New Difference Sequence Spaces Defined by Musielak-Orlicz Function

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Research Article

New Difference Sequence Spaces Defined by

Musielak-Orlicz Function

M. Mursaleen,

1

Sunil K. Sharma,

2

S. A. Mohiuddine,

3

and A. K

JlJçman

4

1_{Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India}

2_{Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal, Jammu and Kashmir 181122, India} 3_{Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} 4_{Department of Mathematics, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia}

Correspondence should be addressed to A. Kılıc¸man; [email protected]

Received 17 March 2014; Revised 11 July 2014; Accepted 11 July 2014; Published 22 July 2014 Academic Editor: Feyzi Bas¸ar

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 generalized𝐵_∧𝜇-difference operator on𝑛-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 C¸ olak [2] as follows:𝑍(Δ𝜇) = {𝑥 = (𝑥_𝑘) ∈ 𝜔 : (Δ𝜇𝑥_𝑘) ∈ 𝑧} for 𝑧 = ℓ_∞, 𝑐, and𝑐₀, where𝜇 is a nonnegative integer and

Δ𝜇𝑥_𝑘= Δ𝜇−1𝑥_𝑘− Δ𝜇−1𝑥_𝑘+1, Δ0𝑥_𝑘= 𝑥_𝑘 ∀𝑘 ∈ N (1) or equivalent to the following binomial representation:

Δ𝜇𝑥_𝑘=∑𝜇

V=0

(−1)V_(𝜇V)𝑥_𝑘+V. (2) These sequence spaces were generalized by Et and Basarir [3] taking𝑧 = ℓ_∞(𝑝), 𝑐(𝑝), and 𝑐₀(𝑝).

Dutta [4] introduced the following difference sequence spaces using a new difference operator:

𝑍 (Δ_(𝜂)) = {𝑥 = (𝑥_𝑘) ∈ 𝜔 : Δ_(𝜂)𝑥 ∈ 𝑧} for 𝑧 = ℓ_∞, 𝑐, 𝑐₀, (3) whereΔ_(𝜂)𝑥 = (Δ_(𝜂)𝑥_𝑘) = (𝑥_𝑘− 𝑥_𝑘−𝜂) for all 𝑘, 𝜂 ∈ N.

In [5], Dutta introduced the sequence spaces 𝑐(‖⋅, ⋅‖, Δ𝜇_(𝜂), 𝑝), 𝑐₀(‖⋅, ⋅‖, Δ𝜇_(𝜂), 𝑝), ℓ_∞(‖⋅, ⋅‖, Δ𝜇_(𝜂), 𝑝), 𝑚(‖⋅, ⋅‖, Δ𝜇_(𝜂), 𝑝), and 𝑚₀(‖⋅, ⋅‖, Δ𝜇_(𝜂), 𝑝), where 𝜂, 𝜇 ∈ N and

Δ𝜇_(𝜂)𝑥_𝑘 = (Δ𝜇_(𝜂)𝑥_𝑘) = (Δ𝜇−1_(𝜂)𝑥_𝑘 − Δ𝜇−1_(𝜂)𝑥_𝑘−𝜂) and Δ0

(𝜂)𝑥𝑘 = 𝑥𝑘

for all𝑘, 𝜂 ∈ N, which is equivalent to the following binomial representation:

Δ𝜇_(𝜂)𝑥_𝑘 =∑𝜇

V=0(−1) V

(𝜇V)𝑥𝑘−𝜂V. (4)

The difference sequence spaces have been studied by authors [6–14] and references therein. Bas¸ar and Altay [15] introduced the generalized difference matrix𝐵 = (𝑏_𝑚𝑘) for all 𝑘, 𝑚 ∈ N, which is a generalization ofΔ₍₁₎-difference operator by

𝑏_𝑚𝑘={{_{_{ { 𝑟, 𝑘 = 𝑚 𝑠, 𝑘 = 𝑚 − 1 0, (𝑘 > 𝑚) or (0 ≤ 𝑘 < 𝑚 − 1) . (5)

Bas¸arir and Kayikc¸i [16] defined the matrix𝐵𝜇(𝑏_𝑚𝑘𝜇 ) which reduced the difference matrixΔ𝜇₍₁₎ in case𝑟 = 1, 𝑠 = −1. The generalized𝐵𝜇-difference operator is equivalent to the following binomial representation:

𝐵𝜇_{𝑥 = 𝐵}𝜇_(𝑥 𝑘) = 𝜇 ∑ V=0(𝜇V)𝑟 𝜇−V_𝑠V_𝑥 𝑘−V. (6)

Volume 2014, Article ID 691632, 9 pages http://dx.doi.org/10.1155/2014/691632

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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

𝐸_∧= {𝑥 = (𝑥_𝑘) ∈ 𝜔 : (∧_𝑘𝑥_𝑘) ∈ 𝐸} . (7) An Orlicz function𝑀 is a function, 𝑀 : [0, ∞) → [0, ∞), which is continuous, nondecreasing, and convex with𝑀(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 exists𝐾 > 2 and 𝑥₀≥ 0 such that 𝑀(2𝑥) ≤ 𝐾𝑀(𝑥) for all 𝑥 ≥ 𝑥₀. TheΔ₂-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:

ℓ_𝑀= {𝑥 ∈ 𝜔 :∑∞

𝑘=1

𝑀 (󵄨󵄨󵄨󵄨𝑥_𝜌𝑘󵄨󵄨󵄨󵄨) < ∞} (8) which is called an Orlicz sequence space. The spaceℓ_𝑀is a Banach space with the norm

‖𝑥‖ = inf {𝜌 > 0 :∑∞

𝑘=1

𝑀 (󵄨󵄨󵄨󵄨𝑥_𝜌𝑘󵄨󵄨󵄨󵄨) ≤ 1} . (9) It is shown in [17] that every Orlicz sequence space ℓ_𝑀 contains a subspace isomorphic toℓ_𝑝(𝑝 ≥ 1).

A sequence M = (𝑀_𝑘) of Orlicz function is called a Musielak-Orlicz function; see [18,19]. A sequenceN = (𝑁_𝑘) defined by

𝑁𝑘(V) = sup {|V| 𝑢 − 𝑀𝑘(𝑢) : 𝑢 ≥ 0} , 𝑘 = 1, 2, . . . , (10)

is called the complimentary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space𝑡_Mand its subspaceℎ_Mare defined as follows:

𝑡_M = {𝑥 ∈ 𝜔 : 𝐼_𝑀(𝑐𝑥) < ∞ for some 𝑐 > 0} , ℎ_M= {𝑥 ∈ 𝜔 : 𝐼_𝑀(𝑐𝑥) < ∞ ∀𝑐 > 0} , (11) where𝐼_Mis a convex modular defined by

𝐼_M(𝑥) =∑∞

𝑘=1

𝑀_𝑘(𝑥_𝑘) , 𝑥 = (𝑥_𝑘) ∈ 𝑡_𝑀. (12) We consider𝑡_Mequipped with the Luxemburg norm

‖𝑥‖ = inf {𝑘 > 0 : 𝐼𝑀(𝑥_𝑘) ≤ 1} (13)

or equipped with the Orlicz norm

‖𝑥‖0= inf {1_𝑘(1 + 𝐼𝑀(𝑘𝑥)) : 𝑘 > 0} . (14)

By a lacunary sequence𝜃 = (𝑖_𝑟), 𝑟 = 0, 1, 2, . . ., where 𝑖₀ = 0, we will mean an increasing sequence of nonnegative integers ℎ𝑟 = (𝑖𝑟− 𝑟𝑟−1) → ∞ (𝑟 → ∞). The intervals determined

by𝜃 are denoted by 𝐼_𝑟 = (𝑖_𝑟−1, 𝑖_𝑟] and the ratio 𝑖_𝑟/𝑖_𝑟−1 will

be denoted by𝑞_𝑟. The space of lacunary strongly convergent sequences𝑁_𝜃was defined by Freedman et al. [20] as follows:

𝑁_𝜃 ={_{ { 𝑥 = (𝑥_𝑘) : lim_{𝑟 → ∞}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 󵄨󵄨󵄨󵄨𝑥𝑘− 𝐿󵄨󵄨󵄨󵄨 = 0, for some 𝐿 } } } . (15) The concept of 2-normed spaces was initially developed by G¨ahler [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. Let𝑛 ∈ N and 𝑋 be linear space over the field K, whereK 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, . . . , 𝑥_𝑛)‖ = 0 if and only if 𝑥₁, 𝑥₂, 𝑥₃, . . . ,𝑥_𝑛 are linearly dependent in𝑋;

(2)‖(𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)‖ is invariant under permutation; (3)‖(𝛼𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)‖ = |𝛼|‖(𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)‖ for any 𝛼 ∈

K;

(4)‖(𝑥 + 𝑥󸀠, 𝑥₂, . . . , 𝑥_𝑛)‖ ≤ ‖(𝑥, 𝑥₂, . . . , 𝑥_𝑛)‖ + ‖(𝑥󸀠_{, 𝑥}

2, . . . , 𝑥𝑛)‖

is called an 𝑛-norm on 𝑋 and the pair (𝑋, ‖⋅, . . . , ⋅‖) is called an𝑛-normed space over the field K. For example, we may take 𝑋 = R𝑛 being equipped with the Euclidean 𝑛-norm‖(𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)‖_𝐸 = the volume of the 𝑛-dimensional parallelepiped spanned by the vectors 𝑥₁, 𝑥₂, . . . , 𝑥_𝑛 which may be given explicitly by the formula

󵄩󵄩󵄩󵄩(𝑥1, 𝑥2, . . . , 𝑥𝑛)󵄩󵄩󵄩󵄩𝐸= 󵄨󵄨󵄨󵄨󵄨det(𝑥𝑖𝑗)󵄨󵄨󵄨󵄨󵄨, (16)

where𝑥_𝑖= (𝑥₁, 𝑥₂, 𝑥₃, . . . , 𝑥_𝑛) ∈ R𝑛for each𝑖 = 1, 2, 3, . . . , 𝑛 and‖ ⋅ ‖_𝐸denotes the Euclidean norm. Let(𝑋, ‖⋅, . . . , ⋅‖) be an 𝑛-normed space of dimension 𝑑 ≥ 𝑛 ≥ 2 and {𝑎₁, 𝑎₂, . . . , 𝑎_𝑛} linearly independent set in𝑋. Then the following function ‖(⋅, . . . , ⋅)‖_∞on𝑋𝑛−1defined by

󵄩󵄩󵄩󵄩(𝑥1, 𝑥2, . . . , 𝑥𝑛)󵄩󵄩󵄩󵄩∞

= max {󵄩󵄩󵄩󵄩(𝑥1, 𝑥2, . . . , 𝑥𝑛−1, 𝑎𝑖)󵄩󵄩󵄩󵄩 : 𝑖 = 1, 2, . . . , 𝑛}

(17) defines an(𝑛−1) norm on 𝑋 with respect to {(𝑎₁, 𝑎₂, . . . , 𝑎_𝑛)}. A sequence (𝑥_𝑘) in an 𝑛-normed space (𝑋, ‖⋅, . . . , ⋅‖) is said to converge to some𝐿 ∈ 𝑋 if

lim

𝑘 → ∞󵄩󵄩󵄩󵄩(𝑥𝑘− 𝐿, 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩 = 0,

for every𝑧₁, . . . , 𝑧_𝑛−1∈ 𝑋.

(18) A sequence(𝑥_𝑘) in a normed space (𝑋, ‖⋅, . . . , ⋅‖) is said to be Cauchy if lim 𝑘 → ∞ 𝑝 → ∞󵄩󵄩󵄩󵄩󵄩(𝑥𝑘 − 𝑥_𝑝, 𝑧₁, . . . , 𝑧_𝑛−1_{)󵄩󵄩󵄩󵄩󵄩 = 0,} for every𝑧₁, . . . , 𝑧_𝑛−1∈ 𝑋. (19)

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If every Cauchy sequence in𝑋 converges to some 𝐿 ∈ 𝑋 then𝑋 is said to be complete with respect to the 𝑛-norm. Any complete𝑛-normed space is said to be 𝑛-Banach space.

Let(𝑋, ‖⋅, . . . , ⋅‖) be an 𝑛-normed space and let 𝑠(𝜔 − 𝑥) denote the space of𝑋-valued sequences. Let 𝑝 = (𝑝_𝑘) be any bounded sequence of positive real numbers andM = (𝑀_𝑘) a Musielak-Orlicz function. We define the following sequence spaces in this paper:

𝑤₀𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀𝑘(󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩( 𝐵∧𝜇𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 = 0, 𝜌 > 0}_} } , 𝑤𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = 0, for some𝐿, 𝜌 > 0}_} } , 𝑤_∞𝜃 (M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 < ∞, 𝜌 > 0}_} } ; (20) whenM(𝑥) = 𝑥, we get 𝑤₀𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = 0, 𝜌 > 0}_} } , 𝑤𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = 0, for some𝐿, 𝜌 > 0}_} } , 𝑤𝜃_∞(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑠 (𝑤 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 < ∞, 𝜌 > 0}_} } ; (21) when𝑝_𝑘= 1, for all 𝑘, we get

𝑤𝜃₀(M, 𝐵𝜇_∧, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑤 (𝑠 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) = 0, 𝜌 > 0 } } } , 𝑤𝜃(M, 𝐵𝜇_∧, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥_𝑘) ∈ 𝑤 (𝑠 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀𝑘(󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) = 0, for some𝐿, 𝜌 > 0}_} } , 𝑤𝜃_∞(M, 𝐵_∧𝜇, ‖⋅, . . . , ⋅‖) ={_{ { 𝑥 = (𝑥𝑘) ∈ 𝑤 (𝑠 − 𝑥) : lim_{𝑟 → ∞}_ℎ1 𝑟 × ∑ 𝑘∈𝐼𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) < ∞, 𝜌 > 0 } } } . (22)

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The following inequality will be used throughout the paper. If 0 ≤ 𝑝_𝑘≤ sup 𝑝_𝑘= 𝐻, 𝑘 = max(1, 2𝐻−1_{), then}

󵄨󵄨󵄨󵄨𝑎𝑘+ 𝑏𝑘󵄨󵄨󵄨󵄨𝑝𝑘 ≤ 𝐾 {󵄨󵄨󵄨󵄨𝑎𝑘|𝑝𝑘+󵄨󵄨󵄨󵄨 𝑏𝑘|𝑝𝑘} (23)

for all𝑘 and 𝑎_𝑘, 𝑏_𝑘∈ C. Also |𝑎|𝑝𝑘 _{≤ max(1, |𝑎|}𝐻_{) for all 𝑎 ∈ C.}

2. Main Results

Theorem 1. Let M = (𝑀_𝑘) be a Musielak-Orlicz function and

𝑝 = (𝑝_𝑘) a bounded sequence of positive real numbers; the

spaces 𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖), 𝑤𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖), and

𝑤𝜃

∞(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) are linear over the field of complex

numbersC.

Proof. Let𝑥 = (𝑥_𝑘), 𝑦 = (𝑦_𝑘) ∈ 𝑤₀𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖), and

𝛼, 𝛽 ∈ C. Then there exist positive real numbers 𝜌₁ and𝜌₂ such that lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = 0, lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑦_𝑘 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 = 0. (24)

Define𝜌₃= max(2|𝛼|𝜌₁, 2|𝛽|𝜌₂). Since ‖⋅, . . . , ⋅‖ is an 𝑛-norm on𝑋 and 𝑀󸀠_𝑘𝑠 are nondecreasing and convex functions so by using inequality (23) we have

lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧(𝛼𝑥_𝑘+ 𝛽𝑦_𝑘) 𝜌₃ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ≤ lim_{𝑟 → ∞}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝛼𝑥_𝑘 𝜌₃ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 +󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵_∧𝜇𝛽𝑦_𝑘 𝜌₃ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ≤ 𝐾 lim_{𝑟 → ∞}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 1 2𝑝𝑘𝑀𝑘( 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 + 𝐾 lim_{𝑟 → ∞}_ℎ1 𝑟𝑘∈𝐼∑𝑟 1 2𝑝𝑘𝑀𝑘( 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵_∧𝜇𝑦_𝑘 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 = 0. (25) Thus, we have 𝛼𝑥 + 𝛽𝑦 ∈ 𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). Hence 𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) is a linear space. Similarly, we can prove that 𝑤𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) and 𝑤𝜃_∞(M, 𝐵_∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖) are linear spaces. This completes the proof of the theorem.

Theorem 2. Let M = (𝑀𝑘) be a Musielak-Orlicz function

and𝑝 = (𝑝_𝑘) a bounded sequence of positive real numbers;

the space𝑤₀𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) is a topological linear space

paranormed by 𝑔 (𝑥) = inf{_{ { 𝜌𝑝𝑟/𝑀_: (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } , (26)

where𝑀 = max(1, sup_𝑘𝑝_𝑘 < ∞).

Proof. Clearly 𝑔(𝑥) ≥ 0 for 𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃₀(M, 𝐵𝜇_∧,

𝑝, ‖⋅, . . . , ⋅‖). Since 𝑀_𝑘(0) = 0, we get 𝑔(0) = 0. Again, if 𝑔(𝑥) = 0, then 𝑔 (𝑥) = inf{_{ { 𝜌𝑝𝑟/𝑀_: (1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } = 0. (27)

This implies that, for a given𝜀 > 0, there exist some 𝜌_𝜀 (0 < 𝜌_𝜀< 𝜀) such that (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌_𝜀 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1. (28) Thus (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜀 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌_𝜀 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1 (29)

for each𝑟, and suppose that 𝑥_𝑘 ̸= 0 for each 𝑘 ∈ N. This implies that𝐵𝜇_∧𝑥_𝑘 ̸= 0 for each 𝑘 ∈ N. Let 𝜀 → 0, then

((󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌_𝜀 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 󳨀→ ∞; (30)

(5)

which is a contradiction. Therefore,𝐵𝜇_∧𝑥_𝑘 = 0 for each 𝑘 and thus𝑥_𝑘= 0 for each 𝑘 ∈ N. Let 𝜌₁> 0 and 𝜌₂> 0 be such that

(_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵_∧𝜇𝑥_𝑘 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1, (1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑦_𝑘 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1 for each 𝑟. (31)

Let𝜌 = 𝜌₁+𝜌₂; then by using Minkowski’s inequality, we have

(1 ℎ𝑟_𝑘∈𝐼∑_𝑟𝑀𝑘( 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵_∧𝜇(𝑥_𝑘+ 𝑦_𝑘) 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘+ 𝐵𝜇_∧𝑦_𝑘 𝜌₁+ 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘 × (( 𝜌1 𝜌₁+ 𝜌₂)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 + ( 𝜌2 𝜌1+ 𝜌2) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑦_𝑘 𝜌2 , 𝑧1, . . . , 𝑧𝑛−1) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ (_𝜌 𝜌1 1+ 𝜌2) × (1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 + (_𝜌 𝜌2 1+ 𝜌2) × (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑦_𝑘 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1. (32)

Since𝜌󸀠𝑠 are nonnegative, we have 𝑔 (𝑥 + 𝑦) = inf{_{ { 𝜌𝑝𝑟/𝑀: (1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } ≤ inf{_{ { 𝜌₁𝑝𝑟/𝑀: (1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌1 , 𝑧1, . . . , 𝑧𝑛−1) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } + inf{_{ { 𝜌₂𝑝𝑟/𝑀_{: (}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑦_𝑘 𝜌₂ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } . (33) Therefore,𝑔(𝑥 + 𝑦) ≤ 𝑔(𝑥) + 𝑔(𝑦).

Finally, we prove that the scalar multiplication is contin-uous. Let] be any complex number. By definition,

𝑔 (]𝑥) = inf{_{ { 𝜌𝑝𝑟/𝑀_{: (}1 ℎ𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( ]𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } . (34)

(6)

Then 𝑔 (]𝑥) = inf{_{ { (|]| 𝑡)𝑝𝑟/𝑀_{: (}1 ℎ𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝑡 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } , (35) where𝑡 = 𝜌/|]|. Since |]|𝑝𝑟 ≤ max(1, |]| sup 𝑝

𝑘), we have

𝑔 (]𝑥) = max (1, |]| sup 𝑝𝑘) inf

×{_{ { (𝑡)𝑝𝑟/𝑀: (_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘 × (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵_∧𝜇𝑥_𝑘 𝑡 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ) 1/𝑀 ≤ 1}_} } . (36) So, the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of theorem.

Theorem 3. Let M = (𝑀_𝑘) be a Musielak-Orlicz

func-tion. If sup_𝑘(𝑀_𝑘(𝑥))𝑝𝑘 < ∞ for all fixed 𝑥 > 0, then

𝑤𝜃_{(M, 𝐵}𝜇

∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞𝜃 (M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖).

Proof. Let𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). Then there

exists some positive number𝜌₁such that lim 𝑟 → ∞ 1 ℎ𝑟_𝑘∈𝐼∑_𝑟𝑀𝑘( 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌1 , 𝑧1, . . . , 𝑧𝑛−1) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = 0. (37) Define𝜌 = 2𝜌₁. Since𝑀_𝑘is nondecreasing and convex and by using inequality (23), we have

lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 = lim_{𝑟 → ∞}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 + 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ≤ 𝐾{_{ { lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘₂1_𝑝 𝑘( 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇∧𝑥𝑘− 𝐿 𝜌₁ , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 + lim_{𝑟 → ∞}1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘 1 2𝑝𝑘(󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩( 𝐿 𝜌₁, 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩) 𝑝𝑘} } } < 𝐾{_{ { lim 𝑟 → ∞ 1 ℎ𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌1 , 𝑧1, . . . , 𝑧𝑛−1) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 + lim_{𝑟 → ∞}_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩(}_𝜌𝐿 1, 𝑧1, . . . , 𝑧𝑛−1) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩)𝑝𝑘}} } . (38) Hence𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃_∞(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). This completes the proof of the theorem.

Theorem 4. Let M = (𝑀_𝑘) be a Musielak-Orlicz function and

0 < ℎ = inf 𝑝_𝑘. Then 𝑤_∞𝜃 (M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃0(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) (39) if and only if lim 𝑟 → ∞ 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(𝑡)𝑝𝑘 _{= ∞, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑡 > 0.} (40) Proof. Let𝑤𝜃_∞(M, 𝐵_∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖).

Suppose (40) does not hold. Therefore there are a subinterval 𝐼_𝑟(𝑚) of the set of intervals𝐼_𝑟 and a number𝑛₀, where𝑛₀ = ‖(𝐵𝜇_∧𝑥_𝑘/𝜌, 𝑧₁, . . . , 𝑧_𝑛−1)‖ for all 𝑘, such that

1 ℎ_𝑟(𝑚)_𝑘∈𝐼∑

𝑟(𝑚)

𝑀(𝑛₀)𝑝𝑘_{≤ 𝑁 < ∞, 𝑚 = 1, 2, 3, . . . .}

(41) Let us define𝑥 = (𝑥_𝑘) as follows:

𝐵𝜇_∧𝑥_𝑘 = {𝜌𝑛0, 𝑘 ∈ 𝐼𝑟(𝑚)

0, 𝑘 ∉ 𝐼_𝑟(𝑚). (42) Thus by (41),𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃_∞(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). But 𝑥 = (𝑥_𝑘) ∉ 𝑤𝜃

0(M, 𝐵∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖). Hence (40) must hold.

Conversely, suppose that (40) holds and let𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃 ∞(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Then, 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ≤ 𝑁 < ∞. (43) Suppose that𝑥 = (𝑥_𝑘) ∉ 𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). Then for some number𝜀, 1 > 𝜀 > 0, there is a number 𝑁₀such that, for a subinterval𝐼_𝑟(𝑚)of the set of intervals𝐼_𝑟,

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩(

𝐵𝜇_∧𝑥_𝑘

(7)

We have 𝑀_𝑘(‖(𝐵𝜇_∧𝑥_𝑘/𝜌, 𝑧₁, . . . , 𝑧_𝑛−1)‖) ≥ 𝑀(𝜀)𝑝𝑘_{, which}

contradicts (40) by using (43). Hence we get

𝑤_∞𝜃 (M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤0𝜃(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) . (45)

This completes the proof.

Theorem 5. Let 0 < ℎ = inf 𝑝_𝑘 ≤ sup 𝑝_𝑘 = 𝐻 < ∞. For

any Musielak-Orlicz functionM = (𝑀_𝑘) which satisfies Δ₂

-condition, one has

(i)𝑤𝜃₀(𝐵_∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤₀𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) (ii)𝑤𝜃(𝐵_∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) (iii)𝑤𝜃_∞(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃_∞(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖).

Proof. (i) Let𝑥 = (𝑥_𝑘) ∈ 𝑤₀𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). Then, we have

1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 󳨀→ 0 as 𝑟 󳨀→ ∞. (46) Let𝜀 > 0, and choose 𝛿 with 0 < 𝛿 < 1 such that 𝑀_𝑘< 𝜀 for 0 ≤ 𝑡 ≤ 𝛿. We can write 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵_∧𝜇𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 = _ℎ1 𝑟 𝑘∈𝐼∑𝑟 ‖(𝐵𝜇 ∧𝑥𝑘/𝜌,𝑧1,...,𝑧𝑛−1)‖≤𝛿 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵∧𝜇𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 +_ℎ1 𝑟 𝑘∈𝐼∑𝑟 ‖(𝐵𝜇∧𝑥𝑘/𝜌,𝑧1,...,𝑧𝑛−1)‖>𝛿 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 . (47) For the first summation above, we can write

1 ℎ_𝑟 _𝑘∈𝐼∑ 𝑟 ‖(𝐵𝜇∧𝑥𝑘/𝜌,𝑧1,...,𝑧𝑛−1)‖≤𝛿 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 < max (𝜀, 𝜀ℎ) . (48)

By using continuity of𝑀_𝑘, for the second summation we can write 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩< 1 + (󵄩󵄩󵄩󵄩(𝐵𝜇∧𝑥𝑘/𝜌, 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩) 𝛿 . (49)

Since each𝑀_𝑘is nondecreasing and convex and satisfiesΔ₂ -condition, it follows that

1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ≤ max (𝜀, 𝜀ℎ) + max {1, [2𝑀𝑘(󵄩󵄩󵄩󵄩(𝐵𝜇∧𝑥𝑘/𝜌, 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩) 𝛿 ] ℎ } ×_ℎ1 𝑟𝑘∈𝐼∑𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵_∧𝜇𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 . (50)

Taking limit as𝜀 → 0 and 𝑟 → ∞, it follows that 𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃

0(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Hence 𝑤𝜃0(𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂

𝑤𝜃

0(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Similarly, we can prove (ii) and (iii).

This completes the proof of the theorem.

Theorem 6. Let M = (𝑀_𝑘) be a Musielak-Orlicz function.

Then the following statements are equivalent:

(i)𝑤𝜃_∞(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃_∞(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖); (ii)𝑤𝜃₀(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤_∞𝜃 (M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖); (iii) sup_𝑟(1/ℎ_𝑟) ∑_𝑘∈𝐼_𝑟𝑀_𝑘(𝑡)𝑝𝑘 _{< ∞ for all 𝑡 > 0, where 𝑡 =}

‖𝐵𝜇_∧𝑥_𝑘/𝜌, 𝑧₁, . . . , 𝑧_𝑛−1‖.

Proof. (i)⇒ (ii) Suppose (i) holds. In order to prove (ii) we

have to show that

𝑤𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞𝜃 (M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) . (51)

Let𝑥 = (𝑥_𝑘) ∈ 𝑤₀𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). Then for a given 𝜀 > 0 there exists𝑠 > 𝑠_𝜀such that

1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 < 𝜀. (52) Hence there exists𝐾 > 0 such that

sup 𝑟 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 (󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 < 𝐾. (53) This shows that𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃_∞(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖).

(ii)⇒ (iii) Suppose (ii) holds and (iii) fails to hold. Then for some𝑡 > 0, sup 𝑟 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀𝑘(𝜀)𝑝𝑘 = ∞, (54) and, therefore, we can find a subinterval𝐼_𝑟(𝑚) of the set of intervals𝐼_𝑟such that

1 ℎ_𝑟(𝑚)_𝑘∈𝐼∑ 𝑟(𝑚) 𝑀_𝑘(1 𝑚) 𝑝𝑘 ≥ 𝑚, 𝑚 = 1, 2, 3, . . . . ₍₅₅₎

(8)

Let us define𝑥 = (𝑥_𝑘) as follows: 𝐵𝜇_∧𝑥_𝑘={_{ { 𝜌 𝑚, 𝑘 ∈ 𝐼𝑟(𝑚) 0, 𝑘 ∉ 𝐼_𝑟(𝑚). (56) Thus 𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃₀(𝐵_∧𝜇, 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 = (𝑥_𝑘) ∉ 𝑤𝜃_∞(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Then for 𝑥 = (𝑥𝑘) ∈ 𝑤∞𝜃 (𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). sup 𝑟 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 = ∞. (57) Let𝑡 = ‖𝐵𝜇_∧𝑥_𝑘/𝜌, 𝑧₁, . . . , 𝑧_𝑛−1‖ for each 𝑘, and then by (57) sup_𝑟(1/ℎ_𝑟) ∑_𝑘∈𝐼_𝑟𝑀_𝑘(𝑡)𝑝𝑘 = ∞, which contradicts (iii). Hence

(i) must hold. This completes the proof of the theorem.

Theorem 7. Let M = (𝑀_𝑘) be a Musielak-Orlicz function.

Then the following statements are equivalent:

(i)𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤₀𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖); (ii)𝑤𝜃₀(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊂ 𝑤_∞𝜃 (𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖); (iii) inf_𝑟(1/ℎ_𝑟) ∑_𝑘∈𝐼

𝑟𝑀𝑘(𝑡)

𝑝𝑘 > 0 for all 𝑡 > 0.

Proof. (i)⇒ (ii) is obvious

(ii)⇒ (iii) Let (ii) hold and let (iii) fail to hold. Then inf_𝑟 1

ℎ_𝑟_𝑘∈𝐼∑

𝑟

𝑀𝑘(𝑡)𝑝𝑘= 0 for some 𝑡 > 0, (58) and we can find a subinterval𝐼_𝑟(𝑚) of the set of intervals𝐼_𝑟 such that 1 ℎ_𝑟(𝑚)_𝑘∈𝐼∑ 𝑟(𝑚) 𝑀_𝑘(𝑚)𝑝𝑘 < 1 𝑚, 𝑚 = 1, 2, 3, . . . . (59) Let us define𝑥 = (𝑥_𝑘) as follows:

𝐵𝜇_∧𝑥_𝑘= {𝜌𝑚, 𝑘 ∈ 𝐼𝑟(𝑚)

0, 𝑘 ∉ 𝐼_𝑟(𝑚). (60) Thus by (iii),𝑥 = (𝑥_𝑘) ∈ 𝑤₀𝜃(M, 𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖). But 𝑥 = (𝑥_𝑘) ∉ 𝑤𝜃

∞(𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) which contradict (ii). Hence (iii)

must hold.

(iii) ⇒ (i) Let (iii) hold. Suppose that 𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃 0(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Therefore, 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 󳨀→ 0 as 𝑟 󳨀→ ∞. (61) Again suppose 𝑥 = (𝑥_𝑘) ∉ 𝑤₀𝜃(𝐵𝜇_∧, 𝑝, ‖⋅, . . . , ⋅‖) for some number𝜀 > 0 and a subinterval 𝐼_𝑟(𝑚) of the set of intervals 𝐼_𝑟, we have 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩≥ 𝜀 ∀𝑘. (62)

Then, from properties of the Orlicz function, we can write 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) 𝑝𝑘 ≥ 𝑀_𝑘(𝜀)𝑝𝑘_. ₍₆₃₎

Consequently, by (61), we have lim_{𝑟 → ∞}(1/ℎ_𝑟)∑_𝑘∈𝐼_𝑟𝑀_𝑘 (𝜀)𝑝𝑘 = 0, which contradicts (iii). Hence (i) must hold. This

completes the proof of the theorem.

Theorem 8. (i) If 0 < inf 𝑝_𝑘 ≤ 𝑝_𝑘 ≤ 1 for all 𝑘, then

𝑤𝜃_{(M, 𝐵}𝜇

∧, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖).

(ii) If 1 ≤ 𝑝_𝑘 ≤ sup 𝑝_𝑘 = 𝐻 < ∞, then

𝑤𝜃(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃(M, 𝐵𝜇∧, ‖⋅, . . . , ⋅‖).

Proof. (i) Let𝑥 ∈ 𝑤𝜃(M, 𝐵_∧𝜇, ‖⋅, . . . , ⋅‖). Since 0 < inf 𝑝_𝑘 ≤ 1,

we get 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) ≤ _ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀𝑘(󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 , (64) and hence𝑥 ∈ 𝑤𝜃(M, 𝐵_∧𝜇, 𝑝, ‖⋅, . . . , ⋅‖).

(ii) 1 ≤ 𝑝_𝑘 ≤ sup 𝑝_𝑘 = 𝐻 < ∞ and 𝑥 = (𝑥_𝑘) ∈ 𝑤𝜃(M, 𝐵𝜇∧, 𝑝, ‖⋅, . . . , ⋅‖). Then for each 0 < 𝜀 < 1 there exists a

positive integer𝑠₀such that 1 ℎ_𝑟_𝑘∈𝐼∑ 𝑟 𝑀𝑘(󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ≤ 𝜀 < 1 ∀𝑟 > 𝑠₀. (65)

This implies that 1 ℎ𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇_∧𝑥_𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩) 𝑝𝑘 ≤_ℎ1 𝑟𝑘∈𝐼∑𝑟 𝑀_𝑘(󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩} 󵄩( 𝐵𝜇∧𝑥𝑘− 𝐿 𝜌 , 𝑧1, . . . , 𝑧𝑛−1)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩) . (66)

Therefore𝑥 = (𝑥_𝑘) ∈ 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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