Volume 2011, Article ID 592840,9pages doi:10.1155/2011/592840
Research Article
Some New Double Sequence Spaces Defined by
Orlicz Function in
n
-Normed Space
Ekrem Savas¸
Department of Mathematics, Istanbul Commerce University, Uskudar, 34672 Istanbul, Turkey
Correspondence should be addressed to Ekrem Savas¸,ekremsavas@yahoo.com
Received 1 January 2011; Accepted 17 February 2011
Academic Editor: Alberto Cabada
Copyrightq2011 Ekrem Savas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to introduce and study some new double sequence spaces with respect to an Orlicz function, and also some properties of the resulting sequence spaces were examined.
1. Introduction
We recall that the concept of a 2-normed space was first given in the works of G¨ahler1,2 as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors see, 3, 4. Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces see, e.g., 5– 9. In particular, Savas¸10combined Orlicz function and ideal convergence to define some sequence spaces using 2-norm.
In this paper, we introduce and study some new double-sequence spaces, whose elements are formn-normed spaces, using an Orlicz function, which may be considered as an extension of various sequence spaces ton-normed spaces. We begin with recalling some notations and backgrounds.
Recall in11that an Orlicz functionM:0,∞ → 0,∞is continuous, convex, and nondecreasing function such thatM0 0 andMx > 0 forx > 0, andMx → ∞ as
x → ∞.
Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary12and others. An Orlicz functionMcan always be represented in the following integral form:Mx 0xptdt, wherepis the known kernel ofM, right differential fort≥0,
If convexity of Orlicz functionMis replaced byMxy ≤Mx My, then this function is called Modulus function, which was presented and discussed by Ruckle13and Maddox14.
Remark 1.1. IfMis a convex function and M0 0, thenMλx ≤ λMxfor all λwith 0< λ <1.
Letn∈ andXbe real vector space of dimensiond, wheren≤d. Ann-norm onXis a function·, . . . ,·:X×X× · · · ×X → which satisfies the following four conditions:
ix1, x2, . . . , xn0 if and only ifx1, x2, . . . , xnare linearly dependent,
iix1, x2, . . . , xnare invariant under permutation,
iiiαx1, x2, . . . , xn|α|x1, x2, . . . , xn,α∈,
ivxx, x2, . . . , xn ≤ x, x2, . . . , xnx, x2, . . . , xn.
The pairX,·, . . . ,·is then called ann-normed space3. LetX
d d ≤ nbe equipped with then-norm, thenx
1, x2, . . . , xn−1, xnS :the volume of then-dimensional parallelepiped spanned by the vectors,x1, x2, . . . , xn−1, xnwhich may be given explicitly by the formula
x1, x2, . . . , xn−1, xnS
x1, x2 · · · x1, xn
.
. · · · .
xn, x1 · · · xn, xn
1/2
, 1.1
where ·,·denotes inner product. LetX,·, . . . ,·be ann-normed space of dimensiond≥ nand {a1, a2, . . . , an}a linearly independent set inX. Then, the function ·,·∞ onXn−1 is
defined by
x1, x2, . . . , xn−1, xn∞:max{x1, x2, . . . , xn−1, ai: i1,2, . . . , n}, 1.2
is defines ann−1norm onXwith respect to{a1, a2, . . . , an}see,15.
Definition 1.2 see 7. A sequence xk in n-normed space X,·, . . . ,· is aid to be convergent to anxinXin then-normif
lim
k→ ∞x1, x2, . . . , xn−1, xk−x0, 1.3
for everyx1, x2, . . . , xn−1 ∈X.
Definition 1.3see16. LetXbe a linear space. Then, a mapg :X → is called a paranorm
onXif it is satisfies the following conditions for allx, y∈Xandλscalar:
iigx g−x,
iiigxy≤gx gy,
iv|λn−λ| → 0n → ∞andgxn−x → 0 n → ∞implygλnxn−λx → 0n →
∞.
2. Main Results
LetX,·, . . . ,·be anyn-normed space, and letSn−XdenoteX-valued sequence spaces. ClearlySn−Xis a linear space under addition and scalar multiplication.
Definition 2.1. LetMbe an Orlicz function andX,·, . . . ,·anyn-normed space. Further, let
p pk,lbe a bounded sequence of positive real numbers. Now, we define the following new double sequence space as follows:
lM, p,·, . . . ,·:
x∈Sn−X:
∞,∞
k,l1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l
<∞, ρ >0
,
2.1
for eachz1, z2, . . . , zn−1∈X.
The following inequalities will be used throughout the paper. Letp pk,lbe a double sequence of positive real numbers with 0 < pk,l ≤supk,lpk,l H, and letD max{1,2H−1}. Then, for the factorable sequences{ak}and{bk}in the complex plane, we have as in Maddox
16
|ak,lbk,l|pk,l ≤D
|ak,l|pk,l|bk,l|pk,l
. 2.2
Theorem 2.2. lM, p,·, . . . ,·sequences space is a linear space.
Proof. Now, assume thatx, y∈l M, p,·, . . . ,·andα, β∈. Then,
∞,∞
k,l1
M xk,l ρ1
, z1, z2, . . . , zn−1 pk,l
<∞ for someρ1>0,
∞,∞
k,l1,1
M xk,l ρ2
, z1, z2, . . . , zn−1
pk,l <∞ for someρ2 >0.
Since·, . . . ,·is an-norm onX, andMis an Orlicz function, we get
∞,∞
k,l1,1
M
αxk,lβyk,l max|α|ρ1,βρ2
, z1, z2, . . . , zn−1 pk,l ≤D ∞,∞
k,l1,1
|α|
|α|ρ1βρ2
M xk,l
ρ1 , z1, z2, . . . , zn−1 pk,l
D
∞
k,l1,1
β
|α|ρ1βρ2
M yk,l ρ2
, z1, z2, . . . , zn−1 pk,l
≤DF
∞,∞
k,l1,1
M xk,l ρ1
, z1, z2, . . . , zn−1 pk,l
DF
∞
k,l1,1
M yk,l
ρ2 , z1, z2, . . . , zn−1 pk,l,
2.4
where
F max ⎡ ⎣1,
|α|
|α|ρ1βρ2 H , β
|α|ρ1βρ2
H⎤
⎦, 2.5
and this completes the proof.
Theorem 2.3. lM, p,·, . . . ,· space is a paranormed space with the paranorm defined by g :
lM, p,·, . . . ,· →
gx inf ⎧ ⎨ ⎩ρpk,l/H:
∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l1/M ∗
<∞
⎫ ⎬
⎭, 2.6
where0< pk,l≤suppk,lH,M∗max1, H.
Proof. iClearly,gθ 0 andiig−x gx.iiiLetxk,l, yk,l ∈lM, p,·, . . . ,·, then there existsρ1, ρ2>0 such that
∞,∞
k,l1,1
M xk,l ρ1
, z1, z2, . . . , zn−1 pk,l
<∞,
∞,∞
k,l1,1
M yk,l ρ2
, z1, z2, . . . , zn−1
pk,l <∞.
So, we have
M xk,lyk,l ρ1ρ2
, z1, z2, . . . , zn−1
≤M xk,l
ρ1ρ2, z1, z2, . . . , zn−1
yk,l
ρ1ρ2, z1, z2, . . . , zn−1
≤ ρ1
ρ1ρ2M
xk,l
ρ1 , z1, z2, . . . , zn−1
ρ1
ρ1ρ2
M yk,l ρ2
, z1, z2, . . . , zn−1 ,
2.8
and thus
gxyinf ⎧ ⎨ ⎩
ρ1ρ2 pk,l/H:
∞
k,l1,1
M xk,lyk,l ρ1ρ2
, z1,z2, . . . , zn−1 pk,l
1/M∗⎫⎬
⎭ ≤inf ⎧ ⎨ ⎩ ρ1 pk,l/H:
∞
k,l1,1
M xk,l ρ1
, z1, z2, . . . , zn−1 pk,l
1/M∗⎫⎬
⎭ inf ⎧ ⎨ ⎩ ρ2 pk,l/H :
∞
k1
M yk,l ρ2
, z1, z2, . . . , zn−1 pk,l
1/M∗⎫⎬
⎭.
2.9
ivNow, letλ → 0 andgxn−x → 0n → ∞. Since
gλx inf ⎧ ⎨ ⎩
ρ
|λ|
pk,l/H :
∞
k,l1,1
M λxk,l
ρ , z1, z2, . . . , zn−1
pk,l
1/M∗
<∞
⎫ ⎬ ⎭.
2.10
This gives usgλxn → 0 n → ∞.
Theorem 2.4. If0< pk,l< qk,l<∞for eachkandl, thenlM, p,·, . . . ,·⊆lM, q,·, . . . ,·.
Proof. Ifx∈lM, p,·, . . . ,·, then there exists someρ >0 such that
∞,∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l
<∞. 2.11
This implies that
M xk,l
ρ , z1, z2, . . . , zn−1
for sufficiently large values ofkandl. SinceMis nondecreasing, we are granted
∞,∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
qk,l
≤ ∞,∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l
<∞.
2.13
Thus,x∈lM, q,·, . . . ,·. This completes the proof.
The following result is a consequence of the above theorem.
Corollary 2.5. iIf0< pk,l<1for eachkandl, then
lM, p,·, . . . ,·⊆lM,·, . . . ,·, 2.14
iiIfpk,l≥1for eachkandl, then
lM,·, . . . ,·⊆lM, p,·, . . . ,·. 2.15
Theorem 2.6. u uk,l∈l∞⇒ux∈lM, p,·, . . . ,·, wherel∞ is the double space of bounded
sequences andux uk,lxk,l.
Proof. u uk,l∈l∞. Then, there exists anA >1 such that|uk,l| ≤Afor eachk,l. We want to showuk,lxk,l∈lM, p,·, . . . ,·. But
∞,∞
k,l1,1
M uk,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
pk,l
∞,∞
k,l1,1
M |uk,l| xk,l
ρ , z1, z2, . . . , zn−2, zn−1
pk,l
≤KAH
∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−2, zn−1
pk,l
,
2.16
and this completes the proof.
Theorem 2.7. LetM1andM2be Orlicz function. Then, we have
lM1, p,·, . . . ,· l
M2, p,·, . . . ,·
⊆lM1M2, p,·, . . . ,·
Proof. We have
M1M2
xk,l
ρ , z1, z2, . . . , zn−1
pk,l
M1
xk,l
ρ , z1, z2, . . . , zn−1
M2
xk,l
ρ , z1, z2, . . . , zn−1
pk,l
≤D
M1
xk,l
ρ , z1, z2, . . . , zn−1
pk,l
D
M2
xk,l
ρ , z1, z2, . . . , zn−1
pk,l
.
2.18
Letx∈lM1, p,·, . . . ,·lM2, p,·, . . . ,·; when adding the above inequality fromk, l 0,0 to∞,∞we getx∈l”M
1M2, p,·, . . . ,·and this completes the proof.
Definition 2.8 see 10. LetX be a sequence space. Then, X is called solid ifαkxk ∈ X wheneverxk∈Xfor all sequencesαkof scalars with|αk| ≤1 for allk∈ .
Definition 2.9. Let X be a sequence space. Then, X is called monotone if it contains the canonical preimages of all its step spacessee,17.
Theorem 2.10. The sequence spacelM, p,·, . . . ,·is solid.
Proof. Letxk,l∈lM, p,·, . . . ,·; that is,
∞,∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l <∞. 2.19
Letαk,lbe double sequence of scalars such that|αk,l| ≤1 for allk, l∈ × . Then, the result follows from the following inequality:
∞,∞
k,l1,1
M αk,lxk,l
ρ , z1, z2, . . . , zn−1
pk,l
≤ ∞,∞
k,l1,1
M xk,l
ρ , z1, z2, . . . , zn−1
pk,l
, 2.20
and this completes the proof.
We have the following result in view of Remark1.1and Theorem2.10.
Corollary 2.11. The sequence spacelM, p,·, . . . ,·is monotone.
Definition 2.12see18. LetA am,n,k,ldenote a four-dimensional summability method that maps the complex double sequences xinto the double-sequenceAx, where themnth term toAxis as follows:
Axm,n
∞,∞
k,l1,1
am,n,k,lxk,l. 2.21
Definition 2.13. LetA am,n,k,lbe a nonnegative matrix. LetMbe an Orlicz function andpk,l a factorable double sequence of strictly positive real numbers. Then, we define the following sequence spaces:
ω0M, A, p,·, . . . ,·
x∈Sn−1: lim m,n→ ∞,∞
∞,∞
k,l1,1
M am,n,k,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
pk,l 0
.
2.22
for each z1, z2, . . . , zn−1 ∈ X. If x−le ∈ ω0M, A, p,·, . . . ,·, then we say xis ω0M, A,
p,·, . . . ,·summable tol, wheree 1,1, . . ..
If we takeMx xandpk,l1 for allk, l, then we have
ω0A, p,·, . . . ,·
x∈Sn−1: lim m,n→ ∞
∞,∞
k,l1,1
am,n,k,lxk,l, z1, z2, . . . , zn−2, zn−10
.
2.23
Theorem 2.14. ω0M, A, p,·, . . . ,·is linear spaces.
Proof. This can be proved by using the techniques similar to those used in Theorem2.2.
Theorem 2.15. 1If0<infpk,l≤pk,l<1, then
ω0M, A, p,·, . . . ,·⊂ω0M, A,·, . . . ,·. 2.24
2If1≤pk,l≤suppk,l<∞, then
ω0M, A,·, . . . ,·⊂ω0M, A, p,·, . . . ,·. 2.25
Proof. 1Letx∈ω0M, A, p,·, . . . ,·; since 0<infpk,l≤1, we have
∞,∞
k,l1,1
M am,n,k,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
≤∞,∞
k,l1
M am,n,k,lxk
ρ , z1, z2, . . . , zn−2, zn−1
pk,l
,
2.26
and hencex∈ω0M, A,·, . . . ,·.
2Letpk,l≥1 for eachk, land supk,lpk,l<∞. Letx∈ω0M, A,·, . . . ,·. Then, for each 0< <1, there exists a positive integer such that
∞,∞
k,l1,1
M am,n,k,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
for allm, n≥ . This implies that
∞,∞
k,l1,1
M am,n,k,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
pk,l
≤ ∞
k,l1
M am,n,k,lxk,l
ρ , z1, z2, . . . , zn−2, zn−1
.
2.28
Thus,x∈ω0M, A, p,·, . . . ,·, and this completes the proof.
Acknowledgments
The author wishes to thank the referees for their careful reading of the paper and for their helpful suggestions.
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