In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function ℳ=(Mk). We also study some topological properties and prove some inclusion relations between these spaces.
1. Introduction and Preliminaries
An Orlicz function M is a function, which is continuous, nondecreasing, and convex with M(0)=0, M(x)>0 for x>0 and M(x)→∞ as x→∞.
Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x=(xk); then
(1)ℓM={x∈w:∑k=1∞M(|xk|ρ)<∞},
which is called as an Orlicz sequence space. The space ℓM is a Banach space with the norm
(2)∥x∥=inf{ρ>0:∑k=1∞M(|xk|ρ)≤1}.
It is shown in [1] that every Orlicz sequence space ℓM contains a subspace isomorphic to ℓp(p≥1). The Δ2-condition is equivalent to M(Lx)≤kLM(x) for all values of x≥0 and for L>1. A sequence ℳ=(Mk) of Orlicz functions is called a Musielak-Orlicz function (see [2, 3]). A sequence 𝒩=(Nk) defined by
(3)Nk(v)=sup{|v|u-Mk(u):u≥0},k=1,2,…,
is called the complementary 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:
(4)tℳ={x∈w:Iℳ(cx)<∞forsomec>0},hℳ={x∈w:Iℳ(cx)<∞∀c>0},
where Iℳ is a convex modular defined by
(5)Iℳ(x)=∑k=1∞(Mk)(xk),x=(xk)∈tℳ.
We consider tℳ equipped with the Luxemburg norm
(6)∥x∥=inf{k>0:Iℳ(xk)≤1}
or equipped with the Orlicz norm
(7)∥x∥0=inf{1k(1+Iℳ(kx)):k>0}.
Let X be a linear metric space. A function p : X→ℝ is called paranorm if
p(x)≥0 for all x∈X,
p(-x)=p(x) for all x∈X,
p(x+y)≤p(x)+p(y) for all x,y∈X,
(λn) is a sequence of scalars with λn→λasn→∞ and (xn) is a sequence of vectors with p(xn-x)→0asn→∞; then p(λnxn-λx)→0asn→∞.
A paranorm p for which p(x)=0 implies x=0 is called total paranorm and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [4], Theorem 10.4.2, pp. 183). For more details about sequence spaces, see [5–12] and references therein.
A sequence of positive integers θ=(kr) is called lacunary if k0=0, 0<kr<kr+1 and hr=kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr) and qr=kr/kr-1. The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al. [13] as
(8)Nθ={x∈w:limr→∞1hr∑k∈Ir|xk-l|=0,forsomel}.
Strongly almost convergent sequence was introduced and studied by Maddox [14] and Freedman et al. [13]. Parashar and Choudhary [15] have introduced and examined some properties of four sequence spaces defined by using an Orlicz function M, which generalized the well-known Orlicz sequence spaces [C,1,p], [C,1,p]0, and [C,1,p]∞. It may be noted here that the space of strongly summable sequences was discussed by Maddox [16] and recently in [17].
Mursaleen and Noman [18] introduced the notion of λ-convergent and λ-bounded sequences as follows.
Let λ=(λk)k=1∞ be a strictly increasing sequence of positive real numbers tending to infinity; that is,
(9)0<λ0<λ1<⋯,λk⟶∞ask⟶∞,
and it is said that a sequence x=(xk)∈w is λ-convergent to the number L, called the λ-limit of x if Λm(x)→L as m→∞, where
(10)λm(x)=1λm∑k=1m(λk-λk-1)xk.
The sequence x=(xk)∈w is λ-bounded if supm|Λm(x)|<∞. It is well known [18] that if limmxm=a in the ordinary sense of convergence, then
(11)limm(1λm(∑k=1m(λk-λk-1)|xk-a|))=0.
This implies that(12)limm|Λm(x)-a|=limm|1λm∑k=1m(λk-λk-1)(xk-a)|=0,
which yields that limmΛm(x)=a and hence x=(xk)∈w is λ-convergent to a.
The concept of 2-normed spaces was initially developed by Gähler [19] in the mid 1960s, while for that of n-normed spaces one can see Misiak [20]. Since then, many others have studied this concept and obtained various results; see Gunawan ([21, 22]) and Gunawan and Mashadi [23]. Let n∈ℕ and let X be a 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
∥x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent in X;
∥x1,x2,…,xn∥ is invariant under permutation;
∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥ for any α∈𝕂;
∥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, if we may take X=ℝn being equipped with the 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
(13)∥x1,x2,…,xn∥E=|det(xij)|,
where xi=(xi1,xi2,…,xin)∈ℝn for each i=1,2,…,n, leting (X,∥·,…,·∥) be an n-normed space of dimension d≥n≥2 and {a1,a2,…,an} be linearly independent set in X, then the following function ∥·,…,·∥∞ on Xn-1 defined by
(14)∥x1,x2,…,xn-1∥∞=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
(15)limk→∞∥xk-L,z1,…,zn-1∥=0limk→∞foreveryz1,…,zn-1∈X.
A sequence (xk) in an n-normed space (X,∥·,…,·∥) is said to be Cauchy if
(16)limp→∞k→∞∥xk-xp,z1,…,zn-1∥=0limxxp→∞k→∞foreveryz1,…,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 ℳ=(Mk) be a Musielak-Orlicz function, and let p=(pk) be a bounded sequence of positive real numbers. We define the following sequence spaces in the present paper:
(17)w0θ(ℳ,Λ,p,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk=0,(∥Λk(x)ρ,z1,z2z2z2z2z2z2z2z2,∥)ρ>0,s≥0Λk(x)ρ},wθ(ℳ,Λ,p,s,∥·,…,·∥)={Λk(x)-Lρx=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]pk=0,forsomeL,ρ>0,s≥0Λk(x)-Lρ},w∞θ(ℳ,Λ,p,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:supr1hr=×∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk<∞,(∥Λk(x)ρ,z1,z2z2z2⋯,zn-1∥)ρ>0,s≥0Λk(x)ρ}.
If we take ℳ(x)=x, we get
(18)w0θ(Λ,p,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[∥Λk(x)ρ,z1,z2,…,zn-1∥]pk=0,∥Λk(x)ρ,z1,z2z2,⋯,zn-1∥ρ>0,s≥0Λk(x)ρ},wθ(Λ,p,s,∥·,…,·∥)={Λk(x)-Lρx=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[∥Λk(x)-Lρ,z1,z2,…,zn-1∥]pk=0,∥,z1,z2z2…,zn-1∥forsomeL,ρ>0,s≥0Λk(x)-Lρ},w∞θ(Λ,p,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:supr1hr=×∑k∈Irk-s[∥Λk(x)ρ,z1,z2,…,zn-1∥]pk<∞,∥Λk(x)ρ,z1,z2z2z2,…,zn-1∥ρ>0,s≥0Λk(x)ρ}.
If we take p=(pk)=1 for all k∈ℕ, we have
(19)w0θ(ℳ,Λ,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]=0,∥Λk(x)ρ,z1,z2,z1,z2,z1,…,zn-1∥ρ>0,s≥0Λk(x)ρ},wθ(ℳ,Λ,s,∥·,…,·∥)={x=(xk)∈w:limr→∞1hr=×∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]=0,forsomeL,ρ>0,s≥0Λk(x)-Lρ},w∞θ(ℳ,Λ,s,∥·,…,·∥)={Λk(x)ρx=(xk)∈w:supr1hr=×∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]<∞,∥Λk(x)ρ,z1,z2,z1,,z1z1…,zn∥ρ>0,s≥0Λk(x)ρ}.
The following inequality will be used throughout the paper. If 0≤pk≤suppk=H, K=max(1,2H-1), then
(20)|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.
In this paper, we introduce sequence spaces defined by a Musielak-Orlicz function over n-normed spaces. We study some topological properties and prove some inclusion relations between these spaces.
2. Main ResultsTheorem 1.
Let ℳ=(Mk) be a Musielak-Orlicz function, and let p=(pk) be a bounded sequence of positive real numbers, then the spaces w0θ(ℳ,Λ,p,s,∥·,…,·∥),wθ(ℳ,Λ,p,s,∥·,…,·∥), and w∞θ(ℳ,Λ,p,s,∥·,…,·∥) are linear spaces over the field of complex number ℂ.
Proof.
Let x=(xk),lety=(yk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥), and let α,β∈ℂ. In order to prove the result, we need to find some ρ3 such that
(21)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(αx+βy)ρ3,z1,z2,…,zn-1∥)]pk=0.
Since x=(xk),y=(yk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥), there exist positive numbers ρ1,ρ2>0 such that
(22)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk=0,limr→∞1hr∑k∈Irk-s[Mk(∥Λk(y)ρ2,z1,z2,…,zn-1∥)]pk=0.
Define ρ3=max(2|α|ρ1,2|β|ρ2). Since (Mk) is nondecreasing, convex function and by using inequality (20), we have
(23)1hr∑k∈Irk-s[Mk(∥Λk(αx+βy)ρ3,z1,z2,…,zn-1∥)]pk≤1hr∑k∈Irk-s[Mk(∥αΛk(x)ρ3,z1,z2,…,zn-1∥≤1hr1hr1hr=-∑k∈Irk-s+∥βΛk(y)ρ3,z1,z2,…,zn-1∥)]pk≤K1hr∑k∈Ir12pkk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk≤+K1hr∑k∈Ir12pkk-s[Mk(∥Λk(y)ρ2,z1,z2,…,zn-1∥)]pk≤K1hr∑k∈Irk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk≤+K1hr∑k∈Irk-s[Mk(∥Λk(y)ρ2,z1,z2,…,zn-1∥)]pk⟶0asr⟶∞.
Thus, we have αx+βy∈w0θ(ℳ,Λ,p,s,∥·,…,·∥). Hence, w0θ(ℳ,Λ,p,s,∥·,…,·∥) is a linear space. Similarly, we can prove that wθ(ℳ,Λ,p,s,∥·,…,·∥) and w∞θ(ℳ,Λ,p,s,∥·,…,·∥) are linear spaces.
Theorem 2.
Let ℳ=(Mk) be a Musielak-Orlicz function, and let p=(pk) be a bounded sequence of positive real numbers. Then w0θ(ℳ,Λ,p,s,∥·,…,·∥) is a topological linear space paranormed by
(24)g(x)=inf{∑k∈Irk-sρpr/H:=infinf(1hr∑k∈Irk-s[Mk(∥∑k∈Irk-sΛk(x)ρ,(1hr∑k∈Ir(1hr1hr1hr=∑k∈Irz1,z2,…,zn-1∑k∈Irk-s∥)]pk)1/H≤1ef22(efsdf1hr∑k∈Irk-s},
where H=max(1,supkpk)<∞.
Proof.
Clearly g(x)≥0 for x=(xk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥). Since Mk(0)=0, we get g(0)=0. Again if g(x)=0, then
(25)inf{(1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk)ρpr/H:infinf(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(x)ρ,(1hr∑k∈Ir(1hrz21hrz21hrz1,z2,…,zn-11hr∑k∈Irk-sΛk(x)ρ∥)]pk)1/H≤1(1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk)}=0.
This implies that for a given ϵ>0, there exist some ρϵ(0<ρϵ<ϵ) such that
(26)(1hr∑k∈Irk-s[Mk(∥Λk(x)ρϵ,z1,z2,…,zn-1∥)]pk)1/H≤1.
Thus,
(27)(1hr∑k∈Irk-s[Mk(∥Λk(x)ϵ,z1,z2,…,zn-1∥)]pk)1/H≤(1hr∑k∈Irk-s[Mk(∥Λk(x)ρϵ,z1,z2,…,zn-1∥)]pk)1/H.
Suppose that (xk)≠0 for each k∈ℕ. This implies that Λk(x)≠0 for each k∈ℕ. Let ϵ→0, then
(28)∥Λk(x)ϵ,z1,z2,…,zn-1∥⟶∞.
It follows that
(29)(1hr∑k∈Irk-s[Mk(∥Λk(x)ϵ,z1,z2,…,zn-1∥)]pk)1/H⟶∞,
which is a contradiction. Therefore, Λk(x)=0 for each k, and thus (xk)=0 for each k∈ℕ. Let ρ1>0 and ρ2>0 be the case such that
(30)(1hr∑k∈Irk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk)1/H≤1,(1hr∑k∈Irk-s[Mk(∥Λk(y)ρ2,z1,z2,…,zn-1∥)]pk)1/H≤1.
Let ρ=ρ1+ρ2; then, by using Minkowski's inequality, we have
(31)(1hr∑k∈Irk-s[Mk(∥Λk(x+y)ρ,z1,z2,…,zn-1∥)]pk)1/H≤(1hr∑k∈Irk-s[Mk({z1,z2,…,zn-11hr∑k∈Irk-sΛk(x)+Λk(y)ρ1+ρ2∥}∥{z1,z2,…,zn-11hr∑k∈Irk-sΛk(x)+Λk(y)ρ1+ρ2∥}Λk(x)+Λk(y)ρ1+ρ2,1hr1hr1hr1hr1hr1hr1hr∑k∈Ir∑k∈Irk-sz1,z2,…,zn-11hr∑k∈Irk-sΛk(x)+Λk(y)ρ1+ρ2∥)]pk)1/H≤({[∥Λk(y)ρ2,z1,z2,…,zn-11hr∑k∈Irk-s∥]}1hr∑k∈Irk-s[Λk(y)ρ2{[∥Λk(y)ρ2,z1,z2,…,zn-11hr∑k∈Irk-s∥]}Mk(ρ1ρ1+ρ2)1hr∑k∈Irk-s1hr∑k∈Ir×[∥Λk(x)ρ1,z1,z2,…,zn-1∥]+(ρ2ρ1+ρ2)1hr∑k∈Irk-s1hr-∑k∈Ir×[∥Λk(y)ρ2,z1,z2,…,zn-11hr∑k∈Irk-s∥]]pk)1/H≤(ρ1ρ1+ρ2)1hr1hr×(1hr∑k∈Irk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk)1/H+(ρ2ρ1+ρ2)1hr1hr×(1hr∑k∈Irk-s[Mk(∥Λk(y)ρ2,z1,z2,…,zn-1∥)]pk)1/H≤1.
Since ρ,ρ1, and ρ2 are nonnegative, we have
(32)g(x+y)=inf{1hr∑k∈Irk-s{z1,z2,…,zn-11hr∑k∈Irk-sΛk(x+y)ρ∥)]pk)1/H}ρpr/H:infinf(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(x+y)ρ,infinfinfinfiniinfinnfinz1,z2,…,zn-11hr∑k∈Irk-sΛk(x+y)ρ∥)]pk)1/H≤1}≤inf{1hr∑k∈Irk-s{z1,z2,…,zn-11hr∑k∈Irk-sΛk(x+y)ρ∥)]pk)1/H}Λk(x)ρ1(ρ1)pr/H:infinfinf(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(x)ρ1,infinfinfinfinfinfinfinfinfiz1,z2,…,zn-11hr∑k∈Irk-sΛk(x)ρ1∥)]pk)1/H≤1}≤f+inf{{z1,z2,…,zn-11hr∑k∈Irk-sΛk(x+y)ρ∥)]pk)1/H}Λk(y)ρ21hr∑k∈Irk-s(ρ2)pr/H:=infinf(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(y)ρ2,infinfinfininifinfinfinfinfz1,z2,…,zn-11hr∑k∈Irk-sΛk(y)ρ2∥)]pk)1/H≤1}.
Therefore, g(x+y)≤g(x)+g(y). Finally we prove that the scalar multiplication is continuous. Let μ be any complex number. By definition,
(33)g(μx)=inf{1hr∑k∈Irk-sΛk(μx)ρρpr/H:=innii(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(μx)ρ,(1hr∑k∈Irk-s(1hr=∑k∈Iriik-sz1,z2,…,zn-11hr∑k∈Irk-sΛk(μx)ρ∥)]pk)1/Hinin&fin≤11hr∑k∈Irk-s}.
Thus,
(34)g(μx)=inf{∑k∈Ir1hrΛk(x)k-st(|μ|t)pr/H:=inin(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-s1hr∑k∈Irk-sΛk(x)t,(1hr∑k∈Ir(1hr1hr1hr∑k∈Iriik-sz1,z2,…,zn-1Λ1hr∑k∈Ir∥)]pk)1/H=in&finfin≤11hr∑k∈Irk-s1hr∑k∈Irk-s},
where 1/t=ρ/|μ|. Since |μ|pr≤max(1,|μ|suppr), we have
(35)g(μx)≤max(1,|μ|suppr)×inf{Λk(x)t1hr∑k∈Irk-stpr/H:=inf==(1hr∑k∈Irk-s[Mk(∥1hr∑k∈Irk-sΛk(x)t,(1hr∑k∈Ir=k-s(1hr=∑k∈Iriik-sz1,z2,…,zn-11hr∑k∈Irk-sΛk(μx)ρ∥)]pk)1/H=n&fn&=≤11hr∑k∈Irk-s}.
So the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.
Theorem 3.
Let ℳ=(Mk) be a Musielak-Orlicz function. If supk[Mk(x)]pk<∞ for all fixed x>0, then w0θ(ℳ,Λ,p,s,∥·,…,·∥)⊆w∞θ(ℳ,Λ,p,s,∥·,…,·∥).
Proof.
Let x=(xk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥); then there exists positive number ρ1 such that
(36)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)ρ1,z1,z2,…,zn-1∥)]pk=0.
Define ρ=2ρ1. Since (Mk) is nondecreasing and convex and by using inequality (20), we have
(37)supr1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk=supr1hr∑k∈Irk-s[Mk(∥Λk(x)+L-Lρ,=supr1hr∑k∈Irk-s=supr∑k∈Irk-sz1,z2,…,zn-1Λk(x)+L-Lρ∥)]pk≤Ksupr1hr∑k∈Irk-s12pk[Mk(∥Λk(x)-Lρ1,z1,z2,…,zn-1{[Mk(∥Λk(x)-Lρ1,}∥{[Mk(∥Λk(x)-Lρ1,})]pkKsupr+Ksupr1hr∑k∈Irk-s12pk[Mk(∥Lρ1,z1,z2,…,zn-1∥)]pk≤Ksupr1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ1,z1,z2,…,zn-1∥)]pkKsupr+Ksupr1hr∑k∈Irk-s[Mk(∥Lρ1,z1,z2,…,zn-1∥)]pk<∞.
Hence, x=(xk)∈w∞θ(ℳ,Λ,p,s,∥·,…,·∥).
Theorem 4.
Let 0<infpk=h≤pk≤suppk=H<∞ and let ℳ=(Mk),ℳ'=(Mk′) be Musielak-Orlicz functions satisfying Δ2-condition, then one has
w0θ(ℳ',Λ,p,s,∥·,…,·∥)⊂w0θ(ℳ∘ℳ',Λ,p,s,∥·,…,·∥);
wθ(ℳ',Λ,p,s,∥·,…,·∥)⊂wθ(ℳ∘ℳ',Λ,p,s,∥·,…,·∥);
w∞θ(ℳ',Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ∘ℳ',Λ,p,s,∥·,…,·∥).
Proof.
Let x=(xk)∈w0θ(ℳ′,Λ,p,s,∥·,…,·∥), then we have
(38)limr→∞1hr∑k∈Irk-s[Mk′(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk=0.
Let ϵ>0 and choose δ with 0<δ<1 such that Mk(t)<ϵ for 0≤t≤δ. Let (yk)=Mk′[∥Λk(x)/ρ,z1,z2,…,zn-1∥] for all k∈ℕ. We can write
(39)1hr∑k∈Irk-s(Mk[yk])pk=1hr∑yk≤δk∈Irk-s(Mk[yk])pk+1hr∑yk≥δk∈Irk-s(Mk[yk])pk.
So, we have
(40)1hr∑yk≤δk∈Irk-s(Mk[yk])pk≤[Mk(1)]H1hr∑yk≤δk∈Irk-s(Mk[yk])pk≤[Mk(2)]H1hr∑yk≤δk∈Irk-s(Mk[yk])pk.
For yk>δ,yk<yk/δ<1+yk/δ. Since (Mk)′s are nondecreasing and convex, it follows that
(41)Mk(yk)<Mk(1+ykδ)<12Mk(2)+12Mk(2ykδ).
Since ℳ=(Mk) satisfies Δ2-condition, we can write
(42)Mk(yk)<12TykδMk(2)+12TykδMk(2)=TykδMk(2).
Hence,
(43)1hr∑yk≥δk∈Irk-sMk[yk]pk≤max(1,(TMk(2)δ)H)1hr∑yk≤δk∈Irk-s[yk]pk.
From (40) and (43), we have x=(xk)∈w0θ(ℳ∘ℳ',Λ,p,s,∥·,…,·∥). This completes the proof of (i). Similarly we can prove that
(44)wθ(ℳ′,Λ,p,s,∥·,…,·∥)⊂wθ(ℳ∘ℳ′,Λ,p,s,∥·,…,·∥),w∞θ(ℳ′,Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ∘ℳ′,Λ,p,s,∥·,…,·∥).
Theorem 5.
Let 0<h=infpk=pk<suppk=H<∞. Then for a Musielak-Orlicz function ℳ=(Mk) which satisfies Δ2-condition, one has
w0θ(Λ,p,s,∥·,…,·∥)⊂w0θ(ℳ,Λ,p,s,∥·,…,·∥);
wθ(Λ,p,s,∥·,…,·∥)⊂wθ(ℳ,Λ,p,s,∥·,…,·∥);
w∞θ(Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ,Λ,p,s,∥·,…,·∥).
Proof.
It is easy to prove, so we omit the details.
Theorem 6.
Let ℳ=(Mk) be a Musielak-Orlicz function and let 0<h=infpk. Then w∞θ(ℳ,Λ,p,s,∥·,…,·∥)⊂w0θ(Λ,p,s,∥·,…,·∥) if and only if
(45)limr→∞1hr∑k∈Irk-s(Mk(t))pk=∞
for some t>0.
Proof.
Let w∞θ(ℳ,Λ,p,s,∥·,…,·∥)⊂w0θ(Λ,p,s,∥·,…,·∥). Suppose that (45) does not hold. Therefore, there are subinterval Ir(j) of the set of interval Ir and a number t0>0, where
(46)t0=∥Λk(x)ρ,z1,z2,…,zn-1∥∀k,
such that
(47)1hr(j)=∑k∈Ir(j)k-s(Mk(t0))pk≤K<∞,m=1,2,3,….
Let us define x=(xk) as follows:
(48)Λk(x)={ρt0,k∈Ir(j)0,k∉Ir(j).
Thus, by (47),x∈w∞θ(ℳ,Λ,p,s,∥·,…,·∥). But x∉w∞0(Λ,p,s,∥·,…,·∥). Hence, (45) must hold.
Conversely, suppose that (45) holds and let x∈w∞θ(ℳ,Λ,p,s,∥·,…,·∥). Then for each r,
(49)1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk≤K<∞.
Suppose that x∉w0θ(Λ,p,s,∥·,…,·∥). Then for some number ϵ>0, there is a number k0 such that for a subinterval Ir(j), of the set of interval Ir,
(50)∥Λk(x)ρ,z1,z2,…,zn-1∥>ϵfork≥k0.
From properties of sequence of Orlicz functions, we obtain
(51)[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk≥Mk(ϵ)pk,
which contradicts (45), by using (49). Hence, we get
(52)w∞θ(ℳ,Λ,p,s,∥·,…,·∥)⊂w0θ(Λ,p,s,∥·,…,·∥).
This completes the proof.
Theorem 7.
Let ℳ=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:
w∞θ(Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ,Λ,p,s,∥·,…,·∥);
w0θ(Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ,Λ,p,s,∥·,…,·∥);
supr1/hr∑k∈Irk-s(Mk(t))pk<∞forallt>0.
Proof.
(i) ⇒ (ii). Let (i) hold. To verify (ii), it is enough to prove
(53)w0θ(Λ,p,s,∥·,…,·∥)⊂w∞θ(ℳ,Λ,p,s,∥·,…,·∥).
Let x=(xk)∈w0θ(Λ,p,s,∥·,…,·∥). Then for ϵ>0, there exists r≥0, such that
(54)1hr∑k∈Irk-s[∥Λk(x)ρ,z1,z2,…,zn-1∥]pk<ϵ.
Hence, there exists K>0 such that
(55)supr1hr∑k∈Irk-s[∥Λk(x)ρ,z1,z2,…,zn-1∥]pk<K.
So, we get x=(xk)∈w∞θ(ℳ,Λ,p,s,∥·,…,·∥).
(ii) ⇒ (iii). Let (ii) hold. Suppose (iii) does not hold. Then for some t>0(56)supr1hr∑k∈Irk-s(Mk(t))pk=∞,
and therefore we can find a subinterval Ir(j), of the set of interval Ir, such that
(57)1hr(j)∑k∈Ir(j)k-s(Mk(1j))pk>j,j=1,2,3,…
Let us define x=(xk) as follows:
(58)Λk(x)={ρj,k∈Ir(j)0,k∉Ir(j).
Then x=(xk)∈w0θ(Λ,p,s,∥·,…,·∥). But by (57), x∉w∞θ(ℳ,Λ,p,s,∥·,…,·∥), which contradicts (ii). Hence, (iii) must holds.
(iii) ⇒ (i). Let (iii) hold and suppose that x=(xk)∈w∞θ(Λ,p,s,∥·,…,·∥). Suppose that x=(xk)∉w∞θ(ℳ,Λ,p,s,∥·,…,·∥); then
(59)supr1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk=∞.
Let t=∥Λk(x)/ρ,z1,z2,…,zn-1∥ for each k; then by (59),
(60)supr1hr∑k∈Irk-s(Mk(t))pk=∞,
which contradicts (iii). Hence, (i) must hold.
Theorem 8.
Let ℳ=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:
w0θ(ℳ,Λ,p,s,∥·,…,·∥)⊂w0θ(Λ,p,s,∥·,…,·∥);
w0θ(ℳ,Λ,p,s,∥·,…,·∥)⊂w∞θ(Λ,p,s,∥·,…,·∥);
infr1/hr∑k∈Irk-s(Mk(t))pk>0forallt>0.
Proof.
(i) ⇒ (ii). It is obvious.
(ii) ⇒ (iii). Let (ii) hold. Suppose that (iii) does not hold. Then
(61)infr1hr∑k∈Irk-s(Mk(t))pk=0forsomet>0,
and we can find a subinterval Ir(j), of the set of interval Ir, such that
(62)1hr(j)∑k∈Ir(j)k-s(Mk(j))pk<1j,j=1,2,3,…
Let us define x=(xk) as follows:
(63)Λk(x)={ρj,k∈Ir(j)0,k∉Ir(j).
Thus, by (62), x=(xk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥), but x=(xk)∉w∞θ(Λ,p,s,∥·,…,·∥), which contradicts (ii). Hence, (iii) must hold.
(iii) ⇒ (i). Let (iii) hold. Suppose that x=(xk)∈w0θ(ℳ,Λ,p,s,∥·,…,·∥). Then
(64)1hr∑k∈Irk-s[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk⟶0(∥Λk(x)ρ,z1,z2,⋯,∥)(∥Λk(x)ρ,∥)asr⟶∞.
Again suppose that x=(xk)∉w0θ(Λ,p,s,∥·,…,·∥); for some number ϵ>0 and a subinterval Ir(j), of the set of interval Ir, we have
(65)∥Λk(x)ρ,z1,z2,…,zn-1∥≥ϵ∀k.
Then from properties of the Orlicz function, we can write
(66)[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]pk≥(Mk(ϵ))pk.
Consequently, by (64), we have
(67)limr→∞1hr∑k∈Irk-s(Mk(ϵ))pk=0,
which contradicts (iii). Hence, (i) must hold.
Theorem 9.
Let 0≤pk≤qk for all k and let (qk/pk) be bounded. Then
(68)wθ(ℳ,Λ,q,s,∥·,…,·∥)⊆wθ(ℳ,Λ,p,s,∥·,…,·∥).
Proof.
Let x=(xk)∈wθ(ℳ,Λ,q,s,∥·,…,·∥); write
(69)tk=[Mk(∥Λk(x)ρ,z1,z2,…,zn-1∥)]qk
and μk=pk/qk for all k∈ℕ. Then 0<μk≤1 for all k∈ℕ. Take 0<μ≤μk for k∈ℕ. Define sequences (uk) and (vk) as follows.
For tk≥1, let uk=tk and vk=0, and for tk<1, let uk=0 and vk=tk. Then clearly for all k∈ℕ, we have
(70)tk=uk+vk,tkμk=ukμk+vkμk.
Now it follows that ukμk≤uk≤tk and vkμk≤vkμ. Therefore,
(71)1hr∑k∈Irtkμk=1hr∑k∈Ir(ukμk+vkμk)≤1hr∑k∈Irtk+1hr∑k∈Irvkμ.
Now for each k,
(72)1hr∑k∈Irvkμ=∑k∈Ir(1hrvk)μ(1hr)1-μ≤(∑k∈Ir[(1hrvk)μ]1/μ)μ×(∑k∈Ir[(1hr)1-μ]1/(1-μ))1-μ=(1hr∑k∈Irvk)μ,
and so
(73)1hr∑k∈Irvkμ≤1hr∑k∈Irtk+(1hr∑k∈Irvk)μ.
Hence, x=(xk)∈wθ(ℳ,Λ,p,s,∥·,…,·∥). This completes the proof of the theorem.
Theorem 10.
(
i) If 0<infpk≤pk≤1 for all k∈ℕ, then
(74)wθ(ℳ,Λ,p,s,∥·,…,·∥)⊆wθ(ℳ,Λ,s,∥·,…,·∥).
(
ii) If 1≤pk≤suppk=H<∞, for all k∈ℕ, then
(75)wθ(ℳ,Λ,s,∥·,…,·∥)⊆wθ(ℳ,Λ,p,s,∥·,…,·∥).
Proof.
(i) Let x=(xk)∈wθ(ℳ,Λ,p,s,∥·,…,·∥); then
(76)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]pk=0.
Since 0<infpk≤pk≤1, this implies that
(77)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]≤limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]pk;
therefore,
(78)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]=0.
Hence,
(79)wθ(ℳ,Λ,p,s,∥·,…,·∥)⊆wθ(ℳ,Λ,s,∥·,…,·∥).
(ii) Let pk≥1 for each k and suppk<∞. Let x=(xk)∈wθ(ℳ,Λ,s,∥·,…,·∥); then for each ρ>0, we have
(80)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,limr→∞1hr∑k∈Irk-slimr→∞1hr=z1,z2,…,zn-1Λk(x)-Lρ∥)]pk=0<1.
Since 1≤pk≤suppk<∞, we have
(81)limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]pk≤limr→∞1hr∑k∈Irk-s[Mk(∥Λk(x)-Lρ,z1,z2,…,zn-1∥)]=0<1.
Therefore, x=(xk)∈wθ(ℳ,Λ,p,s,∥·,…,·∥), for each ρ>0. Hence,
(82)wθ(ℳ,Λ,s,∥·,…,·∥)⊆wθ(ℳ,Λ,p,s,∥·,…,·∥).
This completes the proof of the theorem.
Theorem 11.
If 0<infpk≤pk≤suppk=H<∞, for all k∈ℕ, then
(83)wθ(ℳ,Λ,p,s,∥·,…,·∥)=wθ(ℳ,Λ,s,∥·,…,·∥).
Proof.
It is easy to prove so we omit the details.
Acknowledgment
The authors are very grateful to the referees for their valuable suggestions and comments. The third author also acknowledges the partial support by University Putra Malaysia under the project ERGS 1-2013/5527179.
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