# A-Sequence Spaces in 2-Normed Space Defined by Ideal Convergence and an Orlicz Function

Authors: E. Savaş
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study some newA-sequence spaces using ideal convergence and an Orlicz function in 2-normed space and we give some relations
related to these sequence spaces.


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## Body

## 1. Introduction

LetX and Y be two nonempty subsets of the space w of complex sequences. Let A=(ank),(n,k=1,2,…) be an infinite matrix of complex numbers. We write Ax=(An(x)) if An(x)=∑k=1∞ankxk converges for each n. If x=(xk)∈X⇒Ax=(An(x))∈Y  we say that A defines a (matrix) transformation from X to Y, and we denote it by A:X→Y.The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence. More applications of ideals can be seen in [2–5].The concept of 2-normed space was initially introduced by Gähler [6] as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [7, 8]). Recently a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, [9–12]).Let(X,∥·∥) be a normed space. Recall that a sequence (xn) of elements of X is called  statistically convergent to x∈X if the set A(ɛ)={n∈ℕ:∥xn-x∥≥ɛ} has natural density zero for each ɛ>0.A familyℐ⊂2Y of subsets a nonempty set Y is said to be an ideal in Y if(i)
A,B∈ℐ imply A∪B∈ℐ;(ii)
A∈ℐ,B⊂A imply B∈ℐ, while an admissible ideal ℐ of Y further satisfies {x}∈ℐ for each x∈Y, (see [7, 13]).Givenℐ⊂2ℕ a nontrivial ideal in ℕ. The sequence (xn)n∈ℕ in X is said to be ℐ-convergent to x∈X, if for each ɛ>0 the set A(ɛ)={n∈ℕ:∥xn-x∥≥ɛ} belongs to ℐ, (see, [1, 3]).LetX be a real vector space of dimension d, where 2≤d<∞. A 2-norm on X is a function ∥·,·∥:X×X→ℝ which satisfies(i)
∥x,y∥=0 if and only if x and y are linearly dependent;(ii)
∥x,y∥=∥y,x∥;(iii)
∥αx,y∥=|α|∥x,y∥,
α∈ℝ;(iv)
∥x,y+z∥≤∥x,y∥+∥x,z∥.The pair(X,∥·,·∥) is then called a 2-normed space [7]. As an example of a 2-normed space we may take X=ℝ2 being equipped with the 2-norm ∥x,y∥:= the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula
(1.1)‖x1,x2‖E=abs(|x11x12x21x22|).
Recall that (X,∥·,·∥) is a 2-Banach space if every Cauchy sequence in X is convergent to some x in X.Recall in [14] that an Orlicz function M:[0,∞)→[0,∞) is a continuous, convex, nondecreasing function such that M(0)=0 and M(x)>0 for x>0, and M(x)→∞ as x→∞.Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary [15] and others [16, 17].If convexity of Orlicz functionM is replaced by M(x+y)≤M(x)+M(y) then this function is called modulus function, which was presented and discussed by Ruckle [18] and Maddox [19]. It should be mentioned that notable works involving Orlicz function and modulus function were done in [16, 18–23].In this article, we define some new sequence spaces in 2-normed spaces by using Orlicz function, infinite matrix, generalized difference sequences, and ideals. We introduce and examine certain new sequence spaces using the above tools as also the 2-norm.


## 2. Main Results

LetI be an admissible ideal of ℕ, M be an Orlicz function, (X,∥·,·∥)be a 2-normed space, and A=(an,k) be a nonnegative matrix method. Further, let p=(pk)be a bounded sequence of positive real numbers. By S(2-X), we denote the space of all sequences defined over (X,∥·,·∥).Now we define the following sequence spaces: (2.1)WI(M,Δm,p,‖,⋅,‖)={x∈S(2-X):∀ɛ>0{n∈N:∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]pk≥ɛ}∈Iforsomeρ>0,L∈Xandeachz∈X},W0I(A,M,Δm,p,‖,⋅,‖)={x∈S(2-X):∀ɛ>0{n∈N:∑k=1∞ank[M(‖Δmxkρ,z‖)]pk≥ɛ}∈Iforsomeρ>0,andeachz∈X},W∞(A,M,Δm,p,‖,⋅,‖)={x∈S(2-X):∃K>0s.t.sup⁡n∈N∑k=1∞ank[M(‖Δmxkρ,z‖)]pk≤Kforsomeρ>0,andeachz∈X},W∞I(A,M,Δm,p,‖,⋅,‖)={x∈S(2-X):∃K>0,s.t.{n∈N:∑k=1∞ank[M(‖Δmxkρ,z‖)]pk≥K}∈Iforsomeρ>0,andeachz∈Xn∈N:∑k=1∞ank[M(Δmxkρ,z)]pk≥K},
where  Δmxk=Δm-1xk-Δm-1xk+1.Let us consider a few special cases of the above sets.(1)
IfM(x)=x, for all x∈[0,∞), then the above classes of sequences are denoted by WI(A,Δm,p,∥,·,∥),W0I(A,Δm,p,∥,·,∥),W∞(A,Δm,p,∥,·,∥), and W∞I(A,Δm,p,∥,·,∥), respectively.(2)
Ifpk=1 for all k∈N, then we denote the above classes of sequences by WI(A,M,Δm,∥,·,∥), W0I(A,Δm,∥,·,∥), W∞(A,Δm,∥,·,∥), and W∞I(A,Δm,∥,·,∥), respectively.(3)
IfM(x)=x, for all x∈[0,∞), and pk=1 for all k∈N, then we denote the above spaces by WI(A,Δm,∥,·,∥), W0I(A,Δm,∥,·,∥), W∞(A,Δm,∥,·,∥), and W∞I(A,Δm,∥,·,∥), respectively.(4)
If we takeA=(ank) as
(2.2)ank={1n,ifn≥k,0,otherwise,
then the above classes of sequences are denoted by WI(C,M,Δm,p,∥,·,∥), W0I(C,M,Δm,p,∥,·,∥), W∞(C,M,Δm,p,∥,·,∥), and W∞I(C,M,Δm,p,∥,·,∥) respectively, which were defined and studied by Savaş [24](5)
If we takeA=(ank) is a de la Vallée poussin mean, that is,
(2.3)ank={1λn,ifk∈In=[n-λn+1,n],0,otherwise,
where (λn) is a nondecreasing sequence of positive numbers tending to ∞ and λn+1≤λn+1,λ1=1, then the above classes of sequences are denoted by WI(M,Δm,λ,p,∥,·,∥),W0I(M,Δm,λ,p,∥,·,∥),W∞(M,Δm,λ,p,∥,·,∥), andW∞I(M,Δm,λ,p,∥,·,∥).(6)
By a lacunaryθ=(kr);r=0,1,2,… where k0=0, we will mean an increasing sequence of nonnegative integers with kr-kr-1 as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr] and hr=kr-kr-1. As a final illustration let
(2.4)ank={1hr,ifkr-1<k≤kr,0,otherwise.
Then we denote the above classes of sequences by WI(M,Δm,θ,p,∥,·,∥), W0I(M,Δm,θ,p,∥,·,∥), W∞(M,Δm,θ,p,∥,·,∥), andW∞I(M,Δm,θ,p,∥,·,∥).The following well-known inequality (see [25, p. 190]) will be used in the study.If(2.5)0≤pk≤sup⁡⁡pk=H,D=max⁡(1,2H-1),
then
(2.6)|ak+bk|pk≤D{|ak|pk+|bk|pk},
for all kand ak,bk∈ℂ. Also |a|pk≤max(1,|a|H) for all a∈ℂ.Theorem 2.1.
WI(A,M,Δm,p,∥,·,∥),  W0I(A,M,Δm,p,∥,·,∥),  and W∞I(A,M,Δm,p,∥,·,∥) are linear spaces.Proof.
We will prove the assertion forW0I(A,M,Δm,p,∥,·,∥) only, and the others can be proved similarly. Assume that x,y∈W0I(A,M,Δm,p,∥,·,∥)and α,β∈ℝ.In order to prove the result we need to find some ρ3 such that
(2.7){n∈N:∑k=1∞ank[M(‖αΔmxk+βΔmxkρ3,z‖)]pk≥ɛ}∈Iforsomeρ3>0.
Since x,y∈W0I(A,M,Δm,p,∥,·,∥), there exist some positive ρ1 and ρ2 such that
(2.8){n∈N:∑k=1∞ank[M(‖Δmxkρ1,z‖)]pk≥ɛ}∈Iforsomeρ1>0,{n∈N:∑k=1∞ank[M(‖Δmxkρ2,z‖)]pk≥ɛ}∈Iforsomeρ2>0.
Define ρ3=max(2|α|ρ1,2|β|ρ2). Since M is nondecreasing and convex and also ∥,·,∥is a 2-norm,Δm is linear
(2.9)∑k=1∞ank[M(‖Δm(αxk+βyk)ρ3,z‖)]pk≤∑k=1∞ank[M(‖αΔmxkρ3,z‖+‖βΔmxkρ3,z‖)]pk≤∑k=1∞ank12pk[M(‖Δmxkρ1,z‖+‖Δmxkρ2,z‖)]pk≤∑k=1∞ank[M(‖Δmxkρ1,z‖+‖Δmxkρ2,z‖)]pk≤D∑k=1∞ank[M(‖Δmxkρ1,z‖)]pk+D∑k=1∞ank[M(‖Δmxkρ2,z‖)]pk,
where D=max(1,2H-1). From the above inequality we get
(2.10){n∈N:∑k=1∞ank[M(‖Δm(αxk+βyk)ρ3,z‖)]pk≥ɛ}⊆{n∈N:D∑k=1∞ank[M(‖Δmxkρ1,z‖)]pk≥ɛ2}∪{n∈N:D∑k=1∞ank[M(‖Δmykρ2,z‖)]pk≥ɛ2}.
Two sets on the right-hand side belong to I, and this completes the proof.It is also easy to verify that the spaceW∞(A,M,Δm,p,∥,·,∥) is also a linear space and moreover we have the following.Theorem 2.2.
For any fixedn∈ℕ, W∞(A,M,Δm,p,∥,·,∥) is paranormed space with respect to the paranorm defined by
(2.11)gn(x)=inf⁡z∈X{ρpn/H:(∑k=1∞ank[M(‖Δmxkρ,z‖)]pk)1/H≤1,∀z∈X}.Proof.
The proof is parallel to the proof of the Theorem 2 in [24] and so is omitted.Theorem 2.3.
LetX(A,Δm-1) stand forW0I(A,Δm-1,M,p,∥,·,∥), WI(A,Δm-1,M,p,∥,·,∥), or W∞I(A,Δm-1,M,p,∥,·,∥) and m≥1. Then the inclusion X(A,Δm-1)⊂X(A,Δm) is strict. In general X(A,Δi)⊂X(A,Δm) for all i=1,2,3,…,m-1 and the inclusion is strict.Proof.
We shall give the proof forW0I(A,Δm-1,M,p,∥,·,∥) only. It can be proved in a similar way for W∞I(A,Δm-1,M,p,∥,·,∥), and WI(A,Δm-1,M,p,∥,·,∥). Let x=(xk)∈W0I(A,Δm-1,M,p,∥,·,∥). Then given ɛ>0 we have
(2.12){n∈N:∑k=1∞ank[M(‖Δm-1xkρ,z‖)]pk≥ɛ}∈Iforsomeρ>0.
SinceM is nondecreasing and convex it follows that
(2.13)∑k=1∞ank[M(‖Δmxk2ρ,z‖)]pk=∑k=1∞ank[M(‖Δm-1xk+1-Δm-1xk2ρ,z‖)]pk≤D∑k=1∞ank([12M(‖Δm-1xk+1ρ,z‖)]pk+[12M(‖Δm-1xkρ,z‖)]pk)≤D∑k=1∞ank([M(‖Δm-1xk+1ρ,z‖)]pk+[M(‖Δm-1xkρ,z‖)]pk).
Hence we have
(2.14){n∈N:∑k=1∞ank[M(‖Δmxk2ρ,z‖)]pk≥ɛ}⊆{n∈N:D∑k=1∞ank[M(‖Δm-1xk+1ρ,z‖)]pk≥ɛ2}∪{n∈N:D∑k=1∞ank[M(‖Δm-1xkρ,z‖)]pk≥ɛ2}.
Since the set on the right hand side belongs to I,so does the left hand side. The inclusion is strict as the sequence x=(kr), for example, belongs to W0I(Δm,M,∥,·,∥) but does not belong to W0I(Δm-1,M,∥,·,∥) for M(x)=x, A=(ank)=(C,1) Cesàro matrix and pk=1 for all k.Theorem 2.4.
(i) Let0<infpk≤pk≤1. Then WI(A,Δm,M,p,∥,·,∥)⊂WI(A,Δm,M,∥,·,∥).
(ii)1<pk≤suppk≤∞. Then WI(A,Δm,M,∥,·,∥)⊂WI(A,Δm,M,p∥,·,∥).Proof.
(i) Let(xk)∈WI(A,M,Δm,p,∥,·,∥). Since 0<infpk≤pk≤1, we have
(2.15)∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]≤∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]pk.
So
(2.16){n∈N:∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]≥ɛ}⊆{n∈N:∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]pk≥ɛ}∈I.
(ii) Letpk≥1 for each k, and suppk≤∞. Let (xk)∈WI(A,M,Δm,p,∥,·,∥). Then for each 0<ɛ<1 there exists a positive integer N such that
(2.17)∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]≤ɛ<1,
for all n≥N. This implies that
(2.18)∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]pk≤∑k=1∞ank[M(‖Δmxk-Lρ,z‖)].
So we have
(2.19){n∈N:∑k=1∞ank[M(‖Δmxk-Lρ,z‖)]pk≥ɛ}⊆{n∈N:∑k=1∞ank[(M‖Δmxk-Lρ,z‖)]≥ɛ}∈I.
This completes the proof.The following corollary follows immediately from the above theorem.Corollary 2.5.
LetA=(C,1) Cesàro matrix and let M be an Orlicz function.
(1) If0<infpk≤pk<1, then WI(Δm,M,p,∥,·,∥)⊂WI(Δm,M,∥,·,∥).
(2) If1≤pk≤suppk<∞, then WI(Δm,M,∥,·,∥)⊂WI(Δm,M,p∥,·,∥).Definition 2.6.
LetX be a sequence space. Then X is called solid if (αkxk)∈X whenever (xk)∈X for all sequences (αk) of scalars with |αk|≤1 for all k∈N.Theorem 2.7.
The sequence spacesW0I(A,M,Δm,p,∥,·,∥) and W∞I(A,M,Δm,p,∥,·,∥) are solid.Proof.
We give the proof forW0I(A,M,Δm,p,∥,·,∥) only. Let (xk)∈W0I(A,M,Δm,p,∥,·,∥), and let (αk) be a sequence of scalars such that |αk|≤1 for all k∈N. Then we have
(2.20){n∈N:∑k=1∞ank[M(‖Δm(αkxk)ρ,z‖)]pk≥ɛ}⊆{n∈N:C∑k=1∞ank[(M‖Δmxkρ,z‖)]pk≥ɛ}∈I,
where C=maxk{1,|αk|H}. Hence (αkxk)∈W0I(A,M,Δm,p,∥,·,∥) for all sequences of scalars (αk) with |αk|≤1 for all k∈N whenever (xk)∈W0I(A,M,Δm,p,∥,·,∥).Remark 2.8.
In general it is difficult to predict the solidity ofW0I(A,M,Δm,p,∥,·,∥) and W∞I(A,M,Δm,p,∥,·,∥) when m>0. For this, consider the following example.Example 2.9.
Letm=2,pk=1 for all k, A=(C,1) Cesàro matrix and M(x)=x. Then (xk)=(k)∈W0I(M,Δ2,p,∥,·,∥) but (αkxk)∉W0I(M,Δ2,p,∥,·,∥) when αk=(-1)k for all k∈N. Hence W0I(M,Δ2,p,∥,·,∥) is not solid.



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*Source: 741382-2011-06-30.xml*