AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/691632 691632 Research Article New Difference Sequence Spaces Defined by Musielak-Orlicz Function http://orcid.org/0000-0003-4128-0427 Mursaleen M. 1 Sharma Sunil K. 2 Mohiuddine S. A. 3 Kılıçman A. 4 Başar Feyzi 1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002 India amu.ac.in 2 Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal, Jammu and Kashmir 181122 India mietjammu.in 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589 Saudi Arabia kau.edu.sa 4 Department of Mathematics, University Putra Malaysia (UPM), 43400 Serdang, Selangor Malaysia upm.edu.my 2014 2272014 2014 17 03 2014 11 07 2014 11 07 2014 22 7 2014 2014 Copyright © 2014 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.

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 . It was further generalized by Et and Çolak  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=xkkN 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  taking z=l(p), c(p), and c0(p).

Dutta  introduced the following difference sequence spaces using a new difference operator: (3)Z(Δ(η))={x=(xk)ω:Δ(η)xz}for  z=l,c,c0, where Δ(η)x=(Δ(η)xk)=(xk-xk-η) for all k,ηN.

In , 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,μN and Δ(η)μxk=(Δ(η)μxk)=(Δ(η)μ-1xk-Δ(η)μ-1xk-η) and Δ(η)0xk=xk for all k,ηN, 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  and references therein. Başar and Altay  introduced the generalized difference matrix B=(bmk) for all k,mN, which is a generalization of Δ(1)-difference operator by (5)bmk={r,k=ms,k=m-10,(k>m)or(0k<m-1).

Başarir and Kayikçi  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 x00 such that M(2x)KM(x) for all xx0. The Δ2-condition is equivalent to M(Lx)KLM(x) for all x>x0>0 and for L,K>1.

Lindenstrauss and Tzafriri  used the idea of Orlicz function to define the following sequence space: (8)lM={xω:k=1M(|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=1M(|xk|ρ)1}. It is shown in  that every Orlicz sequence space lM contains a subspace isomorphic to lp(p1).

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):u0},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={xω:IM(cx)<  for  some  c>0},hM={xω:IM(cx)<  c>0}, where IM is a convex modular defined by (12)IM(x)=k=1Mk(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)x0=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.  as follows: (15)Nθ={x=(xk):limr1hrkIr|xk-L|=0,for  some  L}. The concept of 2-normed spaces was initially developed by Gähler  in the mid of 1960’s, while that of n-normed spaces one can see in Misiak . Since then, many others have studied this concept and obtained various results; see Gunawan [23, 24] and Gunawan and Mashadi . For more details about sequence spaces see  and references therein. Let nN and X be 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,x3,,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,,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 dn2 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 LX if (18)limk(xk-L,z1,,zn-1)=0,00000for  every  z1,,zn-1X. A sequence (xk) in a normed space (X,·,,·) is said to be Cauchy if (19)limkp(xk-xp,z1,,zn-1)=0,00000for  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 (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,Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIrMk((Bμxkρ,z1,,zn-1))pk=0,ρ>0x=(xk)s(w-x):limr1hrkIr},wθ(M,Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIrMk((Bμxk-Lρ,z1,,zn-1))pk=0,for  some  L,ρ>0x=(xk)s(w-x):limr1hrkIr},wθ(M,Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIrMk((Bμxkρ,z1,,zn-1))pk<,ρ>0x=(xk)s(w-x):limr1hrkIr}; when M(x)=x, we get (21)w0θ(Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIr((Bμxkρ,z1,,zn-1))pk=0,  ρ>0x=(xk)s(w-x):limr1hrkIr},wθ(Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIr((Bμxk-Lρ,z1,,zn-1))pk=0,for  some  L,ρ>0x=(xk)s(w-x):limr1hrkIr},wθ(Bμ,p,·,,·)={x=(xk)s(w-x):limr1hrkIr×kIr((Bμxkρ,z1,,zn-1))pk<,ρ>0{x=(xk)s(w-x):limr1hrkIr}; when pk=1, for all k, we get (22)w0θ(M,Bμ,·,,·)={x=(xk)w(s-x):limr1hrkIr×kIrMk((Bμxkρ,z1,,zn-1))=0,ρ>0limr1hrkIr},wθ(M,Bμ,·,,·)={x=(xk)w(s-x):limr1hrkIr×kIrMk((Bμxk-Lρ,z1,,zn-1))=0,for  some  L,ρ>0x=(xk)w(s-x):limr1hrkIr},wθ(M,Bμ,·,,·)={x=(xk)w(s-x):limr1hrkIr×kIrMk((Bμxkρ,z1,,zn-1))<,ρ>0x=(xk)w(s-x):limr1hrkIr}. The following inequality will be used throughout the paper. If 0pksuppk=H, k=max(1,2H-1), then (23)|ak+bk|pkK{|ak|pk+|bk|pk} for all k and ak,bkC. Also |a|pkmax(1,|a|H) for all aC.

2. Main Results Theorem 1.

Let M=(Mk) be a Musielak-Orlicz function and p=(pk) a bounded sequence of positive real numbers; the spaces w0θ(M,Bμ,p,·,,·), wθ(M,Bμ,p,·,,·), and wθ(M,Bμ,p,·,,·) are linear over the field of complex numbers C.

Proof.

Let x=(xk), y=(yk)w0θ(M,Bμ,p,·,,·), and α,βC. Then there exist positive real numbers ρ1 and ρ2 such that (24)limr1hrkIrMk((Bμxkρ1,z1,,zn-1))pk=0,limr1hrkIrMk((Bμykρ2,z1,,zn-1))pk=0. Define ρ3=max(2|α|ρ1,2|β|ρ2). Since ·,,· is an n-norm on X and Mks are nondecreasing and convex functions so by using inequality (23) we have (25)limr1hrkIrMk((Bμ(αxk+βyk)ρ3,z1,,zn-1))pklimr1hrkIrMk((Bμαxkρ3,z1,,zn-1)hhhhhhhhhhhhhhh+(Bμβykρ3,z1,,zn-1))pkKlimr1hrkIr12pkMk((Bμxkρ1,z1,,zn-1))pk+Klimr1hrkIr12pkMk((Bμykρ2,z1,,zn-1))pk=0. Thus, we have αx+βyw0θ(M,Bμ,p,·,,·). Hence w0θ(M,Bμ,p,·,,·) is a linear space. Similarly, we can prove that wθ(M,Bμ,p,·,,·) and wθ(M,Bμ,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,Bμ,p,·,,·) is a topological linear space paranormed by (26)g(x)=inf{(1hrkIrMk((Bμxkρ,z1,,zn-1))pk)1/Mρpr/M:hhhhh(1hrkIrMk((Bμxkρ,z1,,zn-1))pk)1/Mhhhhh1(1hrkIrMk((Bμxkρ,z1,,zn-1))pk)1/Mρpr/M}, where M=max(1,supkpk<).

Proof.

Clearly g(x)0 for x=(xk)w0θ(M,Bμ,p,·,,·). Since Mk(0)=0, we get g(0)=0. Again, if g(x)=0, then (27)g(x)=inf{(1hrkIrMk((Bμxkρ,z1,,zn-1))pk)1/Mρpr/M:hhhhh(1hrkIrMk((Bμxkρ,z1,,zn-1))pk)1/Mhhhhh1(1hrkIrMk((Bμxkρ,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)(1hrkIrMk((Bμxkρε,z1,,zn-1))pk)1/M1.

Thus (29)(1hrkIrMk((Bμxkε,z1,,zn-1))pk)1/M(1hrkIrMk((Bμxkρε,z1,,zn-1))pk)1/M1 for each r, and suppose that xk0 for each kN. This implies that Bμxk0 for each kN. Let ε0, then (30)(((Bμxkρε,z1,,zn-1))pk)1/M; which is a contradiction. Therefore, Bμxk=0 for each k and thus xk=0 for each kN. Let ρ1>0 and ρ2>0 be such that (31)(1hrkIrMk((Bμxkρ1,z1,,zn-1))pk)1/M1,(1hrkIrMk((Bμykρ2,z1,,zn-1))pk)1/M11hrkIrMk((Bμykρ2,z1,,zn-1))pkfor  each  r. Let ρ=ρ1+ρ2; then by using Minkowski’s inequality, we have (32)(1hrkIrMk((Bμ(xk+yk)ρ,z1,,zn-1))pk)1/M(1hrkIrMk((Bμxk+Bμykρ1+ρ2,z1,,zn-1))pk)1/M(1hrkIrMkhhhh×((ρ1ρ1+ρ2)(Bμxkρ1,z1,,zn-1)hhhhhhhh+(ρ2ρ1+ρ2)(Bμykρ2,z1,,zn-1))pk1hrkIr)1/M(ρ1ρ1+ρ2)×(1hrkIrMk((Bμxkρ1,z1,,zn-1))pk)1/M+(ρ2ρ1+ρ2)×(1hrkIrMk((Bμykρ2,z1,,zn-1))pk)1/M1.

Since ρs are nonnegative, we have (33)g(x+y)=inf{ρpr/M:(1hrkIrMkhhhhhhhhhhhh×((Bμxkρ,z1,,zn-1))pkkIrMk)1/Mhh1:(1hrkIrMk}inf{ρ1pr/M:(1hrkIrMkhhhhhhhhhhhhh×((Bμxkρ1,z1,,zn-1))pkkIrMk)1/M1kIrMk}+inf{ρ2pr/M:(1hrkIrMkhhhhhhhhhhhhhhh×((Bμykρ2,z1,,zn-1))pkkIrMk)1/M1ρ2pr/M:(1hrkIrMk}. 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:(1hrkIrMkhhhhhhhhhhhhhh×((νBμxkρ,z1,,zn-1))pk1hrkIrMk)1/Mhhhhhh1ρpr/M:(1hrkIrMk}. Then (35)g(νx)=inf{(|ν|t)pr/M:(1hrkIrMkhhhhhhhhhhhhhhhh×((Bμxkt,z1,,zn-1))pk1hrkIrMk)1/M1:(1hrkIrMk}, where t=ρ/|ν|. Since |ν|prmax(1,|ν|suppk), we have (36)g(νx)=max(1,|ν|suppk)inf×{((Bμxkt,z1,,zn-1))pk)1/M(t)pr/M:hhhh(1hrkIrMk×((Bμxkt,z1,,zn-1))pk)1/M1}. 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,Bμ,p,·,,·)wθ(M,Bμ,p,·,,·).

Proof.

Let x=(xk)wθ(M,Bμ,p,·,,·). Then there exists some positive number ρ1 such that (37)limr1hrkIrMk((Bμxk-Lρ1,z1,,zn-1))pk=0. Define ρ=2ρ1. Since Mk is nondecreasing and convex and by using inequality (23), we have (38)limr1hrkIrMk((Bμxkρ,z1,,zn-1))pk=limr1hrkIrMk((Bμxk-L+Lρ,z1,,zn-1))pkK{limr1hrkIrMk12pk((Bμxk-Lρ1,z1,,zn-1))pk+limr1hrkIrMk12pk((Lρ1,z1,,zn-1))pk}<K{limr1hrkIrMk((Bμxk-Lρ1,z1,,zn-1))pk+limr1hrkIrMk((Lρ1,z1,,zn-1))pk}. Hence x=(xk)wθ(M,Bμ,p,·,,·). This completes the proof of the theorem.

Theorem 4.

Let M=(Mk) be a Musielak-Orlicz function and 0<h=infpk. Then (39)wθ(M,Bμ,p,·,,·)w0θ(M,Bμ,p,·,,·) if and only if (40)limr1hrkIrMk(t)pk=,forsomet>0.

Proof.

Let wθ(M,Bμ,p,·,,·)w0θ(M,Bμ,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=(Bμxk/ρ,z1,,zn-1) for all k, such that (41)1hr(m)kIr(m)M(n0)pkN<,m=1,2,3,. Let us define x=(xk) as follows: (42)Bμxk={ρn0,kIr(m)0,kIr(m). Thus by (41), x=(xk)wθ(M,Bμ,p,·,,·). But x=(xk)w0θ(M,Bμ,p,·,,·). Hence (40) must hold.

Conversely, suppose that (40) holds and let x=(xk)wθ(M,Bμ,p,·,,·). Then, (43)1hrkIrMk((Bμxkρ,z1,,zn-1))pkN<. Suppose that x=(xk)w0θ(M,Bμ,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)(Bμxkρ,z1,,zn-1)>εfor  NN0. We have Mk((Bμxk/ρ,z1,,zn-1))M(ε)pk, which contradicts (40) by using (43). Hence we get (45)wθ(M,Bμ,p,·,,·)w0θ(M,Bμ,p,·,,·). This completes the proof.

Theorem 5.

Let 0<h=infpksuppk=H<. For any Musielak-Orlicz function M=(Mk) which satisfies Δ2-condition, one has

w0θ(Bμ,p,·,,·)w0θ(M,Bμ,p,·,,·)

wθ(Bμ,p,·,,·)wθ(M,Bμ,p,·,,·)

wθ(Bμ,p,·,,·)wθ(M,Bμ,p,·,,·).

Proof.

(i) Let x=(xk)w0θ(Bμ,p,·,,·). Then, we have (46)1hrkIr((Bμxkρ,z1,,zn-1))pk0as  r. Let ε>0, and choose δ with 0<δ<1 such that Mk<ε for 0tδ. We can write (47)1hrkIr((Bμxkρ,z1,,zn-1))pk=1hrkIr(Bμxk/ρ,z1,,zn-1)δ((Bμxkρ,z1,,zn-1))pk+1hrkIr(Bμxk/ρ,z1,,zn-1)>δ((Bμxkρ,z1,,zn-1))pk. For the first summation above, we can write (48)1hrkIr(Bμxk/ρ,z1,,zn-1)δ((Bμxkρ,z1,,zn-1))pk<max(ε,εh). By using continuity of Mk, for the second summation we can write (49)(Bμxkρ,z1,,zn-1)<1+((Bμxk/ρ,z1,,zn-1))δ. Since each Mk is nondecreasing and convex and satisfies Δ2-condition, it follows that (50)1hrkIrMk((Bμxkρ,z1,,zn-1))pkmax(ε,εh)+max{1,[2Mk((Bμxk/ρ,z1,,zn-1))δ]h}×1hrkIr((Bμxkρ,z1,,zn-1))pk. Taking limit as ε0 and r, it follows that x=(xk)w0θ(M,Bμ,p,·,,·). Hence w0θ(Bμ,p,·,,·)w0θ(M,Bμ,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:

wθ(Bμ,p,·,,·)wθ(M,Bμ,p,·,,·);

w0θ(Bμ,p,·,,·)wθ(M,Bμ,p,·,,·);

supr(1/hr)kIrMk(t)pk< for all t>0, where t=Bμxk/ρ,z1,,zn-1.

Proof.

(i) (ii) Suppose (i) holds. In order to prove (ii) we have to show that (51)wθ(Bμ,p,·,,·)wθ(M,Bμ,p,·,,·). Let x=(xk)w0θ(Bμ,p,·,,·). Then for a given ε>0 there exists s>sε such that (52)1hrkIr((Bμxkρ,z1,,zn-1))pk<ε. Hence there exists K>0 such that (53)supr1hrkIr((Bμxkρ,z1,,zn-1))pk<K. This shows that x=(xk)wθ(M,Bμ,p,·,,·).

(ii) (iii) Suppose (ii) holds and (iii) fails to hold. Then for some t>0, (54)supr1hrkIrMk(ε)pk=, and, therefore, we can find a subinterval Ir(m) of the set of intervals Ir such that (55)1hr(m)kIr(m)Mk(1m)pkm,m=1,2,3,. Let us define x=(xk) as follows: (56)Bμxk={ρm,kIr(m)0,kIr(m). Thus x=(xk)w0θ(Bμ,p,·,,·). But by (55), x=(xk)wθ(M,Bμ,p,·,,·) which contradicts (ii). Hence (iii) must hold.

(iii) (i) Let (iii) hold. Suppose that x=(xk)wθ(M,Bμ,p,·,,·). Then for x=(xk)wθ(Bμ,p,·,,·). (57)supr1hrkIrMk((Bμxkρ,z1,,zn-1))pk=. Let t=Bμxk/ρ,z1,,zn-1 for each k, and then by (57) supr(1/hr)kIrMk(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,Bμ,p,·,,·)w0θ(Bμ,p,·,,·);

w0θ(M,Bμ,p,·,,·)wθ(Bμ,p,·,,·);

infr(1/hr)kIrMk(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)infr1hrkIrMk(t)pk=0for  some  t>0, and we can find a subinterval Ir(m) of the set of intervals Ir such that (59)1hr(m)kIr(m)Mk(m)pk<1m,m=1,2,3,. Let us define x=(xk) as follows: (60)Bμxk={ρm,kIr(m)0,kIr(m). Thus by (iii), x=(xk)w0θ(M,Bμ,p,·,,·). But x=(xk)wθ(Bμ,p,·,,·) which contradict (ii). Hence (iii) must hold.

(iii) (i) Let (iii) hold. Suppose that x=(xk)w0θ(M,Bμ,p,·,,·). Therefore, (61)1hrkIrMk((Bμxkρ,z1,,zn-1))pk0as  r. Again suppose x=(xk)w0θ(Bμ,p,·,,·) for some number ε>0 and a subinterval Ir(m) of the set of intervals Ir, we have (62)(Bμxkρ,z1,,zn-1)εk. Then, from properties of the Orlicz function, we can write (63)Mk((Bμxkρ,z1,,zn-1))pkMk(ε)pk. Consequently, by (61), we have limr(1/hr)kIrMk(ε)pk=0, which contradicts (iii). Hence (i) must hold. This completes the proof of the theorem.

Theorem 8.

(i) If 0<infpkpk1 for all k, then wθ(M,Bμ,·,,·)wθ(M,Bμ,p,·,,·).

(ii) If 1pksuppk=H<, then wθ(M,Bμ,p,·,,·)wθ(M,Bμ,·,,·).

Proof.

(i) Let xwθ(M,Bμ,·,,·). Since 0<infpk1, we get (64)1hrkIrMk((Bμxk-Lρ,z1,,zn-1))1hrkIrMk((Bμxk-Lρ,z1,,zn-1))pk, and hence xwθ(M,Bμ,p,·,,·).

(ii) 1pksuppk=H< and x=(xk)wθ(M,Bμ,p,·,,·). Then for each 0<ε<1 there exists a positive integer s0 such that (65)1hrkIrMk((Bμxk-Lρ,z1,,zn-1))pkε<1r>s0.

This implies that (66)1hrkIrMk((Bμxk-Lρ,z1,,zn-1))pk1hrkIrMk((Bμxk-Lρ,z1,,zn-1)). Therefore x=(xk)wθ(M,Bμ,·,,·). 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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