Some New Classes of Generalized Difference Strongly Summable 𝑛-Normed Sequence Spaces Defined by Ideal Convergence and Orlicz Function

Abstract

We study some new generalized difference strongly summable n-normed sequence spaces using ideal convergence and an Orlicz function in connection with de la Vallèe Poussin mean. We give some relations related to these sequence spaces also.

1. Introduction

Let ∞, 𝑐, and 𝑐₀ be the Banach space of bounded, convergent, and null sequences 𝑥=(𝑥_𝑘), respectively, with the usual norm ‖𝑥‖=sup_𝑛|𝑥_𝑛|.

A sequence 𝑥∈_∞ is said to be almost convergent if all of its Banach limits coincide.

Let 𝑐̂ denote the space of all almost convergent sequences.

Lorentz in [1] proved that 𝑐̂=𝑥∈𝑙_∞∶lim_𝑚𝑡_𝑚,𝑛(𝑥)existsuniformlyin𝑛, (1) where 𝑡_𝑚,𝑛(𝑥)=𝑥_𝑛+𝑥_𝑛+1+⋯+𝑥_𝑚+𝑛.𝑚+1 (2) The following space of strongly almost convergent sequence was introduced by Maddox in [2]: [𝑐]̂=𝑥∈_∞∶lim_𝑚𝑡_𝑚,𝑛(|𝑥−𝐿𝑒|)existsuniformlyin𝑛forsome𝐿, (3) where 𝑒=(1,1,…).

Let 𝜎 be a one-to-one mapping from the set of positive integers into itself such that 𝜎^𝑚(𝑛)=𝜎^𝑚−1(𝜎(𝑛)), 𝑚=1,2,3,…, where 𝜎^𝑚(𝑛) denotes the 𝑚th iterative of the mapping 𝜎 in 𝑛.

A continuous linear functional 𝜙 on ∞ is said to be an invariant mean or a 𝜎-mean, if and only if it satisfies the following conditions:(1)𝜙(𝑥)≥0, when the sequence 𝑥=(𝑥_𝑛) is such that 𝑥_𝑛≥0 for all 𝑛;(2)𝜙(𝑒)=1, where 𝑒=(1,1,…);(3)𝜙(𝑥_𝜎(𝑛))=𝜙(𝑥), for all 𝑥∈_∞.For a certain kind of mapping 𝜎, we get that every invariant mean 𝜙 extends the functional limit on the space 𝑐, such that 𝜙(𝑥)=lim𝑥 for all 𝑥∈𝑐. Consequently, we get that 𝑐⊂𝑉_𝜎, where 𝑉_𝜎 is the set of bounded sequences with equal 𝜎-means.

Schaefer in [3] proved that 𝑉_𝜎=𝑥∈_∞∶lim_𝑘𝑡_𝑘𝑚(𝑥)=𝐿uniformlyin𝑚forsome𝐿=𝜎−lim𝑥, (4) where 𝑡_𝑘𝑚(𝑥)=𝑥₊𝑥_𝜎(𝑚)+⋯+𝑥_{𝜎𝑘(𝑚)}𝑘+1,𝑡_−1,𝑚=0. (5) Thus we say that a bounded sequence 𝑥=(𝑥_𝑘) is 𝜎-convergent, if and only if 𝑥∈𝑉_𝜎 such that 𝜎^𝑘(𝑛)≠𝑛 for all 𝑛≥0, 𝑘≥1.

Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the 𝜎-convergence leads naturally to the concept of strong 𝜎-convergence.

A sequence 𝑥=(𝑥_𝑘) is said to be strongly 𝜎-convergent (Mursaleen [4]), if there exists a number such that 1𝑘^𝑘_𝑖=1𝑥_{𝜎𝑖(𝑚)}−⟶0,as𝑘⟶∞uniformlyin𝑚. (6) We write [𝑉_𝜎] to denote the set of all strong 𝜎-convergent sequences and when (6) holds, we write [𝑉_𝜎]−lim𝑥=.

Taking 𝜎(𝑚)=𝑚+1, we obtain [𝑉_𝜎]=[𝑐]̂. Then the strong 𝜎-convergence generalizes the concept of strong almost convergence.

We also note that 𝑉_𝜎⊂𝑉_𝜎⊂_∞. (7) The notion of ideal convergence was first introduced by Kostyrko et al. [5] as a generalization of statistical convergence which was later studied by many other authors.

An Orlicz function 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→∞.

Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to construct the sequence space: _𝑀=𝑥_𝑘∈𝑤∶^∞_𝑘=1𝑀|𝑥|𝜌<∞,forsome𝜌>0. (8) The space 𝑀 with the norm ‖𝑥‖=inf𝜌>0∶^∞_𝑘=1𝑀𝑥_𝑘𝜌≤1 (9) becomes a Banach space which is called an Orlicz sequence space.

Kızmaz [7] studied the difference sequence spaces _∞(Δ), 𝑐(Δ), and 𝑐₀(Δ) of crisp sets. The notion is defined as follows: 𝑍(Δ)=𝑥=𝑥_𝑘∶Δ𝑥_𝑘∈𝑍, (10) for 𝑍=_∞,𝑐 and 𝑐₀, where Δ𝑥=(Δ𝑥_𝑘)=(𝑥_𝑘−𝑥_𝑘+1), for all 𝑘∈𝑁.

The above spaces are Banach spaces, normed by ‖𝑥‖_Δ=𝑥₁+sup_𝑘Δ𝑥_𝑘. (11) Later the idea of Kızmaz [7] was applied to introduce different types of difference sequence spaces and study their different properties by many others later on.

The generalized difference notion is defined as follows. For 𝑚≥1 and 𝑛≥1, 𝑍Δ^𝑛_𝑚=𝑥=𝑥_𝑘∶Δ^𝑛_𝑚𝑥_𝑘∈𝑍,for𝑍=_∞,𝑐and𝑐₀. (12) This generalized difference has the following binomial representation: Δ^𝑛_𝑚𝑥_𝑘=^𝑛_𝑟=0(−1)^𝑟𝑛𝑟𝑥_{𝑘+𝑟𝑚}. (13) The concept of 2-normed space was initially introduced by Gähler [8], in the mid of 1960s, while that of 𝑛-normed spaces can be found in Misiak [9]. Since then, many other authors have used this concept and obtained various results. Recently, several various activities have been initiated to study summability, sequence spaces, and related topics in these spaces. The notion of ideal convergence in 2-normed spaces was initially introduced by Gurdal [10]. Later on, it was extended to 𝑛-normed spaces by Gurdal and Sahiner in [11].

2. Definitions and Preliminaries

Let 𝑛∈𝑁 and 𝑋 be a real vector space. A real valued function on 𝑋^𝑛 satisfies the following four properties:(1)‖(𝑧₁,𝑧₂,…,𝑧_𝑛)‖_𝑛=0 if and only if 𝑧₁,𝑧₂,…,𝑧_𝑛 are linearly dependent;(2)‖(𝑧₁,𝑧₂,…,𝑧_𝑛)‖_𝑛 is invariant under permutation;(3)‖(𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝛼𝑧_𝑛)‖_𝑛=|𝛼|‖𝑧₁,𝑧₂,…,𝑧_𝑛‖_𝑛, for all 𝛼∈𝑅;(4)‖(𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥+𝑦)‖_𝑛≤‖(𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑥)‖_𝑛+‖(𝑧₁,𝑧₂,…,𝑧_𝑛−1,𝑦)‖_𝑛is called an 𝑛-norm on 𝑋 and the pair (𝑋,‖⋅,⋅‖_𝑛) is called an 𝑛-normed space.

Let 𝑋 be a nonempty set. Then a family of sets 𝐼⊆2^𝑋 (power sets of 𝑋) is said to be an 𝑖𝑑𝑒𝑎𝑙 if 𝐼 is additive that is 𝐴,𝐵∈𝐼⇒𝐴∪𝐵∈𝐼 and hereditary that is 𝐴∈𝐼,𝐵⊆𝐴⇒𝐵∈𝐼.

A sequence (𝑥_𝑘) in a normed space (𝑋,‖⋅,⋅,⋯‖_𝑛) is said to be 𝐼-convergent to 𝑥₀∈𝑋 with respect to 𝑛-norm, if for each 𝜀>0, the set 𝐸(𝜀)=𝑘∈𝑁∶𝑥_𝑘−𝑥₀,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛≥𝜀,forevery𝑧₁,𝑧₂,…,𝑧_𝑛−1∈𝑋belongsto𝐼. (14) The generalized de la Vallée Poussin mean is defined by 𝑡_𝑛(𝑥)=1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘, (15) where 𝐼_𝑛=[𝑛−𝜆_𝑛+1,𝑛] for 𝑛=1,2….

Then a sequence 𝑥=(𝑥_𝑘) is said to be (𝑉,𝜆)-summable to a number 𝐿, if 𝑡_𝑛(𝑥)→𝐿 as 𝑛→∞, and we write [𝑉,𝜆]₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘,[]=[]=0𝑉,𝜆𝑥∶𝑥−𝑒∈𝑉,𝜆₀forsome,[]∈𝐶𝑉,𝜆_∞=𝑥∶sup_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘<∞ (16) for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by de la Vallée Poussin method.

Maddox introduced and studied the special case, where 𝜆_𝑛=𝑛, for 𝑛=1,2,3,…; the sets [𝑉,𝜆]₀, [𝑉,𝜆], and [𝑉,𝜆]_∞ reduce to the sets 𝑤₀, 𝑤, and 𝑤_∞.

In this paper, we define some new sequence spaces in 𝑛-normed spaces by using Orlicz function with notion of generalized de la Vallèe Poussin mean, generalized difference sequences, and ideals. We will also introduce and examine certain new sequence spaces using the above tools as also the 𝑛-norm.

3. Main Results

Let 𝐼 be an admissible ideal of 𝑁, let 𝑀 be an Orlicz function, and let (𝑋,‖⋅,⋯‖_𝑛) be a 𝑛-normed space. Further, let 𝑠=(𝑠_𝑘) be a bounded sequence of positive real numbers. By 𝑆(𝑛−𝑋), we denote the space of all sequences defined over (𝑋,‖⋅,⋯‖_𝑛).

In this paper, we have introduced the following sequence spaces: 𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼₀,‖⋅,⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼,forsome𝜌>0andeach𝑉𝑧∈𝑋,_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼,‖⋅,⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼,forsome𝜌>0,𝐿∈𝑋andeach𝑉𝑧∈𝑋,_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼_∞,‖⋅,⋅,⋯‖_𝑛=𝑥∶∃𝐾>0,s.t.sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝐾∈𝐼,forsome𝜌>0andeach𝑧∈𝑋. (17) In particular, if we take 𝑟_𝑘=1, for all 𝑘, we have 𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀^𝐼₀,‖⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌≥𝜀,uniformlyin𝑚∈𝐼,forsome𝜌,∀𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑉∈𝑋,_𝜎,𝜆,Δ^𝑞_𝑝,𝑀^𝐼,‖⋅,⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌≥𝜀∈𝐼,forsome𝜌>0,𝐿∈𝑋andeach𝑉𝑧∈𝑋,_𝜎,𝜆,Δ^𝑞_𝑝,𝑀^𝐼_∞,‖⋅,⋅,⋯‖_𝑛=×𝑥∶∃𝐾>0sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌≥𝐾∈𝐼. (18) Similarly, when 𝜎(𝑚)=𝑚+1, then ([𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼₀,‖⋯‖_𝑛), ([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋯‖_𝑛), and ([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼_∞,‖⋯‖_𝑛) are reduced to 𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼₀,‖⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_𝑘+𝑚,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼,forsome𝜌>0andeach𝑧∈𝑋,𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼,‖⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_𝑘+𝑚−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼,forsome𝜌>0,𝐿∈𝑋andeach𝑧∈𝑋,𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼_∞,‖⋅,⋯‖_𝑛=𝑥∶∃𝐾>0,s.t.sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_𝑘+𝑚,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝑘∈𝐼,forsome𝜌>0,andeach𝑧∈𝑋. (19) In particular, if we put 𝑟_𝑘=𝑟, for all 𝑘, then we have the spaces 𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼₀,‖⋅,⋯‖_𝑛=𝑉,𝜆,Δ^𝑞_𝑝,𝑀^𝐼₀,‖⋅,⋅‖_𝑛,𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼,‖⋅,⋯‖_𝑛=𝑉,𝜆,Δ^𝑞_𝑝,𝑀^𝐼,‖⋅,⋅‖_𝑛,𝑉,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼_∞,‖⋅,⋯‖_𝑛=𝑉,𝜆,Δ^𝑞_𝑝,𝑀^𝐼_∞,‖⋅,⋅‖_𝑛. (20) Further when 𝜆_𝑛=𝑛, for 𝑛=1,2,…, the sets ([𝑉,𝜆,Δ^𝑞_𝑝,𝑀]^𝐼₀,‖⋅,⋅,‖_𝑛) and ([𝑉,𝜆,Δ^𝑞_𝑝,𝑀]^𝐼,‖⋅,⋅,‖_𝑛) are reduced to ([𝑐₀(𝑀,Δ^𝑞_𝑝)]^𝐼,‖⋅,⋅,⋯‖_𝑛) and ([𝑐(̂𝑀,Δ^𝑞_𝑝)]^𝐼,‖⋅,⋅,⋯‖_𝑛), respectively.

Now, if we consider 𝑀(𝑥)=𝑥, then we can easily obtain 𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑟^𝐼₀,‖⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚),},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛^𝑟𝑘≥𝜀,uniformlyin𝑚,𝑉∈𝐼_𝜎,𝜆,Δ^𝑞_𝑝,𝑟^𝐼,‖⋅,⋯‖_𝑛=×1𝑥∶∀𝜀>0𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛^𝑟𝑘≥𝜀∈𝐼,for𝐿∈𝑋andeach𝑉𝑧∈𝑋,_𝜎,𝜆,Δ^𝑞_𝑝,𝑟^𝐼_∞,‖⋅,⋯‖_𝑛=𝑥∶∃𝐾>0,s.t.sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛^𝑟𝑘≥𝐾∈𝐼. (21) If 𝑥∈([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋯‖_𝑛), with {(1/𝜆_𝑛)∑_{𝑘∈𝐼𝑛}{𝑀(‖Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1‖_𝑛/𝜌)}^𝑟𝑘≥𝜀}∈𝐼 as 𝑛→∞ uniformly in 𝑚, then we write 𝑥_𝑘→𝐿∈([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋯‖_𝑛).

The following well-known inequality will be used later.

If 0≤𝑟_𝑘≤sup𝑟_𝑘=𝐻 and 𝐶=max(1,2^𝐻−1), then 𝑎_𝑘+𝑏_𝑘^𝑟𝑘≤𝐶𝑎_𝑘^𝑟𝑘+𝑏_𝑘^𝑟𝑘, (22) for all 𝑘 and 𝑎_𝑘,𝑏_𝑘∈𝐶.

Lemma 1. Let 𝑟_𝑘>0 and 𝑠_𝑘>0. Then 𝑐₀(𝑠)⊂𝑐₀(𝑟), if and only if lim_𝑘→∞inf(𝑟_𝑘/𝑠_𝑘)>0, where 𝑐₀(𝑟)={𝑥∶|𝑥_𝑘|^𝑟𝑘→0𝑎𝑠𝑘→∞}.
Note that no other relation between (𝑟_𝑘) and (𝑠_𝑘) is needed in Lemma 1.

Theorem 2. Let lim_𝑘→∞inf𝑟_𝑘>0. Then, 𝑥_𝑘→𝐿 implies 𝑥_𝑘→𝐿∈([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋯‖_𝑛). Let lim_𝑘→∞𝑟_𝑘=𝑟>0. If 𝑥_𝑘→𝐿∈([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋯‖_𝑛), then 𝐿 is unique.

Proof. Let 𝑥_𝑘→𝐿.
By the definition of Orlicz function, we have, for all 𝜀>0, lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌≥𝜀∈𝐼. (23) Since lim_𝑘→∞inf𝑟_𝑘>0, it follows that lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼. (24) And consequently, 𝑥_𝑘→𝐿∈[𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼.
Let lim_𝑘→∞𝑟_𝑘=𝑟>0. Suppose that 𝑥_𝑘→𝐿₁∈[𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼, 𝑥_𝑘→𝐿₂∈[𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼, and (‖𝐿₁−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1‖_𝑛)^𝑟𝑘=𝑎>0.
Now, from (22) and the definition of Orlicz, we have 1𝜆_𝑛lim_{𝑘∈𝐼𝑛}𝑀𝐿₁−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≤𝐶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘+𝐶𝜆_𝑛𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘, (25) since 1𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿₁,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘1≥𝜀∈𝐼,𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)}−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼. (26) Hence, 1𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Ł₁−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼. (27) Further, 𝑀(‖𝐿₁−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1‖_𝑛/𝜌)^𝑟𝑘→𝑀(𝑎/𝜌)^𝑟 as 𝑘→∞, and therefore lim_𝑛→∞1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀𝐿₁−𝐿₂,𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘𝑎=𝑀𝜌^𝑟. (28) From (27) and (28), it follows that 𝑀(𝑎/𝜌)=0 and by the definition of an Orlicz function, we have 𝑎=0.
Hence, 𝐿₁=𝐿₂ and this completes the proof.

Theorem 3. (i) Let 0<inf_𝑘𝑟_𝑘≤𝑟_𝑘≤1. Then, [𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼⊂[𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀]^𝐼. (29) (ii) Let 0<𝑟_𝑘≤sup_𝑘𝑟_𝑘<∞. Then, 𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀^𝐼⊂𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟^𝐼. (30)

Theorem 4. Let 𝑋(𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1) stand for ([𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1,𝑀,𝑟]^𝐼₀,‖⋅,⋅,⋯‖_𝑛), ([𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1,𝑀,𝑟]^𝐼,‖⋅,⋅,⋯‖_𝑛), or ([𝑉_𝜎,𝜆Δ_𝑝^𝑞−1,𝑀,𝑟]^𝐼_∞,‖⋅,⋅,⋯‖_𝑛) and 𝑚≥1. Then the inclusion 𝑋(𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1)⊂𝑋(𝑉_𝜎,𝜆,Δ^𝑞_𝑝) is strict. In general, 𝑋(𝑉_𝜎,𝜆,Δ^𝑖_𝑝)⊂𝑋(𝑉_𝜎,𝜆,Δ^𝑞_𝑝) for all 𝑖=1,2,3,…,𝑝−1 and the inclusion is strict.

Proof. Let us take ([𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1,𝑀,𝑟]^𝐼₀,‖⋅,⋅,⋯‖_𝑛).
Let 𝑥=(𝑥_𝑘)∈([𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1,𝑀,𝑟]^𝐼₀,‖⋅,⋅,⋯‖_𝑛). Then for given 𝜀>0, we have 1𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀∈𝐼,forsome𝜌>0. (31) Since 𝑀 is nondecreasing and convex, it follows that 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘=1𝜆_𝑛×_{𝑘∈𝐼𝑛}𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘+1(𝑚)}−Δ_𝑝^𝑞−1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘1≤𝐷𝜆_{𝑛𝑘∈𝐼𝑛}12𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘+1(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘+12𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘1≤𝐷𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘+1(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘+𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘. (32) Hence we have 1𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘⊆𝐷1≥𝜀𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘+1(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀2∪𝐷1𝑛∈𝑁∶𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ_𝑝^𝑞−1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝑟𝑘≥𝜀2. (33) Since the set on the right hand side belongs to 𝐼, so does the left hand side. The inclusion is strict as the sequence 𝑥=(𝑘^𝑟), for example, belongs to ([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀]^𝐼₀,‖⋅,⋅,⋯‖_𝑛) but does not belong to ([𝑉_𝜎,𝜆,Δ_𝑝^𝑞−1,𝑀]^𝐼₀,‖⋅,⋅,⋯‖_𝑛) for 𝑀(𝑥)=𝑥 and 𝑟_𝑘=1 for all 𝑘.

Theorem 5. ([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼₀,‖⋅,⋅,⋯‖_𝑛) and ([𝑉_𝜎,𝜆,Δ^𝑞_𝑝,𝑀,𝑟]^𝐼,‖⋅,⋅,⋯‖_𝑛) are complete linear topological spaces, with paranorm 𝑔, where 𝑔 is defined by 𝑔(𝑥)=^𝑝𝑞_𝑚=1𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛+inf𝜌^{𝑟𝑘/𝐻}∶sup_𝑚,𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑀Δ^𝑞_𝑝𝑥_{𝜎𝑘(𝑚)},𝑧₁,𝑧₂,…,𝑧_𝑛−1𝑛𝜌^𝐻, (34) where 𝐻=max(1,(sup_𝑘𝑟_𝑘)).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author acknowledges that this research was part of the research project and partially supported by the Universiti Putra Malaysia under Grant no. ERGS 1-2013/5527179. The work of second author was carried out under the Postdoctoral Fellow under National Board of Higher Mathematics, DAE, Project no. NBHM/PDF.50/2011/64.