Research Article Sequence Spaces Defined by Musielak-Orlicz

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 364743, 10 pages http://dx.doi.org/10.1155/2013/364743

Research Article Sequence Spaces Defined by Musielak-Orlicz Function over 𝑛-Normed Spaces M. Mursaleen,1 Sunil K. Sharma,2 and A. KJlJçman3 1

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal 181122, Jammu and Kashmir, India 3 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 2

Correspondence should be addressed to A. Kılıçman; [email protected] Received 21 July 2013; Accepted 16 September 2013 Academic Editor: Abdullah Alotaibi 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 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 󵄨󵄨 󵄨󵄨 󵄨𝑥 󵄨 ℓ𝑀 = {𝑥 ∈ 𝑤 : ∑ 𝑀 ( 󵄨 𝑘 󵄨 ) < ∞} , 𝜌 𝑘=1

(1)

ℎM = {𝑥 ∈ 𝑤 : 𝐼M (𝑐𝑥) < ∞ ∀𝑐 > 0} ,

∞

𝐼M (𝑥) = ∑ (𝑀𝑘 ) (𝑥𝑘 ) ,

which is called as an Orlicz sequence space. The space ℓ𝑀 is a Banach space with the norm (2)

It is shown in [1] that every Orlicz sequence space ℓ𝑀 contains a subspace isomorphic to ℓ𝑝 (𝑝 ≥ 1). The Δ 2 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 𝑁𝑘 (V) = sup {|V| 𝑢 − 𝑀𝑘 (𝑢) : 𝑢 ≥ 0} ,

𝑡M = {𝑥 ∈ 𝑤 : 𝐼M (𝑐𝑥) < ∞ for some 𝑐 > 0} ,

(4)

where 𝐼M is a convex modular defined by

∞

󵄨󵄨 󵄨󵄨 ∞ 󵄨𝑥𝑘 󵄨 ‖𝑥‖ = inf {𝜌 > 0 : ∑ 𝑀 ( 󵄨 󵄨 ) ≤ 1} . 𝜌 𝑘=1

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:

𝑘 = 1, 2, . . . , (3)

𝑘=1

𝑥 = (𝑥𝑘 ) ∈ 𝑡M .

(5)

We consider 𝑡M equipped with the Luxemburg norm 𝑥 ‖𝑥‖ = inf {𝑘 > 0 : 𝐼M ( ) ≤ 1} 𝑘

(6)

or equipped with the Orlicz norm 1 ‖𝑥‖0 = inf { (1 + 𝐼M (𝑘𝑥)) : 𝑘 > 0} . 𝑘

(7)

Let 𝑋 be a linear metric space. A function 𝑝 : 𝑋 → R is called paranorm if (1) 𝑝(𝑥) ≥ 0 for all 𝑥 ∈ 𝑋, (2) 𝑝(−𝑥) = 𝑝(𝑥) for all 𝑥 ∈ 𝑋,

2

Abstract and Applied Analysis (3) 𝑝(𝑥 + 𝑦) ≤ 𝑝(𝑥) + 𝑝(𝑦) for all 𝑥, 𝑦 ∈ 𝑋, (4) (𝜆 𝑛 ) is a sequence of scalars with 𝜆 𝑛 → 𝜆 as 𝑛 → ∞ and (𝑥𝑛 ) is a sequence of vectors with 𝑝(𝑥𝑛 − 𝑥) → 0 as 𝑛 → ∞; then 𝑝(𝜆 𝑛 𝑥𝑛 − 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 [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 } { 1 󵄨 󵄨 𝑁𝜃 = {𝑥 ∈ 𝑤 : lim ∑ 󵄨󵄨󵄨𝑥𝑘 − 𝑙󵄨󵄨󵄨 = 0, for some 𝑙} . (8) 𝑟→∞ℎ 𝑟 𝑘∈𝐼𝑟 } { 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 𝜆convergent and 𝜆-bounded sequences as follows. Let 𝜆 = (𝜆 𝑘 )∞ 𝑘=1 be a strictly increasing sequence of positive real numbers tending to infinity; that is, 0 < 𝜆0 < 𝜆1 < ⋅ ⋅ ⋅ ,

𝜆 𝑘 󳨀→ ∞ as 𝑘 󳨀→ ∞,

(9)

and it is said that a sequence 𝑥 = (𝑥𝑘 ) ∈ 𝑤 is 𝜆-convergent to the number 𝐿, called the 𝜆-limit of 𝑥 if Λ 𝑚 (𝑥) → 𝐿 as 𝑚 → ∞, where 𝜆 𝑚 (𝑥) =

1 𝑚 ∑ (𝜆 − 𝜆 𝑘−1 ) 𝑥𝑘 . 𝜆 𝑚 𝑘=1 𝑘

(10)

The sequence 𝑥 = (𝑥𝑘 ) ∈ 𝑤 is 𝜆-bounded if sup𝑚 |Λ 𝑚 (𝑥)| < ∞. It is well known [18] that if lim𝑚 𝑥𝑚 = 𝑎 in the ordinary sense of convergence, then lim ( 𝑚

𝑚 1 󵄨 󵄨 ( ∑ (𝜆 𝑘 − 𝜆 𝑘−1 ) 󵄨󵄨󵄨𝑥𝑘 − 𝑎󵄨󵄨󵄨)) = 0. 𝜆 𝑚 𝑘=1

(11)

This implies that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 𝑚 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 0, 󵄨 = lim Λ lim − 𝑎 (𝜆 − 𝜆 ) (𝑥 − 𝑎) ∑ (𝑥) 󵄨 󵄨󵄨 󵄨 𝑘 𝑘−1 𝑘 󵄨󵄨 𝑚 󵄨 𝑚 𝑚 󵄨󵄨 𝜆 󵄨󵄨 󵄨󵄨 𝑚 𝑘=1 (12) which yields that lim𝑚 Λ 𝑚 (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 (1) ‖𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ‖ = 0 if and only if 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 are linearly dependent in 𝑋; (2) ‖𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ‖ is invariant under permutation; (3) ‖𝛼𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ‖ = |𝛼| K;

‖𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ‖ for any 𝛼 ∈

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

(13)

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

(14)

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 󵄩 󵄩 lim 󵄩󵄩𝑥𝑘 − 𝐿, 𝑧1 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 = 0

𝑘→∞ 󵄩

(15)

for every 𝑧1 , . . . , 𝑧𝑛−1 ∈ 𝑋. A sequence (𝑥𝑘 ) in an 𝑛-normed space (𝑋, ‖⋅, . . . , ⋅‖) is said to be Cauchy if 󵄩 󵄩 lim 󵄩󵄩󵄩󵄩𝑥𝑘 − 𝑥𝑝 , 𝑧1 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩 = 0

𝑘→∞

𝑝→∞

(16)

for every 𝑧1 , . . . , 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.

Abstract and Applied Analysis

3

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:

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘−𝑠 [󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩] = 0, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 for some 𝐿, 𝜌 > 0, 𝑠 ≥ 0} ,

𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim

𝜃 𝑤∞ (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖)

1

𝑟→∞ℎ 𝑟

= {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : sup

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 −𝑠

𝑟

1 ℎ𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘−𝑠 [󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩] < ∞, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟

𝜌 > 0, 𝑠 ≥ 0} ,

𝜌 > 0, 𝑠 ≥ 0} .

𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim

(18) If we take 𝑝 = (𝑝𝑘 ) = 1 for all 𝑘 ∈ N, we have

1

𝑟→∞ℎ 𝑟

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟

𝑤0𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim

1

𝑟→∞ℎ 𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 󵄩 󵄩 × ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟

= 0, for some 𝐿, 𝜌 > 0, 𝑠 ≥ 0} , 𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖)

𝜌 > 0, 𝑠 ≥ 0} ,

1 = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : sup 𝑟 ℎ𝑟 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] < ∞, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝜌 > 0, 𝑠 ≥ 0} . (17)

𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim

1

𝑟→∞ℎ 𝑟

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 󵄩 󵄩 × ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 = 0, for some 𝐿, 𝜌 > 0, 𝑠 ≥ 0} ,

If we take M(𝑥) = 𝑥, we get

𝜃 𝑤∞ (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖)

= {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : sup

𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖)

𝑟

= {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim

1

𝑟→∞ℎ 𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 × ∑ 𝑘−𝑠 [󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩] = 0, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟

1 ℎ𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 󵄩 󵄩 × ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] < ∞, 󵄩 󵄩󵄩 𝜌 󵄩 𝑘∈𝐼𝑟 𝜌 > 0, 𝑠 ≥ 0} . (19)

𝜌 > 0, 𝑠 ≥ 0} , 𝑤𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) 1 = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : lim 𝑟→∞ℎ 𝑟

The following inequality will be used throughout the paper. If 0 ≤ 𝑝𝑘 ≤ sup 𝑝𝑘 = 𝐻, 𝐾 = max(1, 2𝐻−1 ), then 󵄨󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨󵄨𝑎𝑘 + 𝑏𝑘 󵄨󵄨󵄨 𝑘 ≤ 𝐾 {󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 𝑘 + 󵄨󵄨󵄨𝑏𝑘 󵄨󵄨󵄨 𝑘 }

(20)

for all 𝑘 and 𝑎𝑘 , 𝑏𝑘 ∈ C. Also |𝑎|𝑝𝑘 ≤ max(1, |𝑎|𝐻) for all 𝑎 ∈ C.

4


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.

≤𝐾

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌1 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝑟

+𝐾

󵄩󵄩 Λ (𝑦) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝜌2 󵄩󵄩 𝑟

2. Main Results Theorem 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, 𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), and 𝑤∞ 𝑠, ‖⋅, . . . , ⋅‖) are linear spaces over the field of complex number C. Proof. Let 𝑥 = (𝑥𝑘 ), let 𝑦 = (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 󵄩󵄩 Λ (𝛼𝑥 + 𝛽𝑦) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 𝜌 󵄩 󵄩󵄩 𝑟 𝑘∈𝐼 3 󵄩 lim

𝑟

= 0. (21) Since 𝑥 = (𝑥𝑘 ), 𝑦 = (𝑦𝑘 ) ∈ 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), there exist positive numbers 𝜌1 , 𝜌2 > 0 such that

󳨀→ 0 as 𝑟 󳨀→ ∞. (23) 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 𝜃 𝑤∞ (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 𝑔 (𝑥) { 𝑝𝑟 /𝐻 = inf {𝜌 : { 󵄩󵄩 󵄩󵄩 Λ (𝑥) 1 ( ∑𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟𝑘∈𝐼 󵄩󵄩 𝜌 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 } 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1}, 󵄩󵄩 󵄩 ] } (24)

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0, ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 𝜌1 󵄩󵄩 𝑟 𝑘∈𝐼 lim

𝑟

󵄩󵄩 Λ (𝑦) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] = 0. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 𝜌2 󵄩󵄩 𝑟 𝑘∈𝐼 lim

𝑟

(22) Define 𝜌3 = max(2|𝛼|𝜌1 , 2|𝛽|𝜌2 ). Since (𝑀𝑘 ) is nondecreasing, convex function and by using inequality (20), we have 󵄩󵄩 Λ (𝛼𝑥 + 𝛽𝑦) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 ℎ𝑟 𝑘∈𝐼 𝜌3 󵄩󵄩 󵄩󵄩 𝑟

≤

󵄩󵄩 𝛼Λ (𝑥) 󵄩󵄩 1 󵄩 󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 󵄩󵄩 𝜌3 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

where 𝐻 = max(1, sup𝑘 𝑝𝑘 ) < ∞. 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 { inf {𝜌𝑝𝑟 /𝐻 : { 󵄩󵄩 󵄩󵄩 Λ (𝑥) 1 −𝑠 [ ( ∑𝑘 𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟𝑘∈𝐼 󵄩󵄩 𝜌 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 } 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1} = 0. 󵄩󵄩 󵄩 ] } (25)

𝑟

󵄩󵄩 𝛽Λ (𝑦) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 𝑘 + 󵄩󵄩󵄩󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] 󵄩󵄩 𝜌3 󵄩󵄩 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 1 󵄩 󵄩 ≤ 𝐾 ∑ 𝑝 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩󵄩 𝜌1 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 2 𝑘 𝑟 󵄩󵄩 Λ (𝑦) 󵄩󵄩 𝑝𝑘 1 1 󵄩 󵄩 + 𝐾 ∑ 𝑝 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ℎ𝑟 𝑘∈𝐼 2 𝑘 󵄩󵄩 𝜌2 󵄩󵄩 𝑟

This implies that for a given 𝜖 > 0, there exist some 𝜌𝜖 (0 < 𝜌𝜖 < 𝜖) such that 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩󵄩 𝜌𝜖 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

1/𝐻

≤ 1.

𝑟

(26)


5

Thus,

≤(

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩󵄩 𝜖 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

1/𝐻

𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 ≤ ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩󵄩 𝜌𝜖 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

𝜌1 1 ) ∑ 𝑘−𝑠 [𝑀𝑘 ( ℎ𝑟 𝑘∈𝐼 𝜌1 + 𝜌2 𝑟 [ 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝜌2 󵄩 󵄩 ×[󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩]+( ) 󵄩󵄩 𝜌1 󵄩󵄩 𝜌1 + 𝜌2

1/𝐻

󵄩󵄩 󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 Λ 𝑘 (𝑦) 󵄩󵄩 × [󵄩󵄩󵄩󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩]] ) 󵄩 𝜌2 󵄩󵄩 󵄩]] [󵄩󵄩

.

𝑟

(27) Suppose that (𝑥𝑘 ) ≠ 0 for each 𝑘 ∈ N. This implies that Λ 𝑘 (𝑥) ≠ 0 for each 𝑘 ∈ N. Let 𝜖 → 0, then 󵄩󵄩 󵄩󵄩 Λ (𝑥) 󵄩 󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 󳨀→ ∞. 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝜖

(28)

≤(

𝜌1 ) 𝜌1 + 𝜌2

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 × ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩󵄩 𝜌1 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝑟

+(

It follows that

𝑟

𝜌2 ) 𝜌1 + 𝜌2

󵄩󵄩 Λ (𝑦) 󵄩󵄩 𝑝𝑘 1/𝐻 1 󵄩󵄩 𝑘 󵄩 −𝑠 ×( ∑𝑘 [𝑀𝑘 (󵄩󵄩󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ℎ𝑟𝑘∈𝐼 󵄩󵄩 𝜌2 󵄩󵄩

1/𝐻

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 󵄩 𝜖

1/𝐻

𝑟

(29)

≤ 1.

(31)

󳨀→ ∞, which is a contradiction. Therefore, Λ 𝑘 (𝑥) = 0 for each 𝑘, and thus (𝑥𝑘 ) = 0 for each 𝑘 ∈ N. Let 𝜌1 > 0 and 𝜌2 > 0 be the case such that

Since 𝜌, 𝜌1 , and 𝜌2 are nonnegative, we have

𝑔 (𝑥 + 𝑦) 1/𝐻

(

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌1 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

≤ 1,

𝑟

󵄩󵄩 Λ (𝑦) 󵄩󵄩 𝑝𝑘 1/𝐻 1 󵄩󵄩 𝑘 󵄩 −𝑠 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1. ( ∑ 𝑘 [𝑀𝑘 (󵄩󵄩󵄩 ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝜌2 󵄩󵄩 𝑟

(30)

{ { = inf {𝜌𝑝𝑟 /𝐻 : { { 󵄩󵄩 1 󵄩󵄩 Λ (𝑥 + 𝑦) −𝑠 [ ( ∑𝑘 𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 𝜌 󵄩󵄩 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 } 󵄩󵄩 } 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1} } 󵄩󵄩 󵄩 ] }

Let 𝜌 = 𝜌1 + 𝜌2 ; then, by using Minkowski’s inequality, we have 󵄩󵄩 Λ (𝑥 + 𝑦) 󵄩󵄩 𝑝𝑘 1/𝐻 1 󵄩󵄩 𝑘 󵄩 −𝑠 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ( ∑ 𝑘 [𝑀𝑘 (󵄩󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝜌 󵄩󵄩 󵄩󵄩 𝑟

󵄩󵄩 Λ (𝑥) + Λ (𝑦) 1 󵄩 𝑘 ≤ ( ∑ 𝑘−𝑠 [𝑀𝑘 ( 󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 𝜌1 + 𝜌2 󵄩󵄩 𝑟 [ 󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) 󵄩󵄩 󵄩 ]

{ { 𝑝 /𝐻 ≤ inf {(𝜌1 ) 𝑟 : { { 󵄩󵄩 󵄩󵄩 Λ (𝑥) 1 −𝑠 [ ( ∑𝑘 𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝜌1 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 } 󵄩󵄩 } 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1} } 󵄩󵄩 󵄩 ] }

6

Abstract and Applied Analysis { ×inf {𝑡𝑝𝑟 /𝐻 : {

𝑝𝑟 /𝐻 { { : +inf {(𝜌2 ) { {

󵄩󵄩 󵄩󵄩 Λ (𝑥) 1 −𝑠 [ ( ∑𝑘 𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝑡 𝑟 󵄩 [

󵄩󵄩 󵄩󵄩 Λ (𝑦) 1 −𝑠 [ ( ∑𝑘 𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝜌2 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 } 󵄩󵄩 } 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) ≤ 1} . } 󵄩󵄩 󵄩 ] } (32) 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,

󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) 󵄩󵄩 󵄩 ]

} ≤ 1} . (35) } 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 𝜃 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖).

𝑔 (𝜇𝑥) = inf{𝜌𝑝𝑟 /𝐻 : 󵄩󵄩 1 󵄩󵄩 Λ (𝜇𝑥) −𝑠 ( ∑ 𝑘 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 󵄩󵄩 𝜌 ℎ𝑟 𝑘∈𝐼𝑟 󵄩

Proof. Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); then there exists positive number 𝜌1 such that 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 lim , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 𝜌1 󵄩󵄩 𝑟 𝑘∈𝐼𝑟 (36)

≤ 1} .

Define 𝜌 = 2𝜌1 . Since (𝑀𝑘 ) is nondecreasing and convex and by using inequality (20), we have 󵄩󵄩 𝑝𝑘 󵄩󵄩󵄩 Λ (𝑥) 1 󵄩 sup ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩 𝜌 𝑟 ℎ𝑟 󵄩 󵄩󵄩 𝑘∈𝐼𝑟 󵄩󵄩 Λ (𝑥) + 𝐿 − 𝐿 1 󵄩 = sup ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 󵄩󵄩 𝜌 𝑟 ℎ𝑟 𝑘∈𝐼𝑟 󵄩󵄩 𝑝𝑘 󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩󵄩 󵄩 1 1 󵄩󵄩 Λ (𝑥) − 𝐿 ≤ 𝐾sup ∑ 𝑘−𝑠 𝑝 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑘 󵄩󵄩 2 𝜌1 𝑟 ℎ𝑟 𝑘∈𝐼𝑟 󵄩󵄩 𝑝𝑘 󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 )] 󵄩󵄩 󵄩󵄩 𝐿 󵄩󵄩 𝑝𝑘 1 1 󵄩 󵄩 + 𝐾sup ∑ 𝑘−𝑠 𝑝 [𝑀𝑘 (󵄩󵄩󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 𝑘 󵄩 2 𝜌 𝑟 ℎ𝑟 󵄩 󵄩󵄩 1 𝑘∈𝐼𝑟

󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ) 󵄩󵄩 󵄩

(33) Thus, 𝑔 (𝜇𝑥) { 󵄨 󵄨 𝑝𝑟 /𝐻 = inf { (󵄨󵄨󵄨𝜇󵄨󵄨󵄨 𝑡) : { 󵄩󵄩 󵄩󵄩 Λ (𝑥) 1 ( ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩󵄩 𝑘 , ℎ𝑟 𝑘∈𝐼 󵄩󵄩 𝑡 𝑟 󵄩 [ 󵄩󵄩 𝑝𝑘 1/𝐻 󵄩󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩󵄩)] ) 󵄩󵄩 󵄩 ] } ≤ 1} , }

≤ 𝐾sup 𝑟

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝜌1 𝑟

(34) + 𝐾sup 𝑟

where 1/𝑡 = 𝜌/|𝜇|. Since |𝜇|𝑝𝑟 ≤ max(1, |𝜇|sup 𝑝𝑟 ), we have 𝑔 (𝜇𝑥) 󵄨 󵄨sup 𝑝𝑟 ) ≤ max (1, 󵄨󵄨󵄨𝜇󵄨󵄨󵄨

󵄩󵄩 𝐿 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩󵄩 𝜌1 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝑟

< ∞. (37) 𝜃 Hence, 𝑥 = (𝑥𝑘 ) ∈ 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖).


7

Theorem 4. Let 0 < inf 𝑝𝑘 = ℎ ≤ 𝑝𝑘 ≤ sup 𝑝𝑘 = 𝐻 < ∞ and let M = (𝑀𝑘 ), M󸀠 = (𝑀𝑘󸀠 ) be Musielak-Orlicz functions satisfying Δ 2 -condition, then one has (i) 𝑤0𝜃 (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ‖⋅, . . . , ⋅‖);

⊂

𝑤0𝜃 (M ∘ M󸀠 , Λ, 𝑝, 𝑠,

(ii) 𝑤𝜃 (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ‖⋅, . . . , ⋅‖);

⊂

𝑤𝜃 (M ∘ M󸀠 , Λ, 𝑝, 𝑠,

⊂

𝜃 𝑤∞ (M

(iii)

𝜃 𝑤∞ (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖)

‖⋅, . . . , ⋅‖).

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 𝑤𝜃 (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃 (M ∘ M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ,

∘ M , Λ, 𝑝, 𝑠,

𝜃 (M ∘ M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) . ⊂ 𝑤∞

Proof. Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤0𝜃 (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), then we have 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0. ∑ 𝑘−𝑠 [𝑀𝑘󸀠 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩 󵄩󵄩 𝜌 󵄩 𝑟 𝑘∈𝐼 lim

𝑟

(38) Let 𝜖 > 0 and choose 𝛿 with 0 < 𝛿 < 1 such that 𝑀𝑘 (𝑡) < 𝜖 for 0 ≤ 𝑡 ≤ 𝛿. Let (𝑦𝑘 ) = 𝑀𝑘󸀠 [‖Λ 𝑘 (𝑥)/𝜌, 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 ‖] for all 𝑘 ∈ N. We can write 1 1 𝑝 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 = ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 ℎ𝑟 𝑘∈𝐼 ℎ𝑟 𝑘∈𝐼𝑟 𝑟

𝑦𝑘 ≤𝛿

1 𝑝 + ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 . ℎ𝑟 𝑘∈𝐼𝑟

Theorem 5. Let 0 < ℎ = inf 𝑝𝑘 = 𝑝𝑘 < sup 𝑝𝑘 = 𝐻 < ∞. Then for a Musielak-Orlicz function M = (𝑀𝑘 ) which satisfies Δ 2 -condition, one has (i) 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); (ii) 𝑤𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); 𝜃 𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). (iii) 𝑤∞

Proof . It is easy to prove, so we omit the details. (39)

𝑦𝑘 ≥𝛿

Theorem 6. Let M = (𝑀𝑘 ) be a Musielak-Orlicz function 𝜃 and let 0 < ℎ = inf 𝑝𝑘 . Then 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝜃 𝑤0 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) if and only if lim

So, we have 1 𝑝 𝐻 1 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 ≤ [𝑀𝑘 (1)] ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 ℎ𝑟 𝑘∈𝐼𝑟 ℎ𝑟 𝑘∈𝐼𝑟 𝑦𝑘 ≤𝛿

𝑦𝑘 ≤𝛿

≤ [𝑀𝑘 (2)]

𝐻

(44)

𝜃 (M󸀠 , Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) 𝑤∞

󸀠

1 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 [𝑦𝑘 ]) 𝑘 . ℎ𝑟 𝑘∈𝐼𝑟

Since M = (𝑀𝑘 ) satisfies Δ 2 -condition, we can write 𝑦 1 𝑦 1 𝑦 𝑀𝑘 (𝑦𝑘 ) < 𝑇 𝑘 𝑀𝑘 (2) + 𝑇 𝑘 𝑀𝑘 (2) = 𝑇 𝑘 𝑀𝑘 (2) . 2 𝛿 2 𝛿 𝛿 (42)

=∞

(45)

𝜃 Proof. Let 𝑤∞ (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

󵄩󵄩 Λ (𝑥) 󵄩󵄩 󵄩 󵄩 𝑡0 = 󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 󵄩󵄩 𝜌 󵄩󵄩

(40)

𝑦 2𝑦 1 1 𝑀𝑘 (𝑦𝑘 ) < 𝑀𝑘 (1 + 𝑘 ) < 𝑀𝑘 (2) + 𝑀𝑘 ( 𝑘 ) . (41) 𝛿 2 2 𝛿

𝑝𝑘

∑ 𝑘−𝑠 (𝑀𝑘 (𝑡))

for some 𝑡 > 0.

𝑦𝑘 ≤𝛿

For 𝑦𝑘 > 𝛿, 𝑦𝑘 < 𝑦𝑘 /𝛿 < 1 + 𝑦𝑘 /𝛿. Since (𝑀𝑘 )󸀠 𝑠 are nondecreasing and convex, it follows that

1

𝑟→∞ℎ 𝑟 𝑘∈𝐼𝑟

∀𝑘,

(46)

such that 1 𝑝 = ∑ 𝑘−𝑠 (𝑀𝑘 (𝑡0 )) 𝑘 ≤ 𝐾 < ∞, ℎ𝑟(𝑗) 𝑘∈𝐼

𝑚 = 1, 2, 3, . . . .

𝑟(𝑗)

(47) Let us define 𝑥 = (𝑥𝑘 ) as follows: Λ 𝑘 (𝑥) = {

𝜌𝑡0 , 𝑘 ∈ 𝐼𝑟(𝑗) 0, 𝑘 ∉ 𝐼𝑟(𝑗) .

(48)

𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). But 𝑥 ∉ Thus, by (47), 𝑥 ∈ 𝑤∞ 0 𝑤∞ (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). Hence, (45) must hold. Conversely, suppose that (45) holds and let 𝑥 ∈ 𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). Then for each 𝑟,

Hence, 1 𝑝 ∑ 𝑘−𝑠 𝑀𝑘 [𝑦𝑘 ] 𝑘 ℎ𝑟 𝑘∈𝐼𝑟 𝑦𝑘 ≥𝛿

𝑀 (2) 𝐻 1 𝑝 ≤ max (1, (𝑇 𝑘 ) ) ∑ 𝑘−𝑠 [𝑦𝑘 ] 𝑘 . 𝛿 ℎ𝑟 𝑘∈𝐼𝑟 𝑦𝑘 ≤𝛿

(43)

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ≤ 𝐾 < ∞. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 𝑟

(49)

8


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 , 󵄩󵄩 Λ (𝑥) 󵄩󵄩 󵄩󵄩 𝑘 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 > 𝜖 for 𝑘 ≥ 𝑘0 . 󵄩󵄩 󵄩󵄩 𝜌 󵄩󵄩

(50)

Then 𝑥 = (𝑥𝑘 ) ∈ 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). But by (57), 𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), which contradicts (ii). Hence,

must holds. (iii) ⇒ (i). Let (iii) hold and suppose that 𝑥 = 𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). Suppose that 𝑥 = (𝑥𝑘 ) ∉ (𝑥𝑘 ) ∈ 𝑤∞ 𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); then

From properties of sequence of Orlicz functions, we obtain 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 󵄩 󵄩 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ≥ 𝑀𝑘 (𝜖)𝑝𝑘 , 󵄩󵄩 𝜌 󵄩󵄩

sup (51)

Theorem 7. Let M = (𝑀𝑘 ) be a Musielak-Orlicz function. Then the following statements are equivalent: 𝜃 𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); (i) 𝑤∞

(iii) sup𝑟 1/ℎ𝑟 ∑𝑘∈𝐼𝑟 𝑘−𝑠 (𝑀𝑘 (𝑡))𝑝𝑘 < ∞ for all 𝑡 > 0.

Proof . (i) ⇒ (ii). Let (i) hold. To verify (ii), it is enough to prove (53)

Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). Then for 𝜖 > 0, there exists 𝑟 ≥ 0, such that (54)

𝑟

Hence, there exists 𝐾 > 0 such that sup 𝑟

(55)

𝑟

So, we get 𝑥 = (𝑥𝑘 ) ∈ (ii) ⇒ (iii). Let (ii) hold. Suppose (iii) does not hold. Then for some 𝑡 > 0 1 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 (𝑡)) 𝑘 = ∞, ℎ𝑟 𝑘∈𝐼

(56)

𝑟

and therefore we can find a subinterval 𝐼𝑟(𝑗) , of the set of interval 𝐼𝑟 , such that 1 𝑝𝑘 ∑ 𝑘−𝑠 (𝑀𝑘 ( )) > 𝑗, ℎ𝑟(𝑗) 𝑘∈𝐼 𝑗 1

𝑟

1 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 (𝑡)) 𝑘 = ∞, ℎ𝑟 𝑘∈𝐼

(60)

𝑟

which contradicts (iii). Hence, (i) must hold. Theorem 8. Let M = (𝑀𝑘 ) be a Musielak-Orlicz function. Then the following statements are equivalent: 𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); (ii) 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞ 𝑝𝑘 −𝑠 (iii) inf 𝑟 1/ℎ𝑟 ∑𝑘∈𝐼𝑟 𝑘 (𝑀𝑘 (𝑡)) > 0 for all 𝑡 > 0.

Proof. (i) ⇒ (ii). It is obvious. (ii) ⇒ (iii). Let (ii) hold. Suppose that (iii) does not hold. Then 1 𝑝 inf ∑ 𝑘−𝑠 (𝑀𝑘 (𝑡)) 𝑘 = 0 for some 𝑡 > 0, (61) 𝑟 ℎ 𝑟 𝑘∈𝐼 𝑟

and we can find a subinterval 𝐼𝑟(𝑗) , of the set of interval 𝐼𝑟 , such that 1 1 𝑝 ∑ 𝑘−𝑠 (𝑀𝑘 (𝑗)) 𝑘 < , 𝑗 = 1, 2, 3, . . . (62) ℎ𝑟(𝑗) 𝑘∈𝐼 𝑗

𝑗 = 1, 2, 3, . . .

(57)

𝑟(𝑗)

Let us define 𝑥 = (𝑥𝑘 ) as follows: 𝜌 { , 𝑘 ∈ 𝐼𝑟(𝑗) Λ 𝑘 (𝑥) = { 𝑗 {0, 𝑘 ∉ 𝐼𝑟(𝑗) .

Let us define 𝑥 = (𝑥𝑘 ) as follows: Λ 𝑘 (𝑥) = {

𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖).

𝑟

sup

𝑟(𝑗)

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩] < 𝐾. ∑ 𝑘−𝑠 [󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

sup

(59)

(i) 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖);

𝜃 (ii) 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖);

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩] < 𝜖. ∑ 𝑘−𝑠 [󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

𝑟

(52)

This completes the proof.

𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) . 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤∞

𝑟

󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = ∞. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌 󵄩󵄩 ℎ𝑟 𝑘∈𝐼

Let 𝑡 = ‖Λ 𝑘 (𝑥)/𝜌, 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 ‖ for each 𝑘; then by (59),

which contradicts (45), by using (49). Hence, we get 𝜃 𝑤∞ (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊂ 𝑤0𝜃 (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) .

𝑥 ∉ (iii)

(58)

𝜌𝑗, 𝑘 ∈ 𝐼𝑟(𝑗) 0, 𝑘 ∉ 𝐼𝑟(𝑗) .

(63)

Thus, by (62), 𝑥 = (𝑥𝑘 ) ∈ 𝑤0𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), but 𝑥 = 𝜃 (𝑥𝑘 ) ∉ 𝑤∞ (Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), which contradicts (ii). Hence, (iii) must hold. (iii) ⇒ (i). Let (iii) hold. Suppose that 𝑥 = (𝑥𝑘 ) ∈ 𝜃 𝑤0 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). Then 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󳨀→ 0 ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 󵄩󵄩 𝜌 󵄩󵄩 ℎ𝑟 𝑘∈𝐼 (64) 𝑟 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 󵄩󵄩 󵄩󵄩 Λ (𝑥) 󵄩 󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩 ≥ 𝜖 ∀𝑘. 󵄩󵄩 (65) 󵄩󵄩 󵄩󵄩 𝜌


9

Then from properties of the Orlicz function, we can write 󵄩󵄩 Λ (𝑥) 󵄩󵄩 󵄩 󵄩 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩󵄩 𝜌 󵄩󵄩

𝑝𝑘

𝑝

≥ (𝑀𝑘 (𝜖)) 𝑘 .

(66)

𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) .

1 𝑝 lim ∑ 𝑘−𝑠 (𝑀𝑘 (𝜖)) 𝑘 = 0, 𝑟→∞ℎ 𝑟 𝑘∈𝐼

(67)

𝑟

which contradicts (iii). Hence, (i) must hold.

𝑤𝜃 (M, Λ, 𝑞, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) . (68) Proof. Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜃 (M, Λ, 𝑞, 𝑠, ‖⋅, . . . , ⋅‖); write 󵄩󵄩 Λ (𝑥) 󵄩󵄩 𝑞𝑘 󵄩 󵄩 𝑡𝑘 = [𝑀𝑘 (󵄩󵄩󵄩 𝑘 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] 󵄩󵄩 𝜌 󵄩󵄩

(69)

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 𝜇

𝜇

𝜇

𝜇

𝑡𝑘𝑘 = 𝑢𝑘𝑘 + V𝑘𝑘 .

𝜇

(70)

Now it follows that 𝑢𝑘𝑘 ≤ 𝑢𝑘 ≤ 𝑡𝑘 and V𝑘𝑘 ≤ V𝑘 . Therefore, 1 1 𝜇 𝜇 𝜇 ∑𝑡 𝑘 = ∑ (𝑢 𝑘 + V𝑘𝑘 ) ℎ𝑟 𝑘∈𝐼 𝑘 ℎ𝑟 𝑘∈𝐼 𝑘 𝑟

(71)

1 1 𝜇 ≤ ∑𝑡 + ∑V . ℎ𝑟 𝑘∈𝐼 𝑘 ℎ𝑟 𝑘∈𝐼 𝑘 𝑟

Now for each 𝑘, 𝜇 1 1 1 1−𝜇 𝜇 ∑ V𝑘 = ∑ ( V𝑘 ) ( ) ℎ𝑟 𝑘∈𝐼 ℎ𝑟 ℎ𝑟 𝑘∈𝐼 𝑟

𝑟

≤ ( ∑ [( 𝑘∈𝐼𝑟

𝜇 1 V𝑘 ) ] ℎ𝑟

𝜇

1/𝜇

]

1−𝜇

(72)

𝑟

(76) Since 0 < inf 𝑝𝑘 ≤ 𝑝𝑘 ≤ 1, this implies that 󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 1 󵄩 󵄩 lim , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 𝜌 󵄩󵄩 󵄩󵄩 𝑟 𝑘∈𝐼𝑟 󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ; ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩 󵄩󵄩 𝜌 󵄩 𝑟 𝑘∈𝐼

≤ lim

𝑟

(77) therefore, 󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 󵄩󵄩 𝜌 𝑟 𝑘∈𝐼 lim

𝑟

(78) Hence, (79)

(ii) Let 𝑝𝑘 ≥ 1 for each 𝑘 and sup 𝑝𝑘 < ∞. Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖); then for each 𝜌 > 0, we have 󵄩󵄩 Λ (𝑥) − 𝐿 1 󵄩 lim , ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 𝜌 𝑟 𝑘∈𝐼𝑟 (80) 󵄩󵄩 𝑝𝑘 󵄩 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0 < 1. 󵄩󵄩 󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 󵄩󵄩 𝜌 𝑟 𝑘∈𝐼 lim

𝑟

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩󵄩 󵄩󵄩 𝜌 𝑟 𝑘∈𝐼

)

≤ lim

𝑟

𝑟

𝜇

=(

󵄩󵄩 Λ (𝑥) − 𝐿 󵄩󵄩 𝑝𝑘 1 󵄩 󵄩 , 𝑧1 , 𝑧2 , . . . , 𝑧𝑛−1 󵄩󵄩󵄩)] = 0. ∑ 𝑘−𝑠 [𝑀𝑘 (󵄩󵄩󵄩 𝑘 𝑟→∞ℎ 󵄩 󵄩󵄩 𝜌 󵄩 𝑟 𝑘∈𝐼

Since 1 ≤ 𝑝𝑘 ≤ sup 𝑝𝑘 < ∞, we have

)

1−𝜇 1/(1−𝜇)

1 × ( ∑ [( ) ℎ 𝑟 𝑘∈𝐼

(75)

Proof. (i) Let 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖); then

𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) .

𝑟

𝑟

(74)

lim

Theorem 9. Let 0 ≤ 𝑝𝑘 ≤ 𝑞𝑘 for all 𝑘 and let (𝑞𝑘 /𝑝𝑘 ) be bounded. Then

𝜇

𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) . (ii) If 1 ≤ 𝑝𝑘 ≤ sup 𝑝𝑘 = 𝐻 < ∞, for all 𝑘 ∈ N, then

Consequently, by (64), we have

𝑡𝑘 = 𝑢𝑘 + V𝑘 ,

Theorem 10. (i) If 0 < inf 𝑝𝑘 ≤ 𝑝𝑘 ≤ 1 for all 𝑘 ∈ N, then

1 ∑V ) , ℎ𝑟 𝑘∈𝐼 𝑘

=0 < 1.

𝑟

(81)

and so 𝜇

1 1 1 𝜇 ∑ V𝑘 ≤ ∑ 𝑡𝑘 + ( ∑ V𝑘 ) . ℎ𝑟 𝑘∈𝐼 ℎ𝑟 𝑘∈𝐼 ℎ𝑟 𝑘∈𝐼 𝑟

𝑟

(73)

𝑟

Hence, 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖). This completes the proof of the theorem.

Therefore, 𝑥 = (𝑥𝑘 ) ∈ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖), for each 𝜌 > 0. Hence, 𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) ⊆ 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) . This completes the proof of the theorem.

(82)

10


Theorem 11. If 0 < inf 𝑝𝑘 ≤ 𝑝𝑘 ≤ sup 𝑝𝑘 = 𝐻 < ∞, for all 𝑘 ∈ N, then 𝑤𝜃 (M, Λ, 𝑝, 𝑠, ‖⋅, . . . , ⋅‖) = 𝑤𝜃 (M, Λ, 𝑠, ‖⋅, . . . , ⋅‖) .

(83)

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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