We introduce new sequence spaces by using Musielak-Orlicz function and a generalized B∧ μ-difference operator on n-normed space. Some topological properties and inclusion relations are also examined.
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 function M 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 sequence M=(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):limr→∞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:
∥(x1,x2,…,xn)∥=0 if and only if x1,x2,x3,…,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, 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)limk→∞∥(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)limk→∞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):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=0,ρ>0x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir},wθ(M,B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk=0,forsomeL,ρ>0x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir},w∞θ(M,B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)pk<∞,ρ>0x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir};
when M(x)=x, we get
(21)w0θ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir×∑k∈Ir(∥(B∧μxkρ,z1,…,zn-1)∥)pk=0,ρ>0x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir},wθ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir×∑k∈Ir(∥(B∧μxk-Lρ,z1,…,zn-1)∥)pk=0,forsomeL,ρ>0x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir},w∞θ(B∧μ,p,∥·,…,·∥)={x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir×∑k∈Ir(∥(B∧μxkρ,z1,…,zn-1)∥)pk<∞,ρ>0{x=(xk)∈s(w-x):limr→∞1hr∑k∈Ir};
when pk=1, for all k, we get
(22)w0θ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)=0,ρ>0limr→∞1hr∑k∈Ir},wθ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxk-Lρ,z1,…,zn-1)∥)=0,forsomeL,ρ>0x=(xk)∈w(s-x):limr→∞1hr∑k∈Ir},w∞θ(M,B∧μ,∥·,…,·∥)={x=(xk)∈w(s-x):limr→∞1hr∑k∈Ir×∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)<∞,ρ>0x=(xk)∈w(s-x):limr→∞1hr∑k∈Ir}.
The following inequality will be used throughout the paper. If 0≤pk≤suppk=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 ResultsTheorem 1.
Let M=(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.
Let x=(xk), y=(yk)∈w0θ(M,B∧μ,p,∥·,…,·∥), and α,β∈C. Then there exist positive real numbers ρ1 and ρ2 such that
(24)limr→∞1hr∑k∈IrMk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk=0,limr→∞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)limr→∞1hr∑k∈IrMk(∥(B∧μ(αxk+βyk)ρ3,z1,…,zn-1)∥)pk≤limr→∞1hr∑k∈IrMk(∥(B∧μαxkρ3,z1,…,zn-1)∥hhhhhhhhhhhhhhh+∥(B∧μβykρ3,z1,…,zn-1)∥)pk≤Klimr→∞1hr∑k∈Ir12pkMk(∥(B∧μxkρ1,z1,…,zn-1)∥)pk+Klimr→∞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.
Let M=(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,supkpk<∞).
Proof.
Clearly g(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.
Since ρ′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. Let ν 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|suppk), we have
(36)g(νx)=max(1,|ν|suppk)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.
Let M=(Mk) be a Musielak-Orlicz function. If supk(Mk(x))pk<∞ for all fixed x>0, then wθ(M,B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥).
Proof.
Let x=(xk)∈wθ(M,B∧μ,p,∥·,…,·∥). Then there exists some positive number ρ1 such that
(37)limr→∞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)limr→∞1hr∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=limr→∞1hr∑k∈IrMk(∥(B∧μxk-L+Lρ,z1,…,zn-1)∥)pk≤K{limr→∞1hr∑k∈IrMk12pk(∥(B∧μxk-Lρ1,z1,…,zn-1)∥)pk+limr→∞1hr∑k∈IrMk12pk(∥(Lρ1,z1,…,zn-1)∥)pk}<K{limr→∞1hr∑k∈IrMk(∥(B∧μxk-Lρ1,z1,…,zn-1)∥)pk+limr→∞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.
Let M=(Mk) be a Musielak-Orlicz function and 0<h=infpk. Then
(39)w∞θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥)
if and only if
(40)limr→∞1hr∑k∈IrMk(t)pk=∞,forsomet>0.
Proof.
Let w∞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.
Let 0<h=infpk≤suppk=H<∞. For any Musielak-Orlicz function M=(Mk) which satisfies Δ2-condition, one has
w0θ(B∧μ,p,∥·,…,·∥)⊂w0θ(M,B∧μ,p,∥·,…,·∥)
wθ(B∧μ,p,∥·,…,·∥)⊂wθ(M,B∧μ,p,∥·,…,·∥)
w∞θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥).
Proof.
(i) Let x=(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.
Let M=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:
w∞θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥);
w0θ(B∧μ,p,∥·,…,·∥)⊂w∞θ(M,B∧μ,p,∥·,…,·∥);
supr(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)supr1hr∑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)supr1hr∑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)supr1hr∑k∈IrMk(∥(B∧μxkρ,z1,…,zn-1)∥)pk=∞.
Let t=∥B∧μxk/ρ,z1,…,zn-1∥ for each k, and then by (57) supr(1/hr)∑k∈IrMk(t)pk=∞, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.
Theorem 7.
Let M=(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:
w0θ(M,B∧μ,p,∥·,…,·∥)⊂w0θ(B∧μ,p,∥·,…,·∥);
w0θ(M,B∧μ,p,∥·,…,·∥)⊂w∞θ(B∧μ,p,∥·,…,·∥);
infr(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)infr1hr∑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 limr→∞(1/hr)∑k∈IrMk(ε)pk=0, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.
Theorem 8.
(i) If 0<infpk≤pk≤1 for all k, then wθ(M,B∧μ,∥·,…,·∥)⊆wθ(M,B∧μ,p,∥·,…,·∥).
(ii) If 1≤pk≤suppk=H<∞, then wθ(M,B∧μ,p,∥·,…,·∥)⊆wθ(M,B∧μ,∥·,…,·∥).
Proof.
(i) Let x∈wθ(M,B∧μ,∥·,…,·∥). Since 0<infpk≤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≤suppk=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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. ERGS 1-2013/5527179. The authors are grateful also to the anonymous referees for a careful checking of the details and for helpful comments that improved the paper.
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