# New Difference Sequence Spaces Defined by Musielak-Orlicz Function

Authors: M. Mursaleen; Sunil K. Sharma; S. A. Mohiuddine; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce new sequence spaces by using Musielak-Orlicz function and a generalizedB∧μ-difference operator on n-normed space. Some topological properties and inclusion relations are also examined.


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## Body

## 1. Introduction and Preliminaries

The notion of the difference sequence space was introduced by Kızmaz [1]. It was further generalized by Et and Çolak [2] as follows: Z(Δμ)={x=(xk)∈ω:(Δμxk)∈z} for z=l∞,c, and c0, where μ is a nonnegative integer and
(1)Δμxk=Δμ-1xk-Δμ-1xk+1,Δ0xk=xk∀k∈N
or equivalent to the following binomial representation:
(2)Δμxk=∑v=0μ‍(-1)v(μv)xk+v.
These sequence spaces were generalized by Et and Basarir [3] taking z=l∞(p), c(p), and c0(p).Dutta [4] introduced the following difference sequence spaces using a new difference operator:
(3)Z(Δ(η))={x=(xk)∈ω:Δ(η)x∈z}forz=l∞,c,c0,
where Δ(η)x=(Δ(η)xk)=(xk-xk-η) for all k,η∈N.In [5], Dutta introduced the sequence spaces c¯(∥·,·∥,Δ(η)μ,p), c0¯(∥·,·∥,Δ(η)μ,p), l∞(∥·,·∥,Δ(η)μ,p), m(∥·,·∥,Δ(η)μ,p), and m0(∥·,·∥,Δ(η)μ,p), where η,μ∈N and Δ(η)μxk=(Δ(η)μxk)=(Δ(η)μ-1xk-Δ(η)μ-1xk-η) and Δ(η)0xk=xk for all k,η∈N, which is equivalent to the following binomial representation:
(4)Δ(η)μxk=∑v=0μ‍(-1)v(μv)xk-ηv.
The difference sequence spaces have been studied by authors [6–14] and references therein. Başar and Altay [15] introduced the generalized difference matrix B=(bmk) for all k,m∈N, which is a generalization of Δ(1)-difference operator by
(5)bmk={r,k=ms,k=m-10,(k>m)or(0≤k<m-1).Başarir and Kayikçi [16] defined the matrix Bμ(bmkμ) which reduced the difference matrix Δ(1)μ in case r=1, s=-1. The generalized Bμ-difference operator is equivalent to the following binomial representation:
(6)Bμx=Bμ(xk)=∑v=0μ‍(μv)rμ-vsvxk-v.Let∧=(∧k) be a sequence of nonzero scalars. Then, for a sequence space E, the multiplier sequence space E∧, associated with the multiplier sequence ∧, is defined as
(7)E∧={x=(xk)∈ω:(∧kxk)∈E}.
An Orlicz function M 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→∞.We say that an Orlicz functionM satsfies the Δ2-condition if there exists K>2 and x0≥0 such that M(2x)≤KM(x) for all x≥x0. The Δ2-condition is equivalent to M(Lx)≤KLM(x) for all x>x0>0 and for L,K>1.Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following sequence space:
(8)lM={x∈ω:∑k=1∞‍M(|xk|ρ)<∞}
which is called an Orlicz sequence space. The space lM is a Banach space with the norm
(9)∥x∥=inf⁡{ρ>0:∑k=1∞‍M(|xk|ρ)≤1}.
It is shown in [17] that every Orlicz sequence space lM contains a subspace isomorphic to lp(p≥1).A sequenceM=(Mk) of Orlicz function is called a Musielak-Orlicz function; see [18, 19]. A sequence N=(Nk) defined by
(10)Nk(v)=sup⁡{|v|u-Mk(u):u≥0},k=1,2,…,
is called the complimentary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows:
(11)tM={x∈ω:IM(cx)<∞forsomec>0},hM={x∈ω:IM(cx)<∞∀c>0},
where IM is a convex modular defined by
(12)IM(x)=∑k=1∞‍Mk(xk),x=(xk)∈tM.
We consider tM equipped with the Luxemburg norm
(13)∥x∥=inf⁡{k>0:IM(xk)≤1}
or equipped with the Orlicz norm
(14)∥x∥0=inf⁡{1k(1+IM(kx)):k>0}.
By a lacunary sequence θ=(ir), r=0,1,2,…, where i0=0, we will mean an increasing sequence of nonnegative integers hr=(ir-rr-1)→∞(r→∞). The intervals determined by θ are denoted by Ir=(ir-1,ir] and the ratio ir/ir-1 will be denoted by qr. The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al. [20] as follows:
(15)Nθ={x=(xk):lim⁡r→∞1hr∑k∈Ir‍|xk-L|=0,forsomeL}.
The concept of 2-normed spaces was initially developed by Gähler [21] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak [22]. Since then, many others have studied this concept and obtained various results; see Gunawan [23, 24] and Gunawan and Mashadi [25]. For more details about sequence spaces see [26–33] and references therein. Let n∈N and X be linear space over the field K, where K is the field of real or complex numbers of dimension d, where d≥n≥2.A real valued function∥·,…,·∥ on Xn satisfying the following four conditions:(1)
∥(x1,x2,…,xn)∥=0 if and only if x1,x2,x3,…,xn are linearly dependent in X;(2)
∥(x1,x2,…,xn)∥ is invariant under permutation;(3)
∥(αx1,x2,…,xn)∥=|α|∥(x1,x2,…,xn)∥ for any α∈K;(4)
∥(x+x′,x2,…,xn)∥≤∥(x,x2,…,xn)∥+∥(x′,x2,…,xn)∥is called an n-norm on X and the pair (X,∥·,…,·∥) is called an n-normed space over the field K. For example, we may take X=Rn being equipped with the Euclidean n-norm ∥(x1,x2,…,xn)∥E= the volume of the n-dimensional parallelepiped spanned by the vectors x1,x2,…,xn which may be given explicitly by the formula
(16)∥(x1,x2,…,xn)∥E=|det⁡(xij)|,
where xi=(x1,x2,x3,…,xn)∈Rn for each i=1,2,3,…,n and ∥·∥E denotes the Euclidean norm. Let (X,∥·,…,·∥) be an n-normed space of dimension d≥n≥2 and {a1,a2,…,an} linearly independent set in X. Then the following function ∥(·,…,·)∥∞ on Xn-1 defined by
(17)∥(x1,x2,…,xn)∥∞=max⁡{∥(x1,x2,…,xn-1,ai)∥:i=1,2,…,n}
defines an (n-1) norm on X with respect to {(a1,a2,…,an)}.A sequence(xk) in an n-normed space (X,∥·,…,·∥) is said to converge to some L∈X if
(18)lim⁡k→∞∥(xk-L,z1,…,zn-1)∥=0,00000foreveryz1,…,zn-1∈X.
A sequence (xk) in a normed space (X,∥·,…,·∥) is said to be Cauchy if
(19)lim⁡k→∞p→∞⁡∥(xk-xp,z1,…,zn-1)∥=0,00000foreveryz1,…,zn-1∈X.
If every Cauchy sequence in X converges to some L∈X then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.Let(X,∥·,…,·∥) be an n-normed space and let s(ω-x) denote the space of X-valued sequences. Let p=(pk) be any bounded sequence of positive real numbers and M=(Mk) a Musielak-Orlicz function. We define the following sequence spaces in this paper:
(20)w0θ(M,B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=0,ρ>0x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍},wθ(M,B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk=0,forsomeL,ρ>0x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍},w∞θ(M,B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk<∞,ρ>0x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍};
when M(x)=x, we get
(21)w0θ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk=0,ρ>0x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍},wθ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk=0,forsomeL,ρ>0x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍},w∞θ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk<∞,ρ>0{x=(xk)∈s(w-x):lim⁡r→∞1hr∑k∈Ir‍};
when pk=1, for all k, we get
(22)w0θ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)=0,ρ>0lim⁡r→∞1hr∑k∈Ir‍},wθ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)=0,forsomeL,ρ>0x=(xk)∈w(s-x):lim⁡r→∞1hr∑k∈Ir‍},w∞θ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):lim⁡r→∞1hr∑k∈Ir‍×∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)<∞,ρ>0x=(xk)∈w(s-x):lim⁡r→∞1hr∑k∈Ir‍}.
The following inequality will be used throughout the paper. If 0≤pk≤sup⁡⁡pk=H, k=max⁡(1,2H-1), then
(23)|ak+bk|pk≤K{|ak|pk+|bk|pk}
for all k and ak,bk∈C. Also |a|pk≤max⁡(1,|a|H) for all a∈C.


## 2. Main Results

Theorem 1.
LetM=(Mk) be a Musielak-Orlicz function and p=(pk) a bounded sequence of positive real numbers; the spaces w0θ(M,B∧μ,p,∥·,…,·∥), wθ(M,B∧μ,p,∥·,…,·∥), and w∞θ(M,B∧μ,p,∥·,…,·∥) are linear over the field of complex numbers C.Proof.
Letx=(xk), y=(yk)∈w0θ(M,B∧μ,p,∥·,…,·∥), and α,β∈C. Then there exist positive real numbers ρ1 and ρ2 such that
(24)lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk=0,lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μykρ2,z1,…,zn-1)∥)pk=0.
Define ρ3=max⁡(2|α|ρ1,2|β|ρ2). Since ∥·,…,·∥ is an n-norm on X and Mk′s are nondecreasing and convex functions so by using inequality (23) we have
(25)lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μ(αxk+βyk)ρ3,z1,…,zn-1)∥)pk≤lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μαxkρ3,z1,…,zn-1)∥hhhhhhhhhhhhhhh+∥(B∧μβykρ3,z1,…,zn-1)∥)pk≤Klim⁡r→∞1hr∑k∈Ir‍12pkMk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk+Klim⁡r→∞1hr∑k∈Ir‍12pkMk(∥(B∧μykρ2,z1,…,zn-1)∥)pk=0.
Thus, we have αx+βy∈w0θ(M,B∧μ,p,∥·,…,·∥). Hence w0θ(M,B∧μ,p,∥·,…,·∥) is a linear space. Similarly, we can prove that wθ(M,B∧μ,p,∥·,…,·∥) and w∞θ(M,B∧μ,p,∥·,…,·∥) are linear spaces. This completes the proof of the theorem.Theorem 2.
LetM=(Mk) be a Musielak-Orlicz function and p=(pk) a bounded sequence of positive real numbers; the space w0θ(M,B∧μ,p,∥·,…,·∥) is a topological linear space paranormed by
(26)g(x)=inf⁡{(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mρpr/M:hhhhh(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mhhhhh≤1(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mρpr/M},
where M=max⁡(1,sup⁡k⁡pk<∞).Proof.
Clearlyg(x)≥0 for x=(xk)∈w0θ(M,B∧μ,p,∥·,…,·∥). Since Mk(0)=0, we get g(0)=0. Again, if g(x)=0, then
(27)g(x)=inf⁡{(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mρpr/M:hhhhh(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mhhhhh≤1(1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk)1/Mρpr/M}=0.
This implies that, for a given ε>0, there exist some ρε(0<ρε<ε) such that
(28)(1hr∑k∈Ir‍Mk(∥(B∧μxkρε,z1,…,zn-1)∥)pk)1/M≤1.
Thus(29)(1hr∑k∈Ir‍Mk(∥(B∧μxkε,z1,…,zn-1)∥)pk)1/M≤(1hr∑k∈Ir‍Mk(∥(B∧μxkρε,z1,…,zn-1)∥)pk)1/M≤1
for each r, and suppose that xk≠0 for each k∈N. This implies that B∧μxk≠0 for each k∈N. Let ε→0, then
(30)((∥(B∧μxkρε,z1,…,zn-1)∥)pk)1/M⟶∞;
which is a contradiction. Therefore, B∧μxk=0 for each k and thus xk=0 for each k∈N. Let ρ1>0 and ρ2>0 be such that
(31)(1hr∑k∈Ir‍Mk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk)1/M≤1,(1hr∑k∈Ir‍Mk(∥(B∧μykρ2,z1,…,zn-1)∥)pk)1/M≤11hr∑k∈Ir‍Mk(∥(B∧μykρ2,z1,…,zn-1)∥)pkforeachr.
Let ρ=ρ1+ρ2; then by using Minkowski’s inequality, we have
(32)(1hr∑k∈Ir‍Mk(∥(B∧μ(xk+yk)ρ,z1,…,zn-1)∥)pk)1/M≤(1hr∑k∈Ir‍Mk(∥(B∧μxk+B∧μykρ1+ρ2,z1,…,zn-1)∥)pk)1/M≤(1hr∑k∈Ir‍Mkhhhh×((ρ1ρ1+ρ2)∥(B∧μxkρ1,z1,…,zn-1)∥hhhhhhhh+(ρ2ρ1+ρ2)∥(B∧μykρ2,z1,…,zn-1)∥)pk1hr∑k∈Ir‍)1/M≤(ρ1ρ1+ρ2)×(1hr∑k∈Ir‍Mk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk)1/M+(ρ2ρ1+ρ2)×(1hr∑k∈Ir‍Mk(∥(B∧μykρ2,z1,…,zn-1)∥)pk)1/M≤1.
Sinceρ′s are nonnegative, we have
(33)g(x+y)=inf⁡{ρpr/M:(1hr∑k∈IrMkhhhhhhhhhhhh×(∥(B∧μxkρ,z1,…,zn-1)∥)pk∑k∈IrMk)1/Mhh≤1:(1hr∑k∈IrMk}≤inf⁡{ρ1pr/M:(1hr∑k∈IrMkhhhhhhhhhhhhh×(∥(B∧μxkρ1,z1,…,zn-1)∥)pk∑k∈IrMk)1/M≤1∑k∈IrMk}+inf⁡{ρ2pr/M:(1hr∑k∈IrMkhhhhhhhhhhhhhhh×(∥(B∧μykρ2,z1,…,zn-1)∥)pk∑k∈IrMk)1/M≤1ρ2pr/M:(1hr∑k∈IrMk}.
Therefore, g(x+y)≤g(x)+g(y).
Finally, we prove that the scalar multiplication is continuous. Letν be any complex number. By definition,
(34)g(νx)=inf⁡{ρpr/M:(1hr∑k∈IrMkhhhhhhhhhhhhhh×(∥(νB∧μxkρ,z1,…,zn-1)∥)pk1hr∑k∈IrMk)1/Mhhhhhh≤1ρpr/M:(1hr∑k∈IrMk}.
Then
(35)g(νx)=inf⁡{(|ν|t)pr/M:(1hr∑k∈IrMkhhhhhhhhhhhhhhhh×(∥(B∧μxkt,z1,…,zn-1)∥)pk1hr∑k∈IrMk)1/M≤1:(1hr∑k∈IrMk},
where t=ρ/|ν|. Since |ν|pr≤max⁡(1,|ν|sup⁡pk), we have
(36)g(νx)=max⁡(1,|ν|sup⁡pk)inf⁡×{(∥(B∧μxkt,z1,…,zn-1)∥)pk)1/M(t)pr/M:hhhh(1hr∑k∈Ir‍Mk×(∥(B∧μxkt,z1,…,zn-1)∥)pk)1/M≤1}.
So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of theorem.Theorem 3.
LetM=(Mk) be a Musielak-Orlicz function. If sup⁡k⁡(Mk(x))pk<∞ for all fixed x>0, then wθ(M,B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥).Proof.
Letx=(xk)∈wθ(M,B∧μ,p,∥·,…,·∥). Then there exists some positive number ρ1 such that
(37)lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ1,z1,…,zn-1)∥)pk=0.
Define ρ=2ρ1. Since Mk is nondecreasing and convex and by using inequality (23), we have
(38)lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μxk-L+Lρ,z1,…,zn-1)∥)pk≤K{lim⁡r→∞1hr∑k∈Ir‍Mk12pk(∥(B∧μxk-Lρ1,z1,…,zn-1)∥)pk+lim⁡r→∞1hr∑k∈Ir‍Mk12pk(∥(Lρ1,z1,…,zn-1)∥)pk}<K{lim⁡r→∞1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ1,z1,…,zn-1)∥)pk+lim⁡r→∞1hr∑k∈Ir‍Mk(∥(Lρ1,z1,…,zn-1)∥)pk}.
Hence x=(xk)∈w∞θ(M,B∧μ,p,∥·,…,·∥). This completes the proof of the theorem.Theorem 4.
LetM=(Mk) be a Musielak-Orlicz function and 0<h=inf⁡pk. Then
(39)w∞θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥)
if and only if
(40)lim⁡r→∞1hr∑k∈Ir‍Mk(t)pk=∞,forsomet>0.Proof.
Letw∞θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥). Suppose (40) does not hold. Therefore there are a subinterval Ir(m) of the set of intervals Ir and a number n0, where n0=∥(B∧μxk/ρ,z1,…,zn-1)∥ for all k, such that
(41)1hr(m)∑k∈Ir(m)‍M(n0)pk≤N<∞,m=1,2,3,….
Let us define x=(xk) as follows:
(42)B∧μxk={ρn0,k∈Ir(m)0,k∉Ir(m).
Thus by (41), x=(xk)∈w∞θ(M,B∧μ,p,∥·,…,·∥). But x=(xk)∉w0θ(M,B∧μ,p,∥·,…,·∥). Hence (40) must hold.
Conversely, suppose that (40) holds and let x=(xk)∈w∞θ(M,B∧μ,p,∥·,…,·∥). Then,
(43)1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk≤N<∞.
Suppose that x=(xk)∉w0θ(M,B∧μ,p,∥·,…,·∥). Then for some number ε, 1>ε>0, there is a number N0 such that, for a subinterval Ir(m) of the set of intervals Ir,
(44)∥(B∧μxkρ,z1,…,zn-1)∥>εforN≥N0.
We have Mk(∥(B∧μxk/ρ,z1,…,zn-1)∥)≥M(ε)pk, which contradicts (40) by using (43). Hence we get
(45)w∞θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥).
This completes the proof.Theorem 5.
Let0<h=inf⁡⁡pk≤sup⁡⁡pk=H<∞. For any Musielak-Orlicz function M=(Mk) which satisfies Δ2-condition, one has(i)
w0θ(B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥)(ii)
wθ(B∧μ,p,∥·,…,·∥)⊂wθ(M,B∧μ,p,∥·,…,·∥)(iii)
w∞θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥).Proof.
(i) Letx=(xk)∈w0θ(B∧μ,p,∥·,…,·∥). Then, we have
(46)1hr∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk⟶0asr⟶∞.
Let ε>0, and choose δ with 0<δ<1 such that Mk<ε for 0≤t≤δ. We can write
(47)1hr∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk=1hr∑k∈Ir∥(B∧μxk/ρ,z1,…,zn-1)∥≤δ(∥(B∧μxkρ,z1,…,zn-1)∥)pk+1hr∑k∈Ir∥(B∧μxk/ρ,z1,…,zn-1)∥>δ(∥(B∧μxkρ,z1,…,zn-1)∥)pk.
For the first summation above, we can write
(48)1hr∑k∈Ir∥(B∧μxk/ρ,z1,…,zn-1)∥≤δ(∥(B∧μxkρ,z1,…,zn-1)∥)pk<max⁡⁡(ε,εh).
By using continuity of Mk, for the second summation we can write
(49)∥(B∧μxkρ,z1,…,zn-1)∥<1+(∥(B∧μxk/ρ,z1,…,zn-1)∥)δ.
Since each Mk is nondecreasing and convex and satisfies Δ2-condition, it follows that
(50)1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk≤max⁡⁡(ε,εh)+max⁡{1,[2Mk(∥(B∧μxk/ρ,z1,…,zn-1)∥)δ]h}×1hr∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk.
Taking limit as ε→0 and r→∞, it follows that x=(xk)∈w0θ(M,B∧μ,p,∥·,…,·∥). Hence w0θ(B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥). Similarly, we can prove (ii) and (iii). This completes the proof of the theorem.Theorem 6.
LetM=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent: (i)
w∞θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥);(ii)
w0θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥);(iii)
sup⁡r⁡(1/hr)∑k∈IrMk(t)pk<∞ for all t>0, where t=∥B∧μxk/ρ,z1,…,zn-1∥.Proof.
(i)⇒ (ii) Suppose (i) holds. In order to prove (ii) we have to show that
(51)wθ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥).
Let x=(xk)∈w0θ(B∧μ,p,∥·,…,·∥). Then for a given ε>0 there exists s>sε such that
(52)1hr∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk<ε.
Hence there exists K>0 such that
(53)sup⁡r1hr∑k∈Ir‍(∥(B∧μxkρ,z1,…,zn-1)∥)pk<K.
This shows that x=(xk)∈w∞θ(M,B∧μ,p,∥·,…,·∥).
(ii)⇒ (iii) Suppose (ii) holds and (iii) fails to hold. Then for some t>0,
(54)sup⁡r1hr∑k∈Ir‍Mk(ε)pk=∞,
and, therefore, we can find a subinterval Ir(m) of the set of intervals Ir such that
(55)1hr(m)∑k∈Ir(m)‍Mk(1m)pk≥m,m=1,2,3,….
Let us define x=(xk) as follows:
(56)B∧μxk={ρm,k∈Ir(m)0,k∉Ir(m).
Thus x=(xk)∈w0θ(B∧μ,p,∥·,…,·∥). But by (55), x=(xk)∉w∞θ(M,B∧μ,p,∥·,…,·∥) which contradicts (ii). Hence (iii) must hold.
(iii)⇒ (i) Let (iii) hold. Suppose that x=(xk)∉w∞θ(M,B∧μ,p,∥·,…,·∥). Then for x=(xk)∈w∞θ(B∧μ,p,∥·,…,·∥). 							(57)sup⁡r1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=∞.
Let t=∥B∧μxk/ρ,z1,…,zn-1∥ for each k, and then by (57) sup⁡r⁡(1/hr)∑k∈IrMk(t)pk=∞, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.Theorem 7.
LetM=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent: (i)
w0θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(B∧μ,p,∥·,…,·∥);(ii)
w0θ(M,B∧μ,p,∥·,…,·∥)⊂w∞θ(B∧μ,p,∥·,…,·∥);(iii)
inf⁡r⁡(1/hr)∑k∈IrMk(t)pk>0 for all t>0.Proof.
(i)⇒ (ii) is obvious
(ii)⇒ (iii) Let (ii) hold and let (iii) fail to hold. Then
(58)inf⁡r1hr∑k∈Ir‍Mk(t)pk=0forsomet>0,
and we can find a subinterval Ir(m) of the set of intervals Ir such that
(59)1hr(m)∑k∈Ir(m)‍Mk(m)pk<1m,m=1,2,3,….
Let us define x=(xk) as follows:
(60)B∧μxk={ρm,k∈Ir(m)0,k∉Ir(m).
Thus by (iii), x=(xk)∈w0θ(M,B∧μ,p,∥·,…,·∥). But x=(xk)∉w∞θ(B∧μ,p,∥·,…,·∥) which contradict (ii). Hence (iii) must hold.
(iii)⇒ (i) Let (iii) hold. Suppose that x=(xk)∈w0θ(M,B∧μ,p,∥·,…,·∥). Therefore,
(61)1hr∑k∈Ir‍Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk⟶0asr⟶∞.
Again suppose x=(xk)∉w0θ(B∧μ,p,∥·,…,·∥) for some number ε>0 and a subinterval Ir(m) of the set of intervals Ir, we have
(62)∥(B∧μxkρ,z1,…,zn-1)∥≥ε∀k.
Then, from properties of the Orlicz function, we can write
(63)Mk(∥(B∧μxkρ,z1,…,zn-1)∥)pk≥Mk(ε)pk.
Consequently, by (61), we have lim⁡r→∞⁡(1/hr)∑k∈IrMk(ε)pk=0, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.Theorem 8.
(i) If0<inf⁡⁡pk≤pk≤1 for all k, then wθ(M,B∧μ,∥·,…,·∥)⊆wθ(M,B∧μ,p,∥·,…,·∥).
(ii) If1≤pk≤sup⁡pk=H<∞, then wθ(M,B∧μ,p,∥·,…,·∥)⊆wθ(M,B∧μ,∥·,…,·∥).Proof.
(i) Letx∈wθ(M,B∧μ,∥·,…,·∥). Since 0<inf⁡⁡pk≤1, we get
(64)1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)≤1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk,
and hence x∈wθ(M,B∧μ,p,∥·,…,·∥).
(ii)1≤pk≤sup⁡⁡pk=H<∞ and x=(xk)∈wθ(M,B∧μ,p,∥·,…,·∥). Then for each 0<ε<1 there exists a positive integer s0 such that
(65)1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk≤ε<1∀r>s0.
This implies that(66)1hr∑k∈Ir‍Mk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk≤1hr∑k∈IrMk(∥(B∧μxk-Lρ,z1,…,zn-1)∥).
Therefore x=(xk)∈wθ(M,B∧μ,∥·,…,·∥). This completes the proof of the theorem.



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*Source: 691632-2014-07-22.xml*