AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 364743 10.1155/2013/364743 364743 Research Article Sequence Spaces Defined by Musielak-Orlicz Function over n-Normed Spaces http://orcid.org/0000-0003-4128-0427 Mursaleen M. 1 Sharma Sunil K. 2 Kılıçman A. 3 Alotaibi Abdullah 1 Department of Mathematics Aligarh Muslim University Aligarh 202002 India amu.ac.in 2 Department of Mathematics Model Institute of Engineering & Technology Kot Bhalwal 181122 Jammu and Kashmir India mietjammu.in 3 Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Selangor Malaysia upm.edu.my 2013 6 11 2013 2013 21 07 2013 16 09 2013 2013 Copyright © 2013 M. Mursaleen et al. 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.

In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function =(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  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)M={xw:k=1M(|xk|ρ)<}, which is called as an Orlicz sequence space. The space M is a Banach space with the norm (2)x=inf{ρ>0:k=1M(|xk|ρ)1}.

It is shown in  that every Orlicz sequence space M contains a subspace isomorphic to p  (p1). The Δ2-condition is equivalent to M(Lx)kLM(x) for all values of x0 and for L>1. A sequence =(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):u0},  k=1,2,, is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space t and its subspace h are defined as follows: (4)t={xw:I(cx)<  for  some  c>0},h={xw:I(cx)<  c>0}, where I is a convex modular defined by (5)I(x)=k=1(Mk)(xk),x=(xk)t.

We consider t equipped with the Luxemburg norm (6)x=inf{k>0:I(xk)1} or equipped with the Orlicz norm (7)x0=inf{1k(1+I(kx)):k>0}.

Let X be a linear metric space. A function p : X is called paranorm if

p(x)0 for all xX,

p(-x)=p(x) for all xX,

p(x+y)p(x)+p(y) for all x,yX,

(λn) is a sequence of scalars with λnλ      as      n and (xn) is a sequence of vectors with p(xn-x)0      as      n; then p(λnxn-λx)0          as      n.

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 , Theorem 10.4.2, pp. 183). For more details about sequence spaces, see  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.  as (8)Nθ={xw:limr1hrkIr|xk-l|=0,  for  some  l}.

Strongly almost convergent sequence was introduced and studied by Maddox  and Freedman et al. . Parashar and Choudhary  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  and recently in .

Mursaleen and Noman  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<,λkas  k, 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)=1λmk=1m(λk-λk-1)xk.

The sequence x=(xk)w is λ-bounded if supm|Λm(x)|<. It is well known  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|1λmk=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  in the mid 1960s, while for that of n-normed spaces one can see Misiak . Since then, many others have studied this concept and obtained various results; see Gunawan ([21, 22]) and Gunawan and Mashadi . Let 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 dn2. 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 α𝕂;

x+x,x2,,xnx,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=n being equipped with the n-norm x1,x2,,xnE = 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,,xnE=|det(xij)|, where xi=(xi1,xi2,,xin)n for each i=1,2,,n, leting (X,·,,·) be an n-normed space of dimension dn2 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 LX if (15)limkxk-L,z1,,zn-1=0limkfor  every  z1,,zn-1X.

A sequence (xk) in an n-normed space (X,·,,·) is said to be Cauchy if (16)limpkxk-xp,z1,,zn-1=0limxxpkfor  every  z1,,zn-1X.

If every Cauchy sequence in X converges to some LX, 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 =(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(,Λ,p,s,·,,·)={Λk(x)ρx=(xk)w:  limr1hr=×kIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pk=0,(Λk(x)ρ,z1,z2z2z2z2z2z2z2z2,)ρ>0,  s0Λk(x)ρ},wθ(,Λ,p,s,·,,·)={Λk(x)-Lρx=(xk)w:  limr1hr=×kIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]pk=0,  for  some  L,  ρ>0,  s0Λk(x)-Lρ},wθ(,Λ,p,s,·,,·)={Λk(x)ρx=(xk)w:supr1hr=×kIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pk<,(Λk(x)ρ,z1,z2z2z2,zn-1)ρ>0,  s0Λk(x)ρ}.

If we take (x)=x, we get (18)w0θ(Λ,p,s,·,,·)={Λk(x)ρx=(xk)w:limr1hr=×kIrk-s[Λk(x)ρ,z1,z2,,zn-1]pk=0,Λk(x)ρ,z1,z2z2,,zn-1ρ>0,  s0Λk(x)ρ},wθ(Λ,p,s,·,,·)={Λk(x)-Lρx=(xk)w:limr1hr=×kIrk-s[Λk(x)-Lρ,z1,z2,,zn-1]pk=0,,z1,z2z2,zn-1for  some  L,  ρ>0,  s0Λk(x)-Lρ},wθ(Λ,p,s,·,,·)={Λk(x)ρx=(xk)w:supr1hr=×kIrk-s[Λk(x)ρ,z1,z2,,zn-1]pk<,Λk(x)ρ,z1,z2z2z2,,zn-1ρ>0,  s0Λk(x)ρ}.

If we take p=(pk)=1 for all k, we have (19)w0θ(,Λ,s,·,,·)={Λk(x)ρx=(xk)w:limr1hr=×kIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]=0,Λk(x)ρ,z1,z2,z1,z2,z1,,zn-1ρ>0,  s0Λk(x)ρ},wθ(,Λ,s,·,,·)={x=(xk)w:  limr1hr=×kIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]=0,  for  some  L,  ρ>0,  s0Λk(x)-Lρ},wθ(,Λ,s,·,,·)={Λk(x)ρx=(xk)w:supr1hr=×kIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]<,Λk(x)ρ,z1,z2,z1,,z1z1,znρ>0,  s0Λk(x)ρ}.

The following inequality will be used throughout the paper. If 0pksuppk=H, K=max(1,2H-1), then (20)|ak+bk|pkK{|ak|pk+|bk|pk} for all k and ak,bk. Also |a|pkmax(1,|a|H) for all a.

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 Results Theorem 1.

Let =(Mk) be a Musielak-Orlicz function, and let p=(pk) be a bounded sequence of positive real numbers, then the spaces w0θ(,Λ,p,s,·,,·),  wθ(,Λ,p,s,·,,·), and wθ(,Λ,p,s,·,,·) are linear spaces over the field of complex number .

Proof.

Let x=(xk),  let    y=(yk)w0θ(,Λ,p,s,·,,·), and let α,β. In order to prove the result, we need to find some ρ3 such that (21)limr1hrkIrk-s[Mk(Λk(αx+βy)ρ3,z1,z2,,zn-1)]pk=0.

Since x=(xk),  y=(yk)w0θ(,Λ,p,s,·,,·), there exist positive numbers ρ1,ρ2>0 such that (22)limr1hrkIrk-s[Mk(Λk(x)ρ1,z1,z2,,zn-1)]pk=0,limr1hrkIrk-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)1hrkIrk-s[Mk(Λk(αx+βy)ρ3,z1,z2,,zn-1)]pk1hrkIrk-s[Mk(αΛk(x)ρ3,z1,z2,,zn-11hr1hr1hr=-kIrk-s+βΛk(y)ρ3,z1,z2,,zn-1)]pkK1hrkIr12pkk-s[Mk(Λk(x)ρ1,z1,z2,,zn-1)]pk+K1hrkIr12pkk-s[Mk(Λk(y)ρ2,z1,z2,,zn-1)]pkK1hrkIrk-s[Mk(Λk(x)ρ1,z1,z2,,zn-1)]pk+K1hrkIrk-s[Mk(Λk(y)ρ2,z1,z2,,zn-1)]pk0  as  r. Thus, we have αx+βyw0θ(,Λ,p,s,·,,·). Hence, w0θ(,Λ,p,s,·,,·) is a linear space. Similarly, we can prove that wθ(,Λ,p,s,·,,·) and wθ(,Λ,p,s,·,,·) are linear spaces.

Theorem 2.

Let =(Mk) be a Musielak-Orlicz function, and let p=(pk) be a bounded sequence of positive real numbers. Then w0θ(,Λ,p,s,·,,·) is a topological linear space paranormed by (24)g(x)=inf{kIrk-sρpr/H:=infinf(1hrkIrk-s[Mk(kIrk-sΛk(x)ρ,(1hrkIr(1hr1hr1hr=kIrz1,z2,,zn-1kIrk-s)]pk)1/H1ef22(efsdf1hrkIrk-s}, where H=max(1,supkpk)<.

Proof.

Clearly g(x)0 for x=(xk)w0θ(,Λ,p,s,·,,·). Since Mk(0)=0, we get g(0)=0. Again if g(x)=0, then (25)inf{(1hrkIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pk)ρpr/H:infinf(1hrkIrk-s[Mk(1hrkIrk-sΛk(x)ρ,(1hrkIr(1hrz21hrz21hrz1,z2,,zn-11hrkIrk-sΛk(x)ρ)]pk)1/H1(1hrkIrk-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)(1hrkIrk-s[Mk(Λk(x)ρϵ,z1,z2,,zn-1)]pk)1/H1. Thus, (27)(1hrkIrk-s[Mk(Λk(x)ϵ,z1,z2,,zn-1)]pk)1/H(1hrkIrk-s[Mk(Λk(x)ρϵ,z1,z2,,zn-1)]pk)1/H. Suppose that (xk)0 for each k. This implies that Λk(x)0 for each k. Let ϵ0, then (28)Λk(x)ϵ,z1,z2,,zn-1. It follows that (29)(1hrkIrk-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. Let ρ1>0 and ρ2>0 be the case such that (30)(1hrkIrk-s[Mk(Λk(x)ρ1,z1,z2,,zn-1)]pk)1/H1,(1hrkIrk-s[Mk(Λk(y)ρ2,z1,z2,,zn-1)]pk)1/H1. Let ρ=ρ1+ρ2; then, by using Minkowski's inequality, we have (31)(1hrkIrk-s[Mk(Λk(x+y)ρ,z1,z2,,zn-1)]pk)1/H(1hrkIrk-s[Mk({z1,z2,,zn-11hrkIrk-sΛk(x)+Λk(y)ρ1+ρ2}{z1,z2,,zn-11hrkIrk-sΛk(x)+Λk(y)ρ1+ρ2}Λk(x)+Λk(y)ρ1+ρ2,1hr1hr1hr1hr1hr1hr1hrkIrkIrk-sz1,z2,,zn-11hrkIrk-sΛk(x)+Λk(y)ρ1+ρ2)]pk)1/H({[Λk(y)ρ2,z1,z2,,zn-11hrkIrk-s]}1hrkIrk-s[Λk(y)ρ2{[Λk(y)ρ2,z1,z2,,zn-11hrkIrk-s]}Mk(ρ1ρ1+ρ2)1hrkIrk-s1hrkIr×[Λk(x)ρ1,z1,z2,,zn-1]+(ρ2ρ1+ρ2)1hrkIrk-s1hr-kIr×[Λk(y)ρ2,z1,z2,,zn-11hrkIrk-s]]pk)1/H(ρ1ρ1+ρ2)1hr1hr×(1hrkIrk-s[Mk(Λk(x)ρ1,z1,z2,,zn-1)]pk)1/H+(ρ2ρ1+ρ2)1hr1hr×(1hrkIrk-s[Mk(Λk(y)ρ2,z1,z2,,zn-1)]pk)1/H1. Since ρ,  ρ1, and ρ2 are nonnegative, we have (32)g(x+y)=inf{1hrkIrk-s{z1,z2,,zn-11hrkIrk-sΛk(x+y)ρ)]pk)1/H}ρpr/H:infinf(1hrkIrk-s[Mk(1hrkIrk-sΛk(x+y)ρ,infinfinfinfiniinfinnfinz1,z2,,zn-11hrkIrk-sΛk(x+y)ρ)]pk)1/H1}inf{1hrkIrk-s{z1,z2,,zn-11hrkIrk-sΛk(x+y)ρ)]pk)1/H}Λk(x)ρ1(ρ1)pr/H:infinfinf(1hrkIrk-s[Mk(1hrkIrk-sΛk(x)ρ1,infinfinfinfinfinfinfinfinfiz1,z2,,zn-11hrkIrk-sΛk(x)ρ1)]pk)1/H1}f+inf{{z1,z2,,zn-11hrkIrk-sΛk(x+y)ρ)]pk)1/H}Λk(y)ρ21hrkIrk-s(ρ2)pr/H:=infinf(1hrkIrk-s[Mk(1hrkIrk-sΛk(y)ρ2,infinfinfininifinfinfinfinfz1,z2,,zn-11hrkIrk-sΛk(y)ρ2)]pk)1/H1}. 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{1hrkIrk-sΛk(μx)ρρpr/H:=innii(1hrkIrk-s[Mk(1hrkIrk-sΛk(μx)ρ,(1hrkIrk-s(1hr=kIriik-sz1,z2,,zn-11hrkIrk-sΛk(μx)ρ)]pk)1/Hinin&fin11hrkIrk-s}. Thus, (34)g(μx)=inf{kIr1hrΛk(x)k-st(|μ|t)pr/H:=inin(1hrkIrk-s[Mk(1hrkIrk-s1hrkIrk-sΛk(x)t,(1hrkIr(1hr1hr1hrkIriik-sz1,z2,,zn-1Λ1hrkIr)]pk)1/H=in&finfin11hrkIrk-s1hrkIrk-s}, where 1/t=ρ/|μ|. Since |μ|prmax(1,|μ|suppr), we have (35)g(μx)max(1,|μ|suppr)×inf{Λk(x)t1hrkIrk-stpr/H:=inf==(1hrkIrk-s[Mk(1hrkIrk-sΛk(x)t,(1hrkIr=k-s(1hr=kIriik-sz1,z2,,zn-11hrkIrk-sΛk(μx)ρ)]pk)1/H=n&fn&=11hrkIrk-s}. So the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.

Theorem 3.

Let =(Mk) be a Musielak-Orlicz function. If supk[Mk(x)]pk< for all fixed x>0, then w0θ(,Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·).

Proof.

Let x=(xk)w0θ(,Λ,p,s,·,,·); then there exists positive number ρ1 such that (36)limr1hrkIrk-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)supr1hrkIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pk=supr1hrkIrk-s[Mk(Λk(x)+L-Lρ,=supr1hrkIrk-s=suprkIrk-sz1,z2,,zn-1Λk(x)+L-Lρ)]pkKsupr1hrkIrk-s12pk[Mk(Λk(x)-Lρ1,z1,z2,,zn-1{[Mk(Λk(x)-Lρ1,}{[Mk(Λk(x)-Lρ1,})]pkKsupr+Ksupr1hrkIrk-s12pk[Mk(Lρ1,z1,z2,,zn-1)]pkKsupr1hrkIrk-s[Mk(Λk(x)-Lρ1,z1,z2,,zn-1)]pkKsupr+Ksupr1hrkIrk-s[Mk(Lρ1,z1,z2,,zn-1)]pk<. Hence, x=(xk)wθ(,Λ,p,s,·,,·).

Theorem 4.

Let 0<infpk=hpksuppk=H< and let =(Mk),'=(Mk) be Musielak-Orlicz functions satisfying Δ2-condition, then one has

w0θ(',Λ,p,s,·,,·)w0θ(',Λ,p,s,·,,·);

wθ(',Λ,p,s,·,,·)wθ(',Λ,p,s,·,,·);

wθ(',Λ,p,s,·,,·)wθ(',Λ,p,s,·,,·).

Proof.

Let x=(xk)w0θ(,Λ,p,s,·,,·), then we have (38)limr1hrkIrk-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 0tδ. Let (yk)=Mk[Λk(x)/ρ,z1,z2,,zn-1] for all k. We can write (39)1hrkIrk-s(Mk[yk])pk=1hrykδkIrk-s(Mk[yk])pk+1hrykδkIrk-s(Mk[yk])pk. So, we have (40)1hrykδkIrk-s(Mk[yk])pk[Mk(1)]H1hrykδkIrk-s(Mk[yk])pk  [Mk(2)]H1hrykδkIrk-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 =(Mk) satisfies Δ2-condition, we can write (42)Mk(yk)<12TykδMk(2)+12TykδMk(2)=TykδMk(2). Hence, (43)1hrykδkIrk-sMk[yk]pkmax(1,(TMk(2)δ)H)1hrykδkIrk-s[yk]pk. From (40) and (43), we have x=(xk)w0θ(',Λ,p,s,·,,·). This completes the proof of (i). Similarly we can prove that (44)wθ(,Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·),wθ(,Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·).

Theorem 5.

Let 0<h=infpk=pk<suppk=H<. Then for a Musielak-Orlicz function =(Mk) which satisfies Δ2-condition, one has

w0θ(Λ,p,s,·,,·)w0θ(,Λ,p,s,·,,·);

wθ(Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·);

wθ(Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·).

Proof.

It is easy to prove, so we omit the details.

Theorem 6.

Let =(Mk) be a Musielak-Orlicz function and let 0<h=infpk. Then wθ(,Λ,p,s,·,,·)w0θ(Λ,p,s,·,,·) if and only if (45)limr1hrkIrk-s(Mk(t))pk= for some t>0.

Proof.

Let wθ(,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-1k, such that (47)1hr(j)=kIr(j)k-s(Mk(t0))pkK<,m=1,2,3,. Let us define x=(xk) as follows: (48)Λk(x)={ρt0,kIr(j)0,kIr(j). Thus, by (47),    xwθ(,Λ,p,s,·,,·). But xw0(Λ,p,s,·,,·). Hence, (45) must hold.

Conversely, suppose that (45) holds and let xwθ(,Λ,p,s,·,,·). Then for each r, (49)1hrkIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pkK<. Suppose that xw0θ(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  >ϵfor  kk0. From properties of sequence of Orlicz functions, we obtain (51)[Mk(Λk(x)ρ,z1,z2,,zn-1)]pkMk(ϵ)pk, which contradicts (45), by using (49). Hence, we get (52)wθ(,Λ,p,s,·,,·)w0θ(Λ,p,s,·,,·).

This completes the proof.

Theorem 7.

Let =(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:

wθ(Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·);

w0θ(Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·);

supr1/hr  kIrk-s(Mk(t))pk<  forall  t>0.

Proof.

(i) (ii). Let (i) hold. To verify (ii), it is enough to prove (53)w0θ(Λ,p,s,·,,·)wθ(,Λ,p,s,·,,·). Let x=(xk)w0θ(Λ,p,s,·,,·). Then for ϵ>0, there exists r0, such that (54)1hrkIrk-s[Λk(x)ρ,z1,z2,,zn-1]pk<ϵ. Hence, there exists K>0 such that (55)supr1hrkIrk-s[Λk(x)ρ,z1,z2,,zn-1]pk<K. So, we get x=(xk)wθ(,Λ,p,s,·,,·).

(ii) (iii). Let (ii) hold. Suppose (iii) does not hold. Then for some t>0(56)supr1hrkIrk-s(Mk(t))pk=, and therefore we can find a subinterval Ir(j), of the set of interval Ir, such that (57)1hr(j)kIr(j)k-s(Mk(1j))pk>j,  j=1,2,3, Let us define x=(xk) as follows: (58)Λk(x)={ρj,kIr(j)0,kIr(j).

Then x=(xk)w0θ(Λ,p,s,·,,·). But by (57), xwθ(,Λ,p,s,·,,·), which contradicts (ii). Hence, (iii) must holds.

(iii) (i). Let (iii) hold and suppose that x=(xk)wθ(Λ,p,s,·,,·). Suppose that x=(xk)wθ(,Λ,p,s,·,,·); then (59)supr1hrkIrk-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)supr1hrkIrk-s(Mk(t))pk=, which contradicts (iii). Hence, (i) must hold.

Theorem 8.

Let =(Mk) be a Musielak-Orlicz function. Then the following statements are equivalent:

w0θ(,Λ,p,s,·,,·)w0θ(Λ,p,s,·,,·);

w0θ(,Λ,p,s,·,,·)wθ(Λ,p,s,·,,·);

infr1/hr  kIrk-s(Mk(t))pk>0  forall  t>0.

Proof.

(i) (ii). It is obvious.

(ii) (iii). Let (ii) hold. Suppose that (iii) does not hold. Then (61)infr1hrkIrk-s(Mk(t))pk=0for  some  t>0, and we can find a subinterval Ir(j), of the set of interval Ir, such that (62)1hr(j)kIr(j)k-s(Mk(j))pk<1j,j=1,2,3, Let us define x=(xk) as follows: (63)Λk(x)={ρj,kIr(j)0,kIr(j).

Thus, by (62), x=(xk)w0θ(,Λ,p,s,·,,·), but x=(xk)wθ(Λ,p,s,·,,·), which contradicts (ii). Hence, (iii) must hold.

(iii) (i). Let (iii) hold. Suppose that x=(xk)w0θ(,Λ,p,s,·,,·). Then (64)1hrkIrk-s[Mk(Λk(x)ρ,z1,z2,,zn-1)]pk0(Λk(x)ρ,z1,z2,,)(Λk(x)ρ,)as      r. 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-1ϵk. 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)limr1hrkIrk-s(Mk(ϵ))pk=0, which contradicts (iii). Hence, (i) must hold.

Theorem 9.

Let 0pkqk for all k and let (qk/pk) be bounded. Then (68)wθ(,Λ,q,s,·,,·)wθ(,Λ,p,s,·,,·).

Proof.

Let x=(xk)wθ(,Λ,q,s,·,,·); write (69)tk=[Mk(Λk(x)ρ,z1,z2,,zn-1)]qk and μk=pk/qk for all k. Then 0<μk1 for all k. Take 0<μμk for k. Define sequences (uk) and (vk) as follows.

For tk1, let uk=tk and vk=0, and for tk<1, let uk=0 and vk=tk. Then clearly for all k, we have (70)tk=uk+vk,tkμk=ukμk+vkμk. Now it follows that ukμkuktk and vkμkvkμ. Therefore, (71)1hrkIrtkμk=1hrkIr(ukμk+vkμk)1hrkIrtk+1hrkIrvkμ. Now for each k, (72)1hrkIrvkμ=kIr(1hrvk)μ(1hr)1-μ(kIr[(1hrvk)μ]1/μ)μ×(kIr[(1hr)1-μ]1/(1-μ))1-μ=(1hrkIrvk)μ, and so (73)1hrkIrvkμ1hrkIrtk+(1hrkIrvk)μ. Hence, x=(xk)wθ(,Λ,p,s,·,,·). This completes the proof of the theorem.

Theorem 10.

( i) If 0<infpkpk1 for all k, then (74)wθ(,Λ,p,s,·,,·)wθ(,Λ,s,·,,·).

( ii) If 1pksuppk=H<, for all k, then (75)wθ(,Λ,s,·,,·)wθ(,Λ,p,s,·,,·).

Proof.

(i) Let x=(xk)wθ(,Λ,p,s,·,,·); then (76)limr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]pk=0. Since 0<infpkpk1, this implies that (77)limr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]limr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]pk; therefore, (78)limr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]=0. Hence, (79)wθ(,Λ,p,s,·,,·)wθ(,Λ,s,·,,·). (ii) Let pk1 for each k and suppk<. Let x=(xk)wθ(,Λ,s,·,,·); then for each ρ>0, we have (80)limr1hrkIrk-s[Mk(Λk(x)-Lρ,limr1hrkIrk-slimr1hr=z1,z2,,zn-1Λk(x)-Lρ)]pk=0<1. Since 1pksuppk<, we have (81)limr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]pklimr1hrkIrk-s[Mk(Λk(x)-Lρ,z1,z2,,zn-1)]=0<1. Therefore, x=(xk)wθ(,Λ,p,s,·,,·), for each ρ>0. Hence, (82)wθ(,Λ,s,·,,·)wθ(,Λ,p,s,·,,·).

This completes the proof of the theorem.

Theorem 11.

If 0<infpkpksuppk=H<, for all k, then (83)wθ(,Λ,p,s,·,,·)=wθ(,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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