# Strongly Almost LacunaryI-Convergent Sequences

Authors: Adem Kılıçman; Stuti Borgohain
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study some new strongly almost lacunaryI-convergent generalized difference sequence spaces defined by an Orlicz function. We give also some inclusion relations related to these sequence spaces.


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## Body

## 1. Introduction

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was later studied by many other authors.By a lacunary sequence, we mean an increasing integer sequenceθ=(kr) such that k0=0 and hr = kr-kr-1→∞ as r→∞.Throughout this paper, the intervals determined byθ will be denoted by Jr=(kr-1,kr], and the ratio kr/kr-1 will be defined by ϕr.An Orlicz function is a functionM:[0,∞)→[0,∞), which is continuous, nondecreasing, and convex with M(0)=0, M(x)>0, for x>0 and M(x)→∞, as x→∞.Letℓ∞, c, and c0 be the Banach space of bounded, convergent, and null sequences x=(xk), respectively, with the usual norm ∥x∥=supn|xn|.A sequencex∈ℓ∞ is said to be almost convergent if all of its Banach limits coincide. Let c^ denote the space of all almost convergent sequences.Lorentz [2] introduced the following sequence space
(1)c^={x∈ℓ∞:limmtm,n(x)existsuniformlyinn},
where tm,n(x) = (xn+xn+1+⋯+xm+n)/(m+1).The following space of strongly almost convergent sequence was introduced by Maddox [3]:
(2)[c^]={x∈ℓ∞:limmtm,n(|x-Le|)existsuniformlyinnforsomeLlimmtm,n(|x-Le|)},
where e=(1,1,…).Kızmaz [4] studied the difference sequence spaces ℓ∞(Δ), c(Δ), and c0(Δ) of crisp sets. The notion is defined as follows:
(3)Z(Δ)={x=(xk):(Δxk)∈Z},
for Z=ℓ∞,c, and c0, where Δx=(Δxk) = (xk-xk+1), for all k∈N.The above spaces are Banach spaces, normed by(4)∥x∥Δ=|x1|+supk|Δxk|.Tripathy et al. [5] introduced the generalized difference sequence spaces which are defined as, for m≥1 and n≥1,
(5)Z(Δmn)={x=(xk):(Δmnxk)∈Z},forZ=ℓ∞,c,c0.This generalized difference has the following binomial representation:(6)Δmnxk=∑r=0n‍(-1)r(nr)xk+rm.


## 2. Definitions and Preliminaries

Kostyrko et al. [1] introduced the following three definitions.LetX be a nonempty set. Then a family of sets I⊆2X (power sets of X) is said to be 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(xk) in a normed space (X,∥·∥) is said to be I-convergent to x0∈X if for each ε>0, the set
(7)E(ε)={k∈N:∥xk-x0∥≥ε}belongstoI.A sequence(xk) in a normed space (X,∥·∥) is said to be I-bounded if there exists M>0 such that the set {k∈N:∥xk∥>M} belongs to I.Freedman et al. [6] defined the space Nθ. For any lacunary sequence θ=(kr),
(8)Nθ={(xk):limr→∞hr-1∑k∈Jr‍|xk-L|=0,forsomeL}.The spaceNθ is a BK space with the norm
(9)∥(xk)∥θ=suprhr-1∑k∈Jr‍|xk|.The notion of lacunary ideal convergence of real sequences introduced by Tripathy et al. in [7, 8] and Hazarika [9, 10] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties. In [5, 7], the lacunary ideal convergence is defined as follows.Letθ=(kr) be a lacunary sequence. Then a sequence (xk) is said to be lacunary I-convergent if for every ε>0, such that
(10){r∈N:hr-1∑k∈Jr‍|xk-x|≥ε}∈I,we writeIθ-limxk=x.Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct the sequence space:
(11)ℓM={(xk)∈w:∑k=1∞‍M(|x|ρ)<∞,forsomeρ>0}.The spaceℓM with the norm
(12)∥x∥=inf{ρ>0:∑k=1∞‍M(|xk|ρ)≤1}
becomes a Banach space which is called an Orlicz sequence space.In this paper, we defined some new generalized difference lacunaryI-convergent sequence spaces defined by Orlicz function. We also introduce and examine some new sequence spaces and study their different properties.


## 3. Main Results

Esi [12] introduced the strongly almost ideal convergent sequence spaces in 2-normed spaces. In this paper we introduced the strongly almost lacunary ideal convergent sequence spaces using generalized difference operator and Orlicz function.LetI be an admissible ideal of N,  M an Orlicz function, and θ=(kr) a lacunary sequence. Further, let s=(sk) be a bounded sequence of positive real numbers and Δpq a generalized difference operator.For everyε>0 and for some ρ>0, we have introduced the following sequence spaces:
(13)w^I(M,Δpq,s,θ)={{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)-l|ρ)}sk≥ε}(xk):{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)-l|ρ)}sk≥ε}∈I,m∈N,forsomel∈R{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)-l|ρ)}sk≥ε}},w^0I(M,Δpq,s,θ)={{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk≥ε}(xk):{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk≥ε}∈I,m∈N{r∈N:1hr∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk≥ε}},w^∞I(M,Δpq,s,θ)={s.t.1hr∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk≥K}(xk):{r∈N:∃K>0s.t.1hr∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk≥K}∈I,m∈N∑k∈Jr‍{M(|Δpqtkm(x)|ρ)}sk}.Particular Cases. Consider the following.(1)
Ifθ=(2r), we have w^I(M,Δpq,s,θ)=w^I(M,Δpq,s), w^0I(M,Δpq,s,θ)=w^0I(M,Δpq,s), and w^∞I(M,Δpq,s,θ)=w^∞I(M,Δpq,s).(2)
IfM(x)=x, then w^I(M,Δpq,s,θ)=w^I(Δpq,s,θ), w^0I(M,Δpq,s,θ)=w^0I(Δpq,s,θ), and w^∞I(M,Δpq,s,θ)=w^∞I(Δpq,s,θ).(3)
Ifsk=1 for all k∈N, M(x)=x, and θ=(2r), then w^I(M,Δpq,s,θ)=w^I(Δpq), w^0I(M,Δpq,s,θ)=w^I(Δpq), and w^∞I(M,Δpq,s,θ)=w^I(Δpq).Theorem 1.
Let the sequence(sk) be bounded; then w^0I(M,Δpq,s,θ)⊂w^I(M,Δpq,s,θ)⊂w^∞I(M,Δpq,s,θ).Proof.
Letx∈w^I(M,Δpq,s,θ). Then, for some ρ>0, we have
(14)1hr∑k∈Jr‍(M(|Δpqtkm(x)|2ρ))sk≤Dhr∑k∈Jr‍12pk(M(|Δpqtkm(x)-l|ρ))sk+Dhr∑k∈Jr‍12pk(M(|l|ρ))sk≤Dhr∑k∈Jr‍(M(|Δpqtkm(x)-l|ρ))sk+Dmax{1,sup(M(|l|ρ))H},
where supksk=H and D=max(1,2H-1).
Hence,x∈w^∞I(M,Δpq,s,θ).
The inclusionw^0I(M,Δpq,s,θ)⊂w^I(M,Δpq,s,θ) is obvious.Theorem 2.
Let the sequence(sk) be bounded; then w^I(M,Δpq,s,θ), w^0I(M,Δpq,s,θ), and w^∞I(M,Δpq,s,θ) are closed under the operations of addition and scalar multiplication.Theorem 3.
LetM1,M2 be Orlicz functions; then we have(1)
w^0I(M1,Δpq,s,θ)∩w^0I(M2,Δpq,s,θ)⊂w^0I(M1+M2,Δpq,s,θ),(2)
w^I(M1,Δpq,s,θ)∩w^I(M2,Δpq,s,θ)⊂w^I(M1+M2,Δpq,s,θ),(3)
w^∞I(M1,Δpq,s,θ)∩w^∞I(M2,Δpq,s,θ)⊂w^∞I(M1+M2,Δpq,s,θ).Theorem 4.
Let0<sk≤uk for all k∈N, and let (uk/sk) be bounded; then we have w^I(M1,Δpq,u,θ)⊆w^I(M1,Δpq,s,θ).Theorem 5.
Letθ=(kr) be a lacunary sequence with 1<liminfrur≤suprur<∞. Then, for any Orlicz function M, w^I(M,Δpq,s)=w^I(M,Δpq,s,θ).Proof.
Supposeliminfrur>1 then there exists δ>0 such that ur=kr/kr-1≥1+δ for all r≥1.
Then, forx∈w^I(M,Δpq,s), we have
(15){m∈N:∑k∈Jr‍(M(|Δpqtkm(x)-l|ρ))sk≥ε}∈I.
Let(16)Ar=1hr∑k∈Jr‍(M(|Δpqtkm(x)-l|ρ))sk=1hr∑k=1kr‍(M(|Δpqtkm(x)-l|ρ))sk-1hr∑k=1kr-1‍(M(|Δpqtkm(x)-l|ρ))sk=krhr(1kr∑k=1kr‍(M(|Δpqtkm(x)-l|ρ))sk)-kr-1hr(1kr-1∑k=1kr-1‍(M(|Δpqtkm(x)-l|ρ))sk).
Sincehr=kr-kr-1, we have kr/hr≤(1+δ)/δ and kr-1/hr≤1/δ.
So, forε>0 and for some ρ>0,
(17){r∈N:krhr(1kr∑k=1kr‍(M(|Δpqtkm(x)-l|ρ))sk)≥ε}∈I,m∈N,forsomel∈R,{(1kr-1∑k=1kr-1‍(M(|Δpqtkm(x)-l|ρ))sk)r∈N:kr-1hr(1kr-1∑k=1kr-1‍(M(|Δpqtkm(x)-l|ρ))sk)≥ε}∈I,m∈N,forsomel∈R.
Hence,w^I(M,Δpq,s)⊂w^I(M,Δpq,s,θ).
Next, suppose thatlimsuprqr<∞. Then, there exists β>0, such that, qr<β for all r≥1.
Letx∈w^I(M,Δpq,s,θ) and ε>0. There exists R>0 such that for every j≥R,
(18)Aj={r∈N:1hr∑k∈Jj‍(M(|Δpqtkm(x)-l|ρ))pk≥ε}∈I.
LetK>0 such that Aj≤K for all j=1,2,…. Now let n be any integer with kr-1<n≤kr, where r>R. Then,
(19)1n∑k=1n‍(M(|Δpqtkm(x)-l|ρ))sk≤1kr-1∑k=1kr‍(M(|Δpqtkm(x)-l|ρ))pk=1kr-1{∑k∈J1‍(M(|Δpqtkm(x)-l|ρ))pk+∑k∈J2‍(M(|Δpqtkm(x)-l|ρ))pk+⋯+∑k∈Jr‍(M(|Δpqtkm(x)-l|ρ))pk}=k1kr-1A1+k2-k1kr-1A2+⋯+kR-kR-1kr-1AR+⋯+kr-kr-1kr-1Ar=(supj≥1Aj)kRkr-1+supj≥R(Aj)kr-kRkr-1<KkRkr-1+εβ.
Sincekr-1→∞ as r→∞, it follows that
(20){m∈N:1n∑k=1n‍(M(|Δpqtkm(x)-l|ρ))sk≥ε}∈I.
Hence,w^I(M,Δpq,s,θ)⊂w^I(M,Δpq,s).Theorem 6.
Iflimsk>0 and x is strongly almost lacunary convergent to x0, with respect to the Orlicz function M, that is, (xk)→l(w^I(M,Δpq,s,θ)), then x0 is unique.Proof.
Letlimsk=s>0 and suppose that xk→x1(w^I(M,Δpq,s,θ)), xk→x0(w^I(M,Δpq,s,θ)).
Then there existρ1 and ρ2 such that
(21){r∈N:1hr∑k∈Jr‍(M(|Δpqtkm(x)-x0|ρ1))sk≥ε}∈I,m∈N,{r∈N:1hr∑k∈Jr‍(M(|Δpqtkm(x)-x1|ρ1))sk≥ε}∈I,m∈N.
Letρ=max(2ρ1,2ρ2). Then we have
(22)1hr∑k∈Jr‍(M(|x0-x1|ρ))sk≤Dhr∑k∈Jr‍(M(|Δpqtkm(x)-x0|ρ1))sk+Dhr∑k∈Jr‍(M(|Δpqtkm(x)-x1|ρ2))sk,
where supksk=H and D=max(1,2H-1).
Thus, from (21), we get
(23){r∈N:1hr∑k∈Jr‍(M(|x0-x1|ρ1))sk≥ε}∈I.
Further,M(|x0-x1|/ρ)sk→M(|x0-x1|/ρ)s as k→∞, and, therefore,
(24)M(|x0-x1|ρ)sk⟶M(|x0-x1|ρ)s=0.
Hence,x0=x1.


## 4. Conclusion

The concept of lacunaryI-convergence has been studied by various mathematicians. In this paper, we have introduced some fairly wide classes of strongly almost lacunary I-convergent sequences of real numbers using Orlicz function with the generalized difference operator. Giving particular values to the sequence θ=(kr) and M, we obtain some new sequence spaces which are the special cases of the sequence spaces we have defined. There are lots more to be investigated in the future.



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*Source: 642535-2013-11-06.xml*