# Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function

Authors: Sudhir Kumar; Vijay Kumar; S. S. Bhatia
Journal: International Journal of Analysis
 (2014)

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## Abstract

The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting sequence spaces. Further, we introduce a new concept ofSθσΔm(I)-convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces.


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## Body

## 1. Introduction and Background

Letl∞ and C be the Banach spaces of real bounded and convergent sequences with the usual supremum norm.Letσ be the mapping of the set of all positive integers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or σ-mean if and only if(i)
φ(x)≥0, when the sequence x=(xn) has xn≥0, for all n;(ii)
φ(e)=1, where e=(1,1,1,…);(iii)
φ(xσ(n))=φ(x), for all x∈l∞.Ifx=(xk), we write Tx=(Txk)=(xσ(k)). It can be shown by Schaefer [1] that
(1)Vσ={x∈l∞:lim⁡k→∞tkm(x)=L,uniformlyinm,σ-lim⁡x=Llim⁡k→∞},
where tkm(x)=(xm+xσ(m)+⋯+xσk(m))/(k+1).In caseσ is the translation mapping n→n+1, σ-mean is often called a Banach limit and Vσ, the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]:
(2)wσ={x:lim⁡n→∞1n+1∑k=0ntkm(x-l)⟶0,uniformlyinm},[w]σ={x:lim⁡n→∞1n+1∑k=0n‍|tkm(x-l)|⟶0,uniformlyinm},[wσ]={x:lim⁡n→∞1n+1∑k=0n‍tkm(|x-l|)⟶0,uniformlyinm}
and investigated some of its properties.The notion of statistical convergence for number sequences was studied at the initial stage by Fast [5] and later investigated by Connor [6], Fridy [7], Maddox [8], Šalát [9], and many others.Definition 1 (see [5]).
A number sequencex=(xk) is said to be statistically convergent to a number L (denoted by S-lim⁡k→∞xk=L) provided that, for every ϵ>0,
(3)lim⁡n→∞1n|{k≤n:|xk-L|≥ϵ}|=0,
where the vertical bars denote the cardinality of the enclosed set.
By a lacunary sequence, we mean an increasing sequenceθ=(kr) of positive integers such that k0=0 and hr=kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr], where the ratio kr/kr-1 is denoted by qr. The space of lacunary strongly convergent sequence Nθ was defined by Freedman et al. [10] as follows:
(4)Nθ={x=(xk):lim⁡r→∞1hr∑k∈Ir‍|xk-L|=0forsomeL}.Fridy and Orhan [11] generalized the concept of statistical convergence by using lacunary sequence which is called lacunary statistical convergence. Further, lacunary sequences have been studied by Fridy and Orhan [12], Pehlivan and Fisher [13], Et and Gökhan [14], and Tripathy and Dutta [15]. Quite recently, Karakaya [16] combined the approach of lacunary sequence with invariant means and introduced the notion of strong σ-lacunary statistically convergence as follows.Definition 2 (see [16]).
Letθ=(kr) be a lacunary sequence. A sequence x=(xk) is said to be lacunary strong σ-lacunary statistically convergent if, for every ϵ>0,
(5)lim⁡r→∞1hr|{k∈Ir:|tkm(x-L)|≥ϵ}|=0,wwwwwwwwwwwwwuniformlyinm,
where Stθσ denotes the set of all lacunary strong σ-lacunary statistically convergent sequences.
Another interesting generalization of statistical convergence was introduced in [17] with the help of ideals of subsets of N. Let P(N) denote the power set of N. A family of sets I⊂P(N) is called an ideal in N if and only if (i) ∅∈I; (ii) A,B∈I imply A∪B∈I; (iii) A∈I and B⊂A imply B∈I. A nonempty family of sets F⊂P(N) is called a filter on N if and only if (i) ∅∉F; (ii) A,B∈F imply A∩B∈F; (iii) A∈F and B⊃A imply B∈F. An ideal I is called nontrivial if I≠∅ and N∉I. It immediately implies that I⊂P(N) is a nontrivial ideal if and only if the class F=F(I)={N-A:A∈I} is a filter on N. The filter F=F(I) is called the filter associated with the ideal I. A nontrivial ideal I⊂P(N) is called an admissible ideal in N if and only if it contains all singletons, that is, if it contains {{n}:n∈N}. Throughout the present work, I denotes a nontrivial admissible ideal.Definition 3 (see [17]).
LetI⊂P(N) be a nontrivial ideal in N and let (X,ρ) be a metric space. A sequence x=(xk) in X is said to be I-convergent to ξ if, for each ϵ>0, the set A(ϵ)={k∈N:ρ(xk,ξ)≥ϵ}∈I. In this case, we write I-lim⁡k→∞xk=ξ.
Recently, Das et al. [18] unified the idea of lacunary statistical convergence with ideal convergence and presented the following interesting generalization of statistical convergence.Definition 4 (see [18]).
Letθ=(kr) be a lacunary sequence. A sequence x=(xk) of numbers is said to be I-lacunay statistical convergent or Sθ(I)-convergent to L, if, for every ϵ>0 and δ>0,
(6){r∈N:1hr|{k∈Ir:|xk-L|≥ϵ}|≥δ}∈I.
In this case, we write xk→L(Sθ(I)) or Sθ(I)-lim⁡k→∞xk=L. The set of all I-lacunary statistically convergent sequences will be denoted by Sθ(I).Definition 5 (see [18]).
Letθ=(kr) be a lacunary sequence. A sequence x=(xk) of numbers is said to be Nθ(I)-convergent to L if, for every ϵ>0, we have
(7){r∈N:1hr∑k∈Ir‍|xk-L|≥ϵ}∈I.
It is denoted by xk→L(Nθ(I)).
In 1981, Kızmaz [19] introduced the notion of difference sequence space as follows:
(8)Δ(X)={x=(xk):(Δxk)∈X},
for X=l∞,c and c0, where Δxk=xk-xk+1, for all k∈N.
Continuing on this way, the notion was further generalized by Et and Çolak [20] by introducing the sequence spaces as follows:
(9)Δm(X)={x=(xk):Δmxk∈X},
for X=l∞,c, and c0, where m∈N and Δmxk=(Δm-1xk-Δm-1xk+1), so that Δmxk=∑v=0m‍(-1)v(m/v)xk+v. For extensive view in this area, we refer to the series of papers ([21–27]).The notion of modulus function was introduced by Nakano [28] as follows: by a modulus function, we mean a function f from [0,∞) to [0,∞) such that (i) f(x)=0 if and only if x=0, (ii) f(x+y)≤f(x)+f(y), for all x,y≥0, (iii) f is increasing, and (iv) f is continuous from right at 0. It follows that f must be continuous everywhere on [0,∞). A modulus function may be bounded or unbounded. In the recent past the notion of modulus function was investigated from different aspects and sequence spaces have been studied by Ruckle [29], Maddox [30], Et [31], Pehlivan and Fisher [13], Savas [32], Et and Gökhan [14], Kumar et al. [33], and many others.The following well-known lemma is required for establishing a very important result in our paper.Lemma 6.
Letf be a modulus function and let 0<δ<1. Then, for each x>δ, we have f(x)≤(2·f(1)x)/δ.The following inequality will be used throughout the paper. Letp=(pk) be a positive sequence of real numbers with 0<pk≤supkpk=H and C=max⁡(1,2H-1). Then, for all ak, bk∈C, for all k∈N, we have
(10)|ak+bk|pk≤C{|ak|pk+|bk|pk}.Inspired by the above works, we presently introduce some new kind of sequence spaces by using ideal convergence, modulus function, and invariant mean. Further, we also obtain some relevant connections of these spaces withSθσΔm(I)-convergence.


## 2. Main Results

Throughout the paper,I⊂P(N) is considered a nontrivial admissible ideal and w(X) denotes the space of all sequences x=(xk)∈X.Definition 7.
LetI be an admissible ideal, let f be a modulus function, and let p=(pk) be any sequence of strictly positive real numbers. Then, for each ϵ>0, we define the following sequence spaces:
(11)wσf[Δpm,I,θ]0={{r∈N:1hr∑k∈Ir[f(|tkn(Δmx)|)]pk≥ϵ}x∈w(X):{r∈N:1hr∑k∈Ir[f(|tkn(Δmx)|)]pk≥ϵ}∈I},wσf[Δpm,I,θ]={{r∈N:1hr∑k∈Ir‍[f(|tkn(Δmx-l)|)]pk≥ϵ}x∈w(X):∃l>0,{r∈N:1hr∑k∈Ir‍[f(|tkn(Δmx-l)|)]pk≥ϵ}∈I},wσf[Δpm,I,θ]∞={{r∈N:1hr∑k∈Ir‍[f(|tkn(Δmx)|)]pk≥K}x∈w(X):∃K>0,{r∈N:1hr∑k∈Ir‍[f(|tkn(Δmx)|)]pk≥K}∈I}
uniformly in n.Remark 8.
By taking some particular cases, we obtain the following.(i)
If we takem=0, for all k∈N, then the above spaces reduce to wσf[I,p,θ]0, wσf[I,p,θ], and wσf[I,p,θ]∞, respectively.(ii)
If we choosem=0, for all k∈N and σ(n)=n+1, then we obtain w^σf[I,p,θ]0, w^σf[I,p,θ], and w^σf[I,p,θ]∞ instead of wσf[Δpm,I,θ]0, wσf[Δpm,I,θ], and wσf[Δpm,I,θ]∞.(iii)
By takingI=If={E⊂N:ifEisafinitesubset}, m=0, f(x)=x, θ=(2r), and pk=1, for all k∈N, then we obtain [w]σ defined by Mursaleen et al.[3] instead of wσf[Δpm,I,θ] and w^σf[I,p,θ] reduces to [w^] defined in Das and Sahoo [4].Theorem 9.
Letf be a modulus function and p=(pk) is a bounded sequence of strictly positive real numbers; then, wσf[Δpm,I,θ]0, wσf[Δpm,I,θ], and wσf[Δpm,I,θ]∞ are linear spaces over C.Proof.
We will prove the assertion only forwσf[Δpm,I,θ]0 and others can be treated similarly. Suppose that x=(xk), y=(yk)∈wσf[Δpm,I,θ]0. Then, for every ϵ>0 and uniformly in n, the sets
(12)Aθ(ϵ2)={r∈N:1hr∑k∈Ir[f(|tkn(Δmx)|)]pk≥ϵ2},Bθ(ϵ2)={r∈N:1hr∑k∈Ir‍[f(|tkn(Δmy)|)]pk≥ϵ2}
belong to I.
Letα, β∈C and Δm is linear; then,
(13)1hr∑k∈Ir[f(|tkn(Δm(α·x+β·y))|)]pk=1hr∑k∈Ir[f(|tkn(α·Δmx+β·Δmy)|)]pk≤C·(Kα)H·1hr∑k∈Ir[f(|tkn(Δmx)|)]pk+C·(Kβ)H·1hr∑k∈Ir[f(|tkn(Δmy)|)]pk,by(10),
where Kα, Kβ are two positive numbers such that |α|≤Kα and |β|≤Kβ.
Then, for givenϵ>0, we have the following containment:
(14){r∈N:1hr∑k∈Ir[f(|tkn(Δm(α·x+β·y))|)]pk≥ϵ}⊆{r∈N:1hr∑k∈Ir[f(|tkn(Δmx)|)]pk≥ϵ2C·(Kα)H}∪{r∈N:1hr∑k∈Ir[f(|tkn(Δmy)|)]pk≥ϵ2C·(Kβ)H}
uniformly in n.
Sincex,y∈wσf[Δpm,I,θ]0, it follows that the later sets belong to I. By using the property of an ideal the set on the left hand side in the above expression also belongs to I. This completes the proof.Theorem 10.
Form≥1, then the inclusion
(15)wσf[Δpm-1,I,θ]0,L,∞⊂wσf[Δpm,I,θ]0,L,∞
is strict.Proof.
We will prove the result forwσf[Δpm-1,I,θ]0 only. Suppose that x=(xk)∈wσf[Δpm-1,I,θ]0; by definition, for each ϵ>0,
(16){r∈N:1hr∑k∈Ir[f(|tkn(Δm-1xk)|)]pk‍≥ϵ}∈I
uniformly in n.
Sincef is a modulus function, therefore we have the following inequality:
(17)1hr∑k∈Ir[f(|tkn(Δmxk)|)]pk‍≤1hr∑k∈Ir[f(|tkn(Δm-1xk)|)+f(|tkn(Δm-1xk+1)|)]pk≤C·1hr∑k∈Ir[f(|tkn(Δm-1xk)|)]pk+C·1hr∑k∈Ir[f(|tkn(Δm-1xk+1)|)]pkby(10)
uniformly in n.
Now, for givenϵ>0, we have the following containment:
(18){r∈N:1hr∑k∈Ir[f(|tkn(Δmxk)|)]pk≥ϵ}⊆{r∈N:1hr∑k∈Ir[f(|tkn(Δm-1xk)|)]pk≥ϵ2C}∪{r∈N:1hr∑k∈Ir[f(|tkn(Δm-1xk+1)|)]pk≥ϵ2C},
uniformly in n.
Both the sets on the right hand side in the above containment belong toI by (16). It follows that x∈wσf[Δpm,I,θ]0.
SinceI is an admissible ideal and the inclusion is strict as the sequence x=(xk)=(km-1) belongs to x∈wσf[Δpm,I,θ]0, it does not belong to wσf[Δpm-1,I,θ]0, for f(x)=x, t0n(x)=(xn), θ=(2r), and pk=1, for all k∈N.Theorem 11.
Letp=(pk) be a bounded sequence of strictly positive real numbers and f′, f′′ are modulus functions. If
(19)limsup⁡t→∞f′(t)f′′(t)=M>0,
then wσf′[Δpm,I,θ]0⊂wσf′′[Δpm,I,θ]0.Proof.
Assume thatlimsup⁡t→∞(f′(t)/f′′(t))=M; there exists a positive number K>0 such that f′(t)≥K·f′′(t), for all t≥0, which implies that
(20)1hr∑k∈Ir‍[f′(|tkn(Δmx)|)]pk≥(K)H·1hr∑k∈Ir[f′′(|tkn(Δmx)|)]pk‍
uniformly in n.
Thus, for anyϵ>0,
(21){r∈N:1hr∑k∈Ir[f′′(|tkn(Δmx)|)]pk≥ϵ}⊆{r∈N:1hr∑k∈Ir[f′(|tkn(Δmx)|)]pk‍≥ϵ·(K)H}
uniformly in n. Therefore, the above containment gives the result.Theorem 12.
Iff,f′, and f′′ are modulus functions, then (i)
wσf′[Δpm,I,θ]⊂wσf∘f′[Δpm,I,θ],(ii)
wσf′[Δpm,I,θ]∩wσf′′[Δpm,I,θ]⊂wσf′+f′′[Δpm,I,θ].Proof.
(i) Letx=(xk)∈wσf′[Δpm,I,θ] and let ϵ>0 be given. Choose δ∈(0,1) such that f(t)<ϵ, for all 0<t<δ, since x∈wσf′[Δpm,I,θ] such that
(22)Aδ={n∈N:1hr∑k∈Ir[f′(|tkn(Δmx-l)|)]pk≥δ}∈I
uniformly in n.
On the other hand, we have(23)1hr∑k∈Ir[f∘f′(|tkn(Δmx-l)|)]pk‍=1hr∑k∈Ir&[f′(|tkn(Δmx-l)|)]pk<δ‍[f∘f′(|tkn(Δmx-l)|)]pk+1hr∑k∈Ir&[f′(|tkn(Δmx-l)|)]pk≥δ[f∘f′(|tkn(Δmx-l)|)]pk‍≤(ϵ)H+max⁡⁡(1,(2·f(1)δ)H)·1hr∑k∈Ir[f′(|tkn(Δmx-l)|)]pkbyLemma6,
uniformly in n.
Then, for anyη>0,
(24){n∈N:1hr∑k∈Ir[f∘f′(|tkn(Δmx-l)|)]pk≥η}⊆{n∈N:1hr∑k∈Ir[f′(|tkn(Δmx-l)|)]pk≥(η-ϵ)K},
where K=max⁡(1,(2,f(1)/δ)H).
Sincex=(xk)∈wσf′[Δpm,I,θ], so by (22) the latter set belongs to I and therefore the theorem is proved.
(ii) This result can be proved by the following inequality:(25)1hr∑k∈Ir[(f′+f′′)(|tkn(Δmx-l)|)]pk≤Chr∑k∈Ir[f′(|tkn(Δmx-l)|)]pk+Chr∑k∈Ir[f′′(|tkn(Δmx-l)|)]pk,
uniformly in n, where supkpk=H and C=max⁡(1,2H-1).Theorem 13.
Iff is a modulus function and p=(pk) is a sequence of positive real numbers, then wσ[Δpm,I,θ]⊆wσf[Δpm,I,θ].Proof.
This can be proved similarly as in Theorem12(i).Theorem 14.
Letf be a modulus function. If  limsup⁡t→∞(f(t)/t)=M>0, then
(26)wσf[Δpm,I,θ]⊆wσ[Δpm,I,θ].Proof.
Suppose thatx=(xk)∈wσf[Δpm,I,θ]. It is given that limsup⁡t→∞(f(t)/t)=M>0; there exists a constant K>0 such that f(t)≥K·t, for all t≥0,
which implies that(27)1hr∑k∈Ir[f(|tkn(Δmx-l)|)]pk≥(K)H·1hr∑k∈Ir|tkn(Δmx-l)|pk
uniformly in n.
Then, for eachϵ>0,
(28){n∈N:1hr∑k∈Ir‍[|tkn(Δmx-l)|]pk≥ϵ}⊆{n∈N:1hr∑k∈Ir[f(|tkn(Δmx-l)|)]pk‍≥ϵ·(K)H},
uniformly in n.
Sincex∈wσf[Δpm,I,θ], therefore the latter set belongs to I. It follows that x∈wσ[Δpm,I,θ].Theorem 15.
If0<pk≤qk and (qk/pk) are bounded, then wσf[Δqm,I,θ]⊂wσf[Δpm,I,θ].Proof.
The proof of this theorem is easy and so it is omitted.


## 3.SθσΔm(I)-Convergence

In this section, we define the notion ofSθσΔm(I)-convergence with the help of ideal and invariant means and difference sequences. Further, we also establish some relations between SθσΔm(I)-convergence and wσf[Δpm,I,θ]0.Definition 16.
LetI⊆P(N) be a nontrivial ideal. A sequence x=(xk) is said to be Δm(I)-strong lacunary σ-statistically convergent or SθσΔm(I)-convergent to a number l, provided that, for every ϵ>0 and δ>0,
(29){r∈N:1hr|{k∈Ir:|tkn(Δmx-l)|≥ϵ}|≥δ}∈I,wwwwwwwwwwwwwwwwwwwwwiuniformlyinn.
In this case, we write xk→l(SθσΔm(I)) or SθσΔm(I)-lim⁡k→∞xk=l. Let SθσΔm(I) denote the set of all SθσΔm(I)-convergent sequences.Theorem 17.
Letf be a modulus function and let p=(pk) be a sequence of strictly positive real numbers. If 0<infkpk=h≤pk≤supkpk=H<∞, then wσf[Δpm,I,θ]0⊂SθσΔm(I).Proof.
Assume thatx∈wσf[Δpm,I,θ]0 and ϵ>0 is given. Then, we have
(30)1hr∑k∈Ir[f(|tkn(Δmx)|)]pk=1hr∑k∈Ir&|tkn(Δmx)|≥ϵ[f(|tkn(Δmx)|)]pk+1hr∑k∈Ir&|tkn(Δmx)|<ϵ[f(|tkn(Δmx)|)]pk≥1hr∑k∈Ir&|tkn(Δmx)|≥ϵ[f(|tkn(Δmx)|)]pk≥1hr∑k∈Ir[f(ϵ)]pk≥1hr∑k∈Irmin⁡⁡([f(ϵ)]h,[f(ϵ)]H)≥1hr|{k∈Ir:|tkn(Δmx)|≥ϵ}|·min⁡⁡([f(ϵ)]h,[f(ϵ)]H)
uniformly in n.
Then, for everyδ>0, we have the following containment:
(31){r∈N:1hr|{k∈Ir:|tkn(Δmx)|≥ϵ}|≥δ}⊆{r∈N:1hr∑k∈Ir[f(|tkn(Δmx)|)]pk≥K·δ}
uniformly in n, where K=min⁡([f(ϵ)]h,[f(ϵ)]H), since x∈wσf[Δpm,I,θ]0, which implies that x∈SθσΔm(I).Theorem 18.
Letf be a bounded modulus function and 0<infkpk=h≤pk≤supkpk=H<∞; then, SθσΔm(I)⊂wσf[Δpm,I,θ]0.Proof.
Using the same technique of [26, Theorem 3.3], it is easy to prove.Theorem 19.
If0<infkpk=h≤pk≤supkpk=H<∞, then SθσΔm(I)=wσf[Δpm,I,θ]0 if and only if f is bounded.Proof.
This part is the direct consequence of Theorems17 and 18.Conversely. Suppose that f is unbounded defined by f(k)=k, for all k∈N, and let θ=(2r) be a lacunary sequence. We take a fixed set A∈I and define x=(xk) as follows:
(32)xk={km+1,ifr∉A,2r-1+1≤k≤2r-1+[hr],km+1,ifr∈A,2r-1<k≤2r-1+hr,0,otherwise,
where Ir=(2r-1,2r] and hr=2r-2r-1.For givenϵ>0, we have
(33)lim⁡r→∞1hr|{k∈Ir:|tkn(Δmx-0)|≥ϵ}|=lim⁡r→∞1hr|{k∈Ir:|tkn((-1)m+2(m+1)!(k+m2)-0)|≥ϵ}|≤[hr]hr⟶0,
for all r∉A, and uniformly in n.Hence, forδ>0, there exists a positive integer r0 such that
(34)1hr|{k∈Ir:|tkn(Δmx-0)|≥ϵ}|<δwwwwwwwwwiforr∉A,r≥r0.
Now, we have
(35){r∈N:1hr|{k∈Ir:|tkn(Δmx-0)|≥ϵ}|≥δ}⊂(A∪(1,2,…,r0-1)).
Since I is an admissible ideal, it follows that SθσΔm(I)-lim⁡k→∞xk=0.If we takepk=1, for all k=1,2,…, then xk∉wσf[Δpm,I,θ]0. This contradicts the fact that SθσΔm(I)=wσf[Δpm,I,θ]0.



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*Source: 631301-2014-05-28.xml*