research article generalized lacunary statistical ... · also some geometric properties of sequence...

Research ArticleGeneralized Lacunary Statistical Difference SequenceSpaces of Fractional Order

Ugur Kadak

Department of Mathematics, Bozok University, Yozgat, Turkey

Correspondence should be addressed to Ugur Kadak; [email protected]

Received 28 April 2015; Accepted 16 June 2015

Academic Editor: Binod C. Tripathy

Copyright © 2015 Ugur Kadak. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We generalize the lacunary statistical convergence by introducing the generalized difference operator Δ𝛼] of fractional order, where𝛼 is a proper fraction and ] = (]

𝑘) is any fixed sequence of nonzero real or complex numbers. We study some properties of this

operator and investigate the topological structures of related sequence spaces. Furthermore, we introduce some properties of thestronglyCesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relationsof these spaces. We also determine the relationship between lacunary statistical and strong Cesaro difference sequence spaces offractional order.

1. Introduction

By𝜔, we denote the space of all real valued sequences and anysubspace of 𝑤 is called a sequence space. Let ℓ

∞, 𝑐, and 𝑐0 be

the linear spaces of bounded, convergent, and null sequences𝑥 = (𝑥

𝑘) with real or complex terms, respectively, normed

by ‖𝑥‖∞= sup

𝑘|𝑥𝑘|, where 𝑘 ∈ N, the set of positive

integers.With this norm, it is proved that these are all Banachspaces. Also by bs, cs, ℓ1, and ℓ𝑝, we denote the spaces of allbounded, convergent, absolutely summable, and𝑝-absolutelysummable series, respectively.

The concept of difference sequence space was determinedby Kızmaz [1]. Et and Colak generalized difference sequencespaces [2]. Later on Et and Esi [3] generalized these sequencespaces to the following sequence spaces. Let ] = (]

𝑘) be any

fixed sequence of nonzero complex numbers and let 𝑚 be anonnegative integer. Then, Δ𝑚] (𝑋) = {𝑥 = (𝑥𝑘) : (Δ

𝑚

] 𝑥) ∈ 𝑋}

for 𝑋 = ℓ∞, 𝑐 or 𝑐0, where 𝑚 ∈ N, Δ

0]𝑥 = (]𝑘𝑥𝑘), Δ

𝑚

] 𝑥 =

(Δ𝑚−1] 𝑥𝑘 − Δ

𝑚−1] 𝑥𝑘+1), and so Δ

𝑚

] 𝑥𝑘 = ∑𝑚

𝑖=0(−1)𝑖

(𝑚

𝑖) ]𝑘+𝑖𝑥𝑘+𝑖

.These are Banach spaces with the norm defined by ‖𝑥‖

Δ=

∑𝑚

𝑖=1 |]𝑖𝑥𝑖|+ sup𝑘|Δ𝑚

] 𝑥𝑘|. Furthermore Et and Basarir [4] havegeneralized difference sequence spaces. Aydın and Başar [5]have introduced some new difference sequence spaces. Alsothe notion of difference sequence has been extended byMur-saleen [6], Mursaleen and Noman [7], Malkowsky et al. [8],

and Bektaş et al. [9]. Also Tripathy et al. [10, 11] havegeneralized difference sequences by Orlicz functions.

Let 𝜃 = (𝑘𝑟) be the sequence of positive integers such

that 𝑘0 = 0, 0 < 𝑘𝑟 < 𝑘𝑟+1, and ℎ𝑟 = (𝑘𝑟 − 𝑘𝑟−1) → ∞ as𝑟 → ∞. Then 𝜃 is called a lacunary sequence. The intervalsdetermined by 𝜃 will be denoted by 𝐼

𝑟= (𝑘𝑟−1, 𝑘𝑟]. Das and

Mishra [12] introduced lacunary strong almost convergence.Colak et al. [13] have studied lacunary strongly summablesequences. Also some geometric properties of sequencespaces involving lacunary sequence have been examined byKarakaya [14]. Et has generalized Cesaro difference sequencespaces involving lacunary sequences [15].

Besides the Cesaro sequence spaces Ces𝑝and Ces

∞have

been introduced by Shiue [16]. Jagers [17] has determined theKöthe duals of the sequence space Ces

𝑝(1 < 𝑝 < ∞). Later

on the Cesaro sequence spaces 𝑋𝑝and 𝑋

∞of nonabsolute

type are defined by Ng and Lee [18, 19].Themain focus of the present paper is to generalize strong

Cesaro and lacunary statistical difference sequence spacesand investigate their topological structures as well as someinteresting results concerning the operator Δ𝛼].

The rest of this paper is organized, as follows: in Section 2,some required definitions and consequences related to thedifference operator Δ𝛼 are given. Also some new classes ofdifference sequences of fractional order involving lacunary

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2015, Article ID 984283, 6 pageshttp://dx.doi.org/10.1155/2015/984283

2 International Journal of Mathematics and Mathematical Sciences

sequences are determined and some topological propertiesare investigated. Section 3 is devoted to the strong Cesarodifference sequence spaces of fractional order. Prior to statingand proving the main results concerning these spaces, wegive some theorems about the notion of linearity and 𝐵𝐾-space. In final section, we present some theorems related tothe lacunary statistical convergence of difference sequencesof fractional order and examine some inclusion relations ofthese spaces.

2. Some New Difference Sequence Spaces withFractional Order

By Γ(𝛼), we denote theEuler gamma function of a real number𝛼. Using the definition, Γ(𝛼) with 𝛼 ∉ {0, −1, −2, −3, . . .} canbe expressed as an improper integral as follows:

Γ (𝛼) = ∫

∞

0𝑒−𝑡

𝑡𝛼−1𝑑𝑡. (1)

For a positive proper fraction 𝛼, Baliarsingh and Dutta [20,21] (also see [22]) have defined the generalized fractionaldifference operator Δ𝛼 as

Δ𝛼

(𝑥𝑘) =

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)𝑥𝑘+𝑖. (2)

In particular, we have

(i) Δ1/2(𝑥𝑘) = 𝑥𝑘− (1/2)𝑥

𝑘+1 − (1/8)𝑥𝑘+2 − (1/16)𝑥𝑘+3 −(5/128)𝑥

𝑘+4 − (7/256)𝑥𝑘+5 − (21/1024)𝑥𝑘+6 − ⋅ ⋅ ⋅ ,(ii) Δ−1/2(𝑥

𝑘) = 𝑥𝑘+ (1/2)𝑥

𝑘+1 + (3/8)𝑥𝑘+2 + (5/16)𝑥𝑘+3 +(35/128)𝑥

𝑘+4 + (63/256)𝑥𝑘+5 + (231/1024)𝑥𝑘+6 + ⋅ ⋅ ⋅ ,(iii) Δ2/3(𝑥

𝑘) = 𝑥𝑘− (2/3)𝑥

𝑘+1 − (1/9)𝑥𝑘+2 − (4/81)𝑥𝑘+3 −(7/243)𝑥

𝑘+4 − (14/729)𝑥𝑘+5 − (91/6561)𝑥𝑘+6 − ⋅ ⋅ ⋅ .

Theorem 1 (see [20]). (a) For proper fraction 𝛼, Δ𝛼 : 𝜔 → 𝜔defined by (2) is a linear operator.

(b) For 𝛼, 𝛽 > 0, Δ𝛼(Δ𝛽(𝑥𝑘)) = Δ

𝛼+𝛽

(𝑥𝑘) and

Δ𝛼

(Δ−𝛼

(𝑥𝑘)) = 𝑥

𝑘.

Now, we determine the new classes of difference sequencespaces Δ𝛼](𝑋) as follows:

Δ𝛼

] (𝑋) := {𝑥 = (𝑥𝑘) ∈ 𝜔 : (Δ𝛼

]𝑥) ∈𝑋} , (3)

where Δ𝛼](𝑥𝑘) = ∑∞

𝑖=0(−1)𝑖

(Γ(𝛼+1)/𝑖!Γ(𝛼− 𝑖+1))]𝑘+𝑖𝑥𝑘+𝑖

and𝑋 is any sequence spaces.

Theorem 2. For a proper fraction 𝛼, if𝑋 is a linear space, thenΔ𝛼

](𝑋) is also a linear space.

Proof. The proof is straightforward (see [20]).

Theorem 3. If𝑋 is a Banach space with the norm ‖ ⋅ ‖∞, then

Δ𝛼

](𝑋) is also a Banach space with the norm ‖ ⋅ ‖Δ𝛼] defined by

‖𝑥‖Δ𝛼

](𝑋)=

∞

∑

𝑖=0

]𝑖𝑥𝑖 +Δ𝛼

]𝑥∞, (4)

where ‖Δ𝛼]𝑥‖∞ = sup𝑘|Δ𝛼

](𝑥𝑘)|.

Proof. Proof of this theorem is a routine verification, henceomitted.

Remark 4. Without loss of generality, we assume throughoutthat each series given in (4) is convergent. Furthermore, if 𝛼is a positive integer, then these infinite sums in (4) reduce tofinite sums; that is, ∑𝛼

𝑖=0(−1)𝑖

(Γ(𝛼 + 1)/𝑖!Γ(𝛼 − 𝑖 + 1))]𝑘+𝑖𝑥𝑘+𝑖

and ∑𝛼𝑖=0 |]𝑖𝑥𝑖|.

Lemma 5. Let 𝛼 be a proper fraction. If𝑋 ⊂ 𝑌, then Δ𝛼](𝑋) ⊂Δ𝛼

](𝑌).

Proof. Let 𝑋 ⊂ 𝑌 and 𝑥 = (𝑥𝑘) ∈ Δ

𝛼

](𝑋). It is trivial that(Δ𝛼

]𝑥) ∈ 𝑋 implies (Δ𝛼

]𝑥) ∈ 𝑌. Hence𝑥 ∈ Δ𝛼

](𝑌) andΔ𝛼

](𝑋) ⊂

Δ𝛼

](𝑌).

Theorem 6. Let𝑋 be a Banach space and𝐾 a closed subset of𝑋. Then Δ𝛼](𝐾) is also closed subset of Δ

𝛼

](𝑋).

Proof. By using Lemma 5, it is trivial that Δ𝛼](𝐾) ⊂ Δ𝛼

](𝑋).Now we prove that Δ𝛼](𝐾) = Δ

𝛼

](𝐾). Let 𝑥 ∈ Δ𝛼](𝐾); thenthere exists a sequence (𝑥𝑛) ∈ Δ𝛼](𝐾) such that

𝑥𝑛

−𝑥Δ𝛼](𝐾)

→ 0 as 𝑛 → ∞,

i.e.∞

∑

𝑖=0

]𝑖 (𝑥𝑛

𝑖−𝑥𝑖)

+ sup𝑘

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖(𝑥𝑛

𝑘+𝑖−𝑥𝑘+𝑖)

→ 0

i.e. lim𝑛

sup𝑘

]𝑘(𝑥𝑛

𝑘−𝑥𝑘) − 𝛼]

𝑘+1 (𝑥𝑛

𝑘+1 −𝑥𝑘+1)

+𝛼 (𝛼 − 1)

2]𝑘+2 (𝑥

𝑛

𝑘+2 −𝑥𝑘+2) + ⋅ ⋅ ⋅

= 0.

(5)

Thus, we observe that

𝑥𝑛

𝑗−𝑥𝑗

Δ𝛼](𝐾)=

∞

∑

𝑖=0

]𝑖 (𝑥𝑛

𝑖−𝑥𝑖)

+Δ𝛼

] (𝑥𝑛

𝑗) −Δ𝛼

] (𝑥𝑗)∞→ 0,

(6)

as 𝑛 → ∞ in 𝐾. This implies that 𝑥 ∈ Δ𝛼](𝐾).Conversely, let 𝑥 ∈ Δ𝛼](𝐾); then 𝑥 ∈ Δ

𝛼

](𝐾). Since 𝐾 isclosed Δ𝛼](𝐾) = Δ

𝛼

](𝐾), hence Δ𝛼

](𝐾) is a closed subset ofΔ𝛼

](𝑋).

Theorem 7. If 𝑋 is a 𝐵𝐾-space with the norm ‖ ⋅ ‖∞, then

Δ𝛼

](𝑋) is also a 𝐵𝐾-space with the norm given in (4).

International Journal of Mathematics and Mathematical Sciences 3

Proof. It is clear that Δ𝛼](𝑋) is a Banach space (see Theo-rem 3). Suppose that ‖𝑥𝑛

𝑗− 𝑥𝑗‖Δ𝛼

](𝑋)→ 0 for each 𝑗 ∈ N.

One can conclude that |]𝑗(𝑥𝑛

𝑗−𝑥𝑗)| → 0 as 𝑛 → ∞ for each

𝑗 ∈ N and implies ‖Δ𝛼](𝑥𝑛

𝑗)−Δ𝛼

](𝑥𝑗)‖∞ → 0 as 𝑛 → ∞.Thisfollows from the fact that

sup𝑘

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖(𝑥𝑛

𝑘+𝑖−𝑥𝑘+𝑖)

→ 0,

(]𝑘+𝑖̸= 0) ,

(7)

where 𝑃 = ∑∞𝑖=0(−1)

𝑖

(Γ(𝛼 + 1)/𝑖!Γ(𝛼 − 𝑖 + 1)); since 𝛼 > 0 theseries represented by 𝑃 is finite. Hence |𝑥𝑛

𝑗−𝑥𝑗| → 0 as 𝑛 →

∞ for each 𝑗 ∈ N. Therefore Δ𝛼](𝑋) is a Banach space withthe continuous coordinates. This completes the proof.

Definition 8. A sequence 𝑥 = (𝑥𝑘) is said to be Δ𝛼]-strongly

Cesaro convergent if there is a real or complex number 𝐿 suchthat

lim𝑛→∞

1𝑛

𝑛

∑

𝑘=0

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖−𝐿

𝑝

= 0

for some𝐿,

(8)

where 𝑝 is a fixed positive number and 𝛼 is a properfraction. The number 𝐿 is unique when it exists. By Δ𝛼](𝜔𝑝),one denotes the set of all strongly Δ𝛼]-Cesaro convergentsequences. In this case, one writes 𝑥

𝑘→ 𝐿(Δ

𝛼

](𝜔𝑝)).

Theorem 9. The sequence space Δ𝛼](𝜔𝑝) is a Banach space for1 ≤ 𝑝 < ∞ normed by

‖𝑥‖Δ ]1=

∞

∑

𝑖=0

]𝑖𝑥𝑖

+ sup𝑛∈N1

(1𝑛

𝑛

∑

𝑘=0

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

𝑝

)

1/𝑝 (9)

and a complete 𝑝-normed space with the 𝑝-norm

‖𝑥‖Δ ]2

=

∞

∑

𝑖=0

]𝑖𝑥𝑖

𝑝

+ sup𝑛∈N1

1𝑛

𝑛

∑

𝑘=0

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

𝑝

(10)

for 0 < 𝑝 < 1.

Proof. Proof follows by usingTheorem 3.

3. Cesaro Difference Sequence Spaces ofFractional Order

In this section by using the operator Δ𝛼], we introducesome new sequence spaces𝐶(Δ𝛼], 𝑝)𝜃,𝐶[Δ

𝛼

], 𝑝]𝜃,𝐶∞(Δ𝛼

], 𝑝)𝜃,

𝐶∞[Δ𝛼

], 𝑝]𝜃, and 𝑁(Δ𝛼

], 𝑝)𝜃 involving lacunary sequences 𝜃and arbitrary sequence 𝑝 = (𝑝

𝑟) of strictly positive real

numbers.We define the sequence spaces as follows:

𝐶 (Δ𝛼

], 𝑝)𝜃 :={

{

{

𝑥= (𝑥𝑘) ∈ 𝜔 :

∞

∑

𝑟=1

1ℎ𝑟

∑

𝑘∈𝐼𝑟

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

𝑝𝑟


Theorem 11. If 𝑝 = 𝑝𝑟for all 𝑟 ∈ N, then the sequence space

𝐶[Δ𝛼

], 𝑝]𝜃 is a 𝐵𝐾-space with the norm defined by

‖𝑥‖1 =∞

∑

𝑖=0

]𝑖𝑥𝑖 +{

{

{

∞

∑

𝑟=1(

1ℎ𝑟

⋅ ∑

𝑘∈𝐼𝑟

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

)

𝑝

}

}

}

1/𝑝

,

(1 ≤ 𝑝) .

(12)

Also if 𝑝𝑟= 1 for all 𝑟 ∈ N, then the sequence spaces 𝐶

∞[Δ𝛼

],

𝑝]𝜃and𝑁(Δ𝛼], 𝑝)2𝑟 are 𝐵𝐾-spaces with the norm defined by

‖𝑥‖2 =∞

∑

𝑖=0

]𝑖𝑥𝑖

+ sup𝑟

1ℎ𝑟

∑

𝑘∈𝐼𝑟

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

.

(13)

Proof. We give the proof for the space𝐶∞[Δ𝛼

], 𝑝]𝜃 and that ofothers followed by using similar techniques.

Suppose (𝑥𝑛) is a Cauchy sequence in 𝐶∞[Δ𝛼

], 𝑝]𝜃, where𝑥𝑛

= (𝑥𝑛

𝑗) and 𝑥𝑚 = (𝑥𝑚

𝑗) are two elements in 𝐶

∞[Δ𝛼

], 𝑝]𝜃.Then there exists a positive integer 𝑛0(𝜖) such that ‖𝑥

𝑛

−

𝑥𝑚

‖2 → 0 as𝑚, 𝑛 → ∞ for all𝑚, 𝑛 ≥ 𝑛0(𝜖) and for each 𝑗 ∈N.Therefore (𝑥1

𝑖, 𝑥

2𝑖, . . .) and (Δ𝛼](𝑥

1𝑗), Δ𝛼

](𝑥2𝑗), . . .) are Cauchy

sequences in complex field C and 𝐶∞[Δ𝛼

], 𝑝]𝜃, respectively.By using the completeness ofC and𝐶

∞[Δ𝛼

], 𝑝]𝜃, we have thatthey are convergent and suppose that 𝑥𝑛

𝑖→ 𝑥

𝑖in C and

(Δ𝛼

](𝑥𝑛

𝑗)) → 𝑦

𝑗in 𝐶∞[Δ𝛼

], 𝑝]𝜃 for each 𝑗 ∈ N as 𝑛 → ∞.Then we can find a sequence (𝑥

𝑗) such that 𝑦

𝑗= Δ𝛼

](𝑥𝑗) foreach 𝑗 ∈ N. These 𝑥

𝑗’s can be interpreted as

𝑥𝑗=

1]𝑗

𝑗−𝑘

∑

𝑖=1(−1)𝑟

Γ (𝑗 − 𝑖)

(𝑘 − 1)!Γ (𝑗 − 𝑖 − 𝑘 + 1)𝑦𝑖

=1]𝑗

𝑗

∑

𝑖=1

(−1)𝑟

(𝑘 − 1)!Γ (𝑗 + 𝑘 − 𝑖)

Γ (𝑗 − 𝑖 + 1)𝑦𝑖−𝑘,

(𝑦1−𝑘 = 𝑦2−𝑘 = ⋅ ⋅ ⋅ = 𝑦0 = 0)

(14)

for sufficiently large 𝑗; that is, 𝑗 > 2𝑘. Then (Δ𝛼](𝑥𝑛

)) =

(Δ𝛼

](𝑥1𝑗), Δ𝛼

](𝑥2𝑗), . . .) converges to (Δ𝛼](𝑥𝑗)) for each 𝑗 ∈ N as

𝑛 → ∞. Thus ‖𝑥𝑛 −𝑥‖2 → 0 as𝑚 → ∞. Since𝐶∞[Δ𝛼

], 𝑝]𝜃is a Banach space with continuous coordinates, that is, ‖𝑥𝑛 −𝑥‖2 → 0 implies |𝑥

𝑛

𝑗− 𝑥𝑗| → 0 for each 𝑗 ∈ N, as 𝑛 → ∞,

this shows that 𝐶∞[Δ𝛼

], 𝑝]𝜃 is a 𝐵𝐾-space.

Theorem 12. If 𝑝 = 𝑝𝑟for all 𝑟 ∈ N, then the sequence space

𝐶(Δ𝛼

], 𝑝)𝜃 is a 𝐵𝐾-space with the norm defined by

‖𝑥‖3 =∞

∑

𝑖=0

]𝑖𝑥𝑖 +{

{

{

∞

∑

𝑟=1

1ℎ𝑟

⋅ ∑

𝑘∈𝐼𝑟

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

𝑝

}

}

}

1/𝑝

,

(1 ≤ 𝑝) .

(15)

Also if𝑝𝑟= 1 for all 𝑟 ∈ N, then the sequence space𝐶

∞(Δ𝛼

], 𝑝)𝜃is a 𝐵𝐾-space with the norm defined by

‖𝑥‖4 =∞

∑

𝑖=0

]𝑖𝑥𝑖

+ sup𝑟

1ℎ𝑟

∑

𝑘∈𝐼𝑟

∞

∑

𝑖=0(−1)𝑖 Γ (𝛼 + 1)

𝑖!Γ (𝛼 − 𝑖 + 1)]𝑘+𝑖𝑥𝑘+𝑖

.

(16)

Proof. The proof follows fromTheorem 11.

Now, we can present the following theorem, determiningsome inclusion relations without proof, since it is a routineverification.

Theorem 13. Let two positive proper fractions 𝛼 > 𝛽 > 0 and𝑝𝑟= 𝑝 for each 𝑟 ∈ N be given. Then the following inclusions

are satisfied:

(i) 𝐶(Δ𝛽], 𝑝)𝜃 ⊂ 𝐶(Δ𝛼

], 𝑝)𝜃,

(ii) 𝐶[Δ𝛽], 𝑝]𝜃 ⊂ 𝐶[Δ𝛼

], 𝑝]𝜃,(iii) 𝐶(Δ𝛼], 𝑝)𝜃 ⊂ 𝐶(Δ

𝛼

], 𝑞)𝜃, (0 < 𝑝 < 𝑞).

4. Lacunary Statistical Convergence ofDifference Sequences

The concept of statistical convergence from different aspectshas been studied by various mathematicians. The notion ofstatistical convergence was independently introduced by Fast[23] and Schoenberg [24].

Let𝐾 be a subset of the set of natural numbersN.Then theasymptotic density of𝐾 denoted by 𝛿(𝐾) is defined as 𝛿(𝐾) =lim𝑛(1/𝑛)|{𝑘 ≤ 𝑛 : 𝑘 ∈ 𝐾}|, where the vertical bars denote the

cardinality of the enclosed set.A number sequence 𝑥 = (𝑥

𝑘) said to be statistically

convergent to the number 𝐿 if, for each 𝜀 > 0, the set 𝐾(𝜀) ={𝑘 ≤ 𝑛 : |𝑥

𝑘− 𝐿| > 𝜀} has asymptotic density zero; that is,

lim𝑛(1/𝑛)|{𝑘 ≤ 𝑛 : |𝑥

𝑘− 𝐿| ≥ 𝜀}| = 0.

Definition 14. Let 𝜃 = (𝑘𝑟) be a lacunary sequence and 𝛼 a

proper fraction. Then the sequence 𝑥 = (𝑥𝑘) is said to be Δ𝛼]-

lacunary statistically convergent to a real or complex number𝐿 if, for each 𝜀 > 0,

lim𝑟→∞

1ℎ𝑟

{𝑘 ∈ (𝑘𝑟−1, 𝑘𝑟] :Δ𝛼

] (𝑥𝑘) − 𝐿 ≥ 𝜀} = 0, (17)

International Journal of Mathematics and Mathematical Sciences 5

where the vertical bars denote the cardinality of the enclosedset. The set of Δ𝛼]-lacunary statistically convergent sequenceswill be denoted by Δ𝛼](𝑆𝜃). In this case we write 𝑥𝑘 →𝐿(Δ𝛼

](𝑆𝜃)).

In particular, the Δ𝛼]-lacunary statistical convergenceincludes many special cases; that is, in the case 𝛼 = 𝑚 ∈N, Δ𝛼]-lacunary statistical convergence reduces to the Δ

𝑚

] -lacunary statistical convergence defined by [15].

Theorem 15. Let 0 < 𝑝 < ∞. If 𝑥𝑘→ 𝐿(𝑁(Δ

𝛼

], 𝑝)𝜃) for𝑝𝑟= 𝑝, then 𝑥

𝑘→ 𝐿(Δ

𝛼

](𝑆𝜃)). If 𝑥 ∈ Δ𝛼

](ℓ∞) and 𝑥𝑘 →𝐿(Δ𝛼

](𝑆𝜃)), then 𝑥𝑘 → 𝐿(𝑁(Δ𝛼

], 𝑝)𝜃).

Proof. Let 𝑥 = (𝑥𝑘) ∈ 𝐿(𝑁(Δ

𝛼

], 𝑝)𝜃) and 𝜖 > 0 and let ∑∗denote the sum over 𝑘 ∈ (𝑘

𝑟−1, 𝑘𝑟] such that |Δ𝛼

](𝑥𝑘) − 𝐿| ≥ 𝜀.We have that

1ℎ𝑟

∑

∗

Δ𝛼

]𝑥𝑘 −𝐿

𝑝

≥1ℎ𝑟

{𝑘 ∈ (𝑘𝑟−1, 𝑘𝑟] :Δ𝛼

(𝑥𝑘) − 𝐿 ≥ 𝜀} 𝜖𝑝

.

(18)

So we observe, by passing to limit as 𝑟 → ∞, in (18)

lim𝑟→∞

1ℎ𝑟

{𝑘 ∈ (𝑘𝑟−1, 𝑘𝑟] :Δ𝛼

] (𝑥𝑘) − 𝐿 ≥ 𝜀}

≤1𝜖𝑝( lim𝑟→∞

1ℎ𝑟

∑

∗

Δ𝛼

]𝑥𝑘 −𝐿

𝑝

) = 0(19)

which implies that 𝑥𝑘→ 𝐿(Δ

𝛼

](𝑆𝜃)).Suppose that 𝑥 ∈ Δ𝛼](ℓ∞) and 𝑥𝑘 → 𝐿(Δ

𝛼

](𝑆)). Then it isobvious that (Δ𝛼]𝑥) ∈ ℓ∞ and (1/ℎ𝑟)|{𝑘 ∈ (𝑘𝑟−1, 𝑘𝑟] : |Δ

𝛼

](𝑥𝑘)−

𝐿| ≥ 𝜀}| → 0 as 𝑟 → ∞. Let 𝜀 > 0 be given and there exists𝑛0 ∈ N such that

{𝑘 ∈ (𝑘

𝑟−1, 𝑘𝑟] :Δ𝛼

(𝑥𝑘) − 𝐿 ≥ (𝜀/2)

1/𝑝}

ℎ𝑟

≤𝜀

2 (Δ𝛼]𝑥∞+ 𝐿)𝑝,

(20)

where ∑∞𝑖=1 |]𝑖𝑥𝑖| = 𝐿, for all 𝑟 > 𝑛0. Furthermore, we can

writeΔ𝛼

] (𝑥𝑘) − 𝐿 ≤Δ𝛼

] (𝑥𝑘) − 𝐿Δ𝛼]≤ 𝐿+

Δ𝛼

]𝑥∞

= ‖𝑥‖Δ𝛼

](𝑋).

(21)

For 𝑟 > 𝑛0,

1ℎ𝑟

∑

𝑘∈𝐼𝑟

Δ𝛼

]𝑥𝑘 −𝐿

𝑝

=1ℎ𝑟

(∑

𝑘∈𝐿𝑟

Δ𝛼

𝑥𝑘−𝐿

𝑝

+ ∑

𝑘∉𝐿𝑟

Δ𝛼

𝑥𝑘−𝐿

𝑝

)

<1ℎ𝑟

(ℎ𝑟

𝜀

2+ ℎ𝑟

𝜀 ‖𝑥‖𝑝

Δ𝛼

](𝑋)

2 ‖𝑥‖𝑝Δ𝛼

](𝑋)

) = 𝜀,

(22)

where 𝐿𝑟= {𝑘 ∈ (𝑘

𝑟−1, 𝑘𝑟] : |Δ𝛼

(𝑥𝑘) − 𝐿| ≥ (𝜀/2)1/𝑝}. Hence

𝑥𝑘→ 𝐿(𝑁(Δ

𝛼

], 𝑝)𝜃). This completes the proof.

Corollary 16. The following statements hold:

(a) 𝑆 ∩ ℓ∞⊂ Δ𝛼

](𝑆𝜃) ∩ Δ𝛼

](ℓ∞),

(b) Δ𝛼](𝑆𝜃) ∩ Δ𝛼

](ℓ∞) = Δ𝛼

](𝜔𝑝).

Definition 17. Let 𝜃 be a lacunary sequence and 𝛼 a properfraction. Then a sequence 𝑥 = (𝑥

𝑘) is said to be Δ𝛼]-lacunary

statistically Cauchy if there exists a number 𝑁 = 𝑁(𝜀) suchthat

lim𝑟→∞

1ℎ𝑟

{𝑘 ∈ (𝑘𝑟−1, 𝑘𝑟] :Δ𝛼

] (𝑥𝑘 − 𝑥𝑁) ≥ 𝜀} = 0 (23)

for every 𝜀 > 0.

Theorem 18. If 𝑥 = (𝑥𝑘) is a Δ𝛼]-lacunary statistically con-

vergent sequence, then 𝑥 is a Δ𝛼]-lacunary statistically Cauchysequence.

Proof. Assume that 𝑥𝑘→ 𝐿(Δ

𝛼

](𝑆𝜃)) and 𝜀 > 0. Then|Δ𝛼

](𝑥𝑘) − 𝐿| < 𝜀/2 for almost all 𝑘, and if we select 𝑁, then|Δ𝛼

](𝑥𝑁) − 𝐿| < 𝜀/2 holds. Now, we have

Δ𝛼

] (𝑥𝑘) −Δ𝛼

] (𝑥𝑁) <Δ𝛼

] (𝑥𝑘) − 𝐿 +Δ𝛼

] (𝑥𝑁) − 𝐿

< 𝜀

(24)

for almost all 𝑘. Hence 𝑥 is a Δ𝛼]-lacunary statistically Cauchysequence.

Theorem 19. Let 𝛼 be a proper fraction and 0 < inf𝑝𝑟≤ 𝑝𝑟≤

sup𝑝𝑟< ∞. Then𝑁(Δ𝛼], 𝑝)𝜃 ⊂ Δ

𝛼

](𝑆𝜃).

Proof. Suppose that 𝑥 = (𝑥𝑘) ∈ 𝑁(Δ

𝛼

], 𝑝)𝜃 and∑∗ denote thesum over 𝑘 ∈ (𝑘

𝑟−1, 𝑘𝑟] such that |Δ𝛼

](𝑥𝑘) − 𝐿| ≥ 𝜀. Thereforewe have

1ℎ𝑟

∑

𝑘∈𝐼𝑟

Δ𝛼

] (𝑥𝑘) − 𝐿

𝑝𝑟

≥1ℎ𝑟

∑

∗

Δ𝛼

𝑥𝑘−𝐿

𝑝𝑟

≥1ℎ𝑟

∑

∗

𝜀inf𝑝𝑟

≥1ℎ𝑟

{𝑘 ∈ 𝐼𝑟 :Δ𝛼

(𝑥𝑘) − 𝐿 ≥ 𝜀} 𝜀

inf𝑝𝑟 .

(25)

Taking the limit as 𝑟 → ∞,

lim𝑟→∞

1ℎ𝑟

{𝑘 ∈ 𝐼𝑟 :Δ𝛼

] (𝑥𝑘) − 𝐿 ≥ 𝜀}

≤1𝜀inf𝑝𝑟

( lim𝑟→∞

1ℎ𝑟

∑

𝑘∈𝐼𝑟

Δ𝛼

]𝑥𝑘 −𝐿

𝑝𝑟

) = 0(26)

implies that 𝑥 ∈ Δ𝛼](𝑆𝜃). Hence𝑁(Δ𝛼

], 𝑝)𝜃 ⊂ Δ𝛼

](𝑆𝜃).


5. Concluding Remarks

In this paper, certain results on some lacunary statisticaldifference sequence spaces of order 𝑚 (𝑚 ∈ N) have beenextended to the difference sequence spaces of fractional order𝛼. The results presented in this paper not only generalize theearlier works done by several authors [2, 3, 15, 20, 21] butalso give a new perspective regarding the development ofdifference sequences. As a future work we will study certainmatrix transformations of these spaces.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The author expresses his sincere thanks to the referees fortheir valuable suggestions and comments, which improvedthe presentation of this paper.

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