies: 1) x, y = 0 if and only if x and y are linearly dependent, 2) x, y = y, x , 3) = α x, y α x, y , α ∈ , 4)
x, y + z ≤ x, y + x, z .
The pair ( X , .,. ) is called a 2-normed space. As an example of a 2-normed space we may take X = 2 being equiped with the 2-norm x, y = the area os paralelogram spaned by the vectors x and y , which may be given explicitly by the formula x1 , x2
E
x = abs 11 x21
x22 . x22
Then clearly ( X , .,. ) is 2-normed space. Recall that ( X , .,. ) is a 2-Banach space if every cauchy sequence in X is convergent to some x ∈ X . Let σ be a mapping of the positive integers into itself. A continuous linear functional φ on ∞ is said to be an invariant mean or σ -mean if and only if 1) φ ( x ) ≥ 0 , when the sequence x = ( xn ) has, xn ≥ 0 for all n , 2) φ ( e= e (1,1,1, ) , ) 1,=
(
)
3) φ xσ ( n ) = φ ( x ) for all x ∈ ∞ .
If x = ( xk ) , where= Tx
Txk ) (=
( x ( ) ) . It can be shown that σ k
{
}
Vσ = x ∈ ∞ : lim tkn ( x ) = l , uniformly in n k
l= σ − lim x, where tkn ( x ) =
xn + xσ 1 ( n ) + x
σ 2( n )
k +1
++ x
σ k(n)
[9].
In the case σ is the translation mapping n → n + 1 , σ -mean is often called a Banach limit and Vσ the set of bounded sequences of all whose invariant means are equal is the set of almost convergent sequence [10]. By Lacunary sequence = θ (= kr ) , r 0,1, 2, where k0 = 0 we mean an increasing sequence of non negative integers h= I r [ kr −1 − kr ] k − k → ( r r −1 ) ∞ ( r → ∞ ) . The intervals determined by θ are denoted by = r kr and the ratio will be denoted by qr . The space of lacunary strongly convergent sequence Nθ was dekr −1 fined by Freedman et al. [11] as follows: Nθ = x =
( xi ) : rlim →∞
1 hr
∑
k = Ir
xk − l = 0 for some .
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 → ∞.
2603
M. Aiyub
It is well known that if M is convex function and M ( 0 ) = 0 then M ( λ x ) ≤ λ M ( x ) , for all λ with 0 ≤ λ ≤ 1. Lindenstrauss and Tzafriri [12] used the idea of Orlicz function and defined the sequence space which was called an Orlicz sequence space M such as
M= = x
∞
xk < ∞, for some ρ > 0 ρ
( xk ) : ∑M k =1
which was a Banach space with the norm ∞ x x => inf ρ 0 : ∑M k ≤ 1 k =1 ρ
which was called an Orlicz sequence space. The M was closely related to the space p which was an Orlicz p sequence space with M ( t ) = t , for 1 ≤ p < ∞ . Later the Orlicz sequence spaces were investigated by Prashar and Choudhry [13], Maddox [14], Tripathy et al. [15]-[17] and many others. A sequence of function M = ( M k ) of Orlicz function is called a Musielak-Orlicz function [18] [19]. Also a Musielak-Orlicz function Φ = ( Φ k ) is called complementary function of a Musielak-Orlicz function M if
Φ k (= t ) sup { t s − M k ( s ) : s ≥ 0} , for= k 1, 2,3, . For a given Musielak-Orlicz function M , the Musielak-Orlicz sequence space lM and its subspaces M are defined as follow:
lM=
{ x=
M=
xk ∈ ω : I M ( cx ) < ∞, for some c > 0}
{ x=
xk ∈ ω : I M ( cx ) < ∞, for all c > 0}
where I M is a convex modular defined by
I M (= x)
∞
, x ( xk ) ∈ lM . ∑M k ( xk )= k =1
We consider lM equipped with the Luxemburg norm x = inf k > 0 : I M
x ≤ 1 k
or equipped with the Orlicz norm 0 1 inf (1 + I M ( kx ) ) : k > 0 . x = k
The main purpose of this paper is to introduce the following sequence spaces and examine some properties of the resulting sequence spaces. Let M = ( M k ) be a Musielak-Orlicz function, ( X , .,. ) is called a 2-normed space. Let p = ( pk ) be any sequences of positive real numbers, for all k ∈ and u = ( uk ) such that uk ≠ 0 ( k = 1, 2,3, ) . Let s be any real number such that s ≥ 0 . By ω ( 2 − X ) we denote the space of all sequences defined over ( X , .,. ) . Then we define the following sequence spaces: ∞
( )
ωθ , M , p, u , s, .,. ∆ vm σ x = =
1 ( xk ) ∈ ω ( 2 − X ) : sup r , n hr
(
)
tkn ∆ vm xk −s ,z ∑ k uk M k ρ k ∈I r
2604
pk
< ∞ ρ > 0, s ≥ 0
M. Aiyub
( )
ωθ , M , p, u , s, .,. ∆ vm σ
1 = x =( xk ) ∈ ω ( 2 − X ) :: lim r h r 0
(
)
pk
=0 for some l , ρ > 0, s ≥ 0
(
)
pk
=0 ρ > 0, s ≥ 0
tkn ∆ vm xk ,z ∑ k uk M k ρ k ∈I r −s
( )
ωθ , M , p, u , s, .,. ∆ vm σ
1 = x =( xk ) ∈ ω ( 2 − X ) :: lim r h r
tkn ∆ vm xk ,z ∑ k − s uk M k ρ k ∈I r
Definition 1. A sequence space E is said to be solid or normal if (α k xk ) ∈ E whenever ( xk ) ∈ E and for all sequences of scalar (α k ) with α k ≤ 1 [20]. Definition 2. A sequence space E is said to be monotone if it contains the canonical pre-images of all its steps spaces, [20]. Definition 3. If X is a Banach space normed by . , then ∆ m ( X ) is also Banach space normed by
x= ∆
(
m
∑ xk
)
+ f ∆m x .
k =1
Remark 1. The following inequality will be used throughout the paper. Let p = ( pk ) be a positive sequence G , D = max 1, 2G −1 . Then for all ak , bk ∈ for all k ∈ . We of real numbers with 0 < pk ≤ sup pk = have
(
ak + bk
pk
)
(
≤ D ak
pk
+ bk
pk
).
(1)
2. Main Results Theorem 1. Let M = ( M k ) be a Musielak-Orlicz function, p = ( pk ) be a bounded sequence of positive real ∞
( )
( )
number and θ = ( kr ) be a lacunary sequence. Then ωθ , M , p, u , s, .,. ∆ vm , ωθ , M , p, u , s, .,. ∆ vm σ σ 0
( )
and ωθ , M , p, u , s, .,. ∆ vm σ
are linear spaces over the field of complex numbers.
( )
Proof 1. Let x = ( yk ) ∈ ωθ , M , p, u, s, .,. σ ∆ vm ( xk ) , y = need to find some ρ3 such that, 0
(
and α , β ∈ C . In order to prove the result we
)
t ∆ m (α x + β y ) nk v k k 1 −s lim ∑ uk k M k ,z r →∞ h ρ3 r k ∈I r
Since
pk
= 0, uniformly in n.
( xk ) , ( yk ) ∈ ωθ , M , p, u, s, .,. σ ( ∆ vm ) , there exist positive ρ1 , ρ2 0
1 lim r →∞ h r
∑ uk k
k ∈I r
−s
t ∆m x M kn v k , z k ρ
(
)
pk
(
)
pk
such that
= 0 uniformly in n
and
1 lim r →∞ h r
tkn ∆ vm xk −s ,z ∑ uk k M k ρ k ∈I r
Define ρ3 = max ( 2 α ρ1 , 2 β ρ 2 ) . Since
(Mk )
= 0 uniformly in n.
is non decreasing and convex
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M. Aiyub
(
)
tnk ∆ vm (α xk + β yk ) 1 −s u k M ,z ∑ k k hr k∈I r ρ3 1 ≤ hr 1 ≤ hr
∑ uk k
−s
∑ uk k
−s
k ∈I r
k ∈I r
(
)
pk
(
)
t ∆ m (α xk ) tnk ∆ vm ( β yk ) M nk v + z , ,z k ρ3 ρ3
(
)
(
)
(
)
t ∆m x t ∆m y M nk v k , z + nk v k , z k ρ1 ρ2
tnk ∆ vm xk D −s ≤ ,z ∑ uk k M k ρ hr k∈I r → 0, as r → ∞, uniformly in n.
pk
pk
pk
(
)
tnk ∆ vm yk D −s + ∑ uk k M k ,z ρ hr k∈I r
pk
( )
So that (α xk ) + ( β yk ) ∈ ωθ , M , p, u , s, .,. ∆ vm . This completes the proof. Similarly, we can prove that σ 0
( )
ωθ , M , p, u , s, .,. ∆ vm and ωθ , M , p, u , s, .,. ( ∆ vm ) are linear spaces. σ σ Theorem 2. Let M = ( M k ) be a Musielak-Orlicz function, p = ( pk ) be a bounded sequence of positive ∞
( )
real number and θ = ( kr ) be a lacunary sequence. Then ωθ , M , p, u , s, .,. ∆ vm is a topological linear σ space totalparanormed by pk 1 H tnk ∆ vm xk m p H 1 −s r , z ≤ 1 for some ρ , r = 1, 2, g∆ ( x ) = ∑ xk + inf ρ : h ∑ uk k M k ρ k 1 = r k ∈I r 0
(
)
Proof 2. Clearly g ∆ (= x ) g ∆ ( − x ) . Since M k ( 0 ) = 0 , for all k ∈ . we get g ∆ (θ ) = 0 , for x = θ . Let
( )
x = ( xk ) , y = ( yk ) ∈ ωθ , M , p, u, s, .,. σ ∆ vm 0
and let us choose ρ1 > 0 and ρ 2 > 0 such that
(
)
pk
(
)
pk
tnk ∆ vm ( xk ) −1 −s sup hr ∑ uk k M k ,z ρ1 r k ∈I r
≤1 r = 1, 2,3,
and
tnk ∆ vm ( yk ) sup hr−1 ∑ uk k − s M k ,z ρ2 r k ∈I r
≤1 r = 1, 2,3,
Let ρ= ρ1 + ρ 2 , then we have
(
)
tnk ∆ vm ( xk + yk ) −1 −s sup hr ∑ uk k M k ,z ρ r k ∈I r
(
pk
)
tnk ∆ vm ( xk ) −1 −s ≤ sup hr ∑ uk k M k ,z ρ1 + ρ 2 r ρ1 k ∈I r
ρ1
(
)
tnk ∆ vm ( yk ) + sup hr−1 ∑ uk k − s M k ,z ρ1 + ρ 2 r ρ2 k ∈I r
ρ1
2606
pk
pk
≤ 1.
M. Aiyub
Since ρ > 0 , we have g∆ ( x + y ) =
m
∑
= k 1
m
≤∑
= k 1
pk 1 H m ∆ + t x y pr H 1 nk v ( k k) −s ≤ 1 for some ρ > 0, r= 1, 2, xk + yk + inf ρ uk k M k : , z hr k∑ ρ ∈I r H 1 pk tnk ∆ vm ( xk ) pr H 1 −s 1, 2, ≤ 1 for some ρ1 > 0, r = xk + inf ρ1 uk k M k : , z hr k∑ ρ1 ∈I r
m
+∑
= k 1
yk + inf ρ 2pr
1 : hr
H
t ∆m ( y ) ∑ uk k − s M k nk ρv k , z k ∈I r 2
pk
1H
≤ 1 for some ρ 2 > 0, r = 1, 2, .
g∆ ( x + y ) ≤ g∆ ( x ) + g∆ ( y ) .
Finally, we prove that the scalar multiplication is continuous. Let λ be a given non zero scalar in . Then the continuity of the product follows from the following expression. pr inf g ∆ ( λ x= x λ + ) ∑ k ρ k 1 = m
1 : hr
H
p λ ∑ xk + inf ( λ ζ ) r = k 1 = m
where = ζ
ρ > 0. Since λ λ
pr
g ∆ ( λ x ) = max (1, λ pr + inf ρ
H
1 : hr
H
∑ uk k
k ∈I r
1 : hr
t ∆m ( x ) ∑ uk k − s M k nk vζ k , z k ∈I r
≤ max (1, λ
)
t ∆ m ( λ xk ) M k nk v ,z ρ
−s
)
sup pr
pk
r 1, 2, ≤ 1 for some ρ > 0, =
1H
pk
1H
≤ 1 for some ζ > 0, r = 1, 2,
,
sup pr
(
)
tnk ∆ vm xk −s ,z ∑ uk k M k ρ k ∈I r
pk
1H
≤ 1, for some ρ > 0, r = 1, 2, .
This completes the proof of this theorem. Theorem 3. Let M = ( M k ) be a Musielak-Orlicz function, p = ( pk ) be a bounded sequence of positive real number and θ = ( kr ) be a lacunary sequence. Then ∞
( )
( ) ( ∆ ) ⊂ ω
( ) ( ∆ ) is obvious. Let 0
ωθ , M , p, u , s, .,. ∆ vm ⊂ ωθ , M , p, u , s, .,. ∆ vm ⊂ ωθ , M , p, u , s, .,. ∆ vm . σ σ σ 0
m θ m Proof 3. The inclusion ωθ , M , p, u , s, .,. , M , p, u , s, .,. v v σ σ xk ∈ ωθ , M , p, u , s, .,. ∆ vm . Then there exists some positive number ρ1 such that
σ
( )
1 lim r →∞ h r
∑ uk k
k ∈I r
−s
(
)
t ∆ m x − le M nk v k ,z k ρ1
pk
→0
as r → ∞ , uniformly in n . Define ρ = 2 ρ1 . Since M k is non decreasing and convex for all k ∈ , we have
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M. Aiyub
1 hr
∑ uk k
−s
k ∈I r
(
)
t ∆m x M nk v k , z k ρ
pk
(
)
tnk ∆ vm xk − le D −s ≤ u k M ,z ∑ k k hr k∈I r ρ1 le D + ∑ M k ,z hr k∈I r ρ1
(
(
)
where G = sup k (= pk ) , D max 1, 2G − 1 Thus xk ∈ ωθ , M , p, u , s, .,. ∆ vm . σ
( )
pk
pk
)
t ∆ m x − le D M nk v k ≤ ,z ∑ k hr k∈I r ρ1 le + D max 1, M ,z ρ 1
G
pk
by (1).
Theorem 4. Let M = ( M k ) be a Musielak-Orlicz functions. If sup k M k ( z ) ∞ ωθ , M , p, u , s, .,. ∆ vm ⊂ ωθ , M , p, u , s, .,. ∆ vm . σ
( )
σ
( )
Proof 4. Let xk ∈ ωθ , M , p, u , s, .,. ∆ vm σ
( )
(
)
)
(
t ∆ m x − le D M nk v k ≤ ,z ∑ k hr k∈I r ρ le D + ∑ M k ,z ρ hr k∈I r
pk
pk
pk
.
Since sup k M ( z ) < ∞ , we can take the sup k M ( z ) xk ∈ ωθ , M , p, u , s, .,. ∆ vm . pk
σ
< ∞ for all z > 0 , then
by using (1), we have
tnk ∆ vm xk −s ,z ∑ uk k M k ρ k ∈I r
1 hr
pk
pk
( )
= K . Hence we can get
This complete the proof. Theorem 5. Let m ≥ 1 be fixed integer. Then the following statements are equivalent: ∞
( ) ( ∆ ) ⊂ ω ( ∆ ) ⊂ ω
∞
( ) (∆ ) , ( ∆ ).
1) ωθ , M , p, u , s, .,. ∆ vm −1 ⊂ ωθ , M , p, u , s, .,. ∆ vm , σ σ θ
2) ω , M , p, u , s, .,. σ 0
3) ωθ , M , p, u , s, .,. σ
m −1 v
θ
, M , p, u , s, .,. σ
m −1 v
θ
, M , p, u , s, .,. σ
0
0
(
m v
m v
)
Proof 5. Let xk ∈ ωθ , M , p, u , s, .,. ∆ vm −1 . Then there exist ρ > 0 such that σ
1 lim r →∞ h r
(
)
tnk ∆ vm xk ,z ∑ uk k − s M k ρ k ∈I r
Since M k is non decreasing and convex, we have
2608
pk
→ 0.
M. Aiyub
1 hr
∑ uk k
(
)
t ∆m x M nk v k , z k 2ρ
−s
k ∈I r
pk
(
)
1 + hr
∑ uk k
t ∆ m −1 x − ∆ m −1 x nk v k k +1 1 −s ,z = u k M ∑ k k 2 ρ hr k∈I r 1 ≤ hr
∑ uk k
−s
k ∈I r
(
)
t ∆ m −1 x M nk v k , z k 2ρ
pk
(
pk
)
tnk ∆ vm −1 xk D −s ≤ ,z u k M ∑ k k hr k∈I r ρ Taking lim r →∞ , we have
1 hr 0
(
∑ uk k
−s
k ∈I r
pk
(
)
(
)
.
t ∆ m −1 x M nk v k +1 , z k 2ρ
−s
k ∈I r
tnk ∆ vm −1 xk +1 D + ∑ uk k − s M k ,z hr k∈I r ρ
(
)
t ∆m x M nk v k , z k ρ
pk
pk
pk
= 0,
)
i.e. xk ∈ ωθ , M , p, u , s, .,. ∆ vm −1 . The rest of these cases can be proved in similar way. σ Theorem 6. Let M = ( M k ) and T = (Tk ) be two Musielak-Orlicz functions. Then we have ∞
( ) ( ∆ ) ω ( ∆ ) ω
∞
( ) ( ∆ ) ⊂ ω ( ∆ ) ⊂ ω
∞
( ) ( ∆ ). ( ∆ ).
1) ωθ , M , p, u , s, .,. ∆ vm ωθ , T , p, u , s, .,. ∆ vm ⊂ ωθ , M + T , p, u , s, .,. ∆ vm . σ σ σ θ
2) ω , M , p, u , s, .,. σ 0
3) ωθ , M , p, u , s, .,. σ
m v
θ
, T , p, u , s, .,. σ
m v
θ
, T , p, u , s, .,. σ
0
∞
m v
θ
, M + T , p, u , s, .,. σ
m v
θ
, M + T , p, u , s, .,. σ
( )
0
∞
m v
m v
( )
Proof 6. Let xk ∈ ωθ , M , p, u , s, .,. ∆ vm ωθ , T , p, u , s, .,. ∆ vm . Then σ σ
(
)
(
)
(
)
1 sup r , n hr
tnk ∆ vm xk −s ,z ∑ uk k M k ρ k ∈I r
1 sup r , n hr
t ∆m x nk v k −s ,z u k ∑ k Tk ρ k ∈I r
pk
