ISSN: 2217-3412, URL: http://ilirias.com/jma Volume 7 Issue 5(2016), Pages 78-84.
ON (∆m, I) − LACUNARY STATISTICAL CONVERGENCE OF
ORDER α
MIKAIL ET AND HACER S¸ENG ¨UL
Abstract. In this study, using the generalized difference operator ∆m, we introduce the concepts of (∆m, I) −lacunary statistical convergence of order α
and lacunary strong ∆m
p−summability of order α of sequences and give some
relations about these concepts.
1. Introduction
In 1951, Fast [10] introduced the notion of statistical convergence and Schoenberg [18] reintroduced independently in 1959. Later on C¸ olak [2], Fridy [11], ˇSal´at [19], Tripathy [23] and another researchers have studied the concept from the sequence space point of view and linked with the Summability theory.
The notion of I−convergence is a generalization of the statistical convergence. Kostyrko, ˇSal´at and Wilczy´nski [15] introduced the notion of I−convergence. Some further results connected with the notion of I−convergence can be found in ([4],[5], [9],[16],[20],[21]).
By a lacunary sequence we mean an increasing integer sequence θ = (kr) such that hr = (kr− kr−1) → ∞ as r → ∞. Throught this paper the intervals deter-mined by θ will be denoted by Ir= (kr−1, kr] and the ratiokkr−1r will be abbreviated by qr. Recently lacunary sequences have been studied in ([1],[3],[8],[12],[13],[22]).
A non-empty family I ⊆ 2Nis said to be an ideal of N if φ ∈ I, A, B ∈ I implies A ∪ B ∈ I and A ∈ I, B ⊂ A implies B ∈ I.
A non-empty family F ⊆ 2Nis said to be a filter of N if φ /∈ F, A, B ∈ F implies A ∩ B ∈ F and A ∈ F, A ⊂ B implies B ∈ F.
If I is a proper ideal of N (i.e., N /∈ I) , then the family of sets F (I) = {M ⊂ N : ∃A ∈ I : M = N \ A} is a filter of N.
A proper ideal I is said to be admissible if {n} ∈ I for each n ∈ N.
Throughout this study, I will stand for a proper admissible ideal of N and by a sequence we always mean a sequence of real numbers.
2010 Mathematics Subject Classification. 40A05, 40C05, 46A45.
Key words and phrases. Difference sequence; Statistical convergence; Lacunary sequence; Ces`aro summability; I−convergence.
c
2016 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted June 21, 2016. Published October 5, 2016.
ON (∆ , I) − LACUNARY STATISTICAL CONVERGENCE OF ORDER α 79
The notion of difference sequence spaces was introduced by Kızmaz [14] and it was generalized by Et et al. ([6],[7],[9],[17],[21]) such as
∆m(X) = {x = (xk) : (∆mxk) ∈ X} , where X is any sequence space, m ∈ N, ∆0x = (x
k) , ∆mx = ∆m−1xk− ∆m−1xk+1 , and so ∆mxk=P m i=0(−1) i m ixk+i. If x ∈ ∆
m(X) then there exists one and only one y = (yk) ∈ X such that
xk = k−m X i=1 (−1)mk − i − 1 m − 1 yi= k X i=1 (−1)mk + m − i − 1 m − 1 yi−m,
y1−m= y2−m= ... = y0= 0, for sufficiently large k; for example, k > 2m. We use this truth to define in sequences (2.1) , (2.2) and (2.3) .
2. Main Results
In this section, we describe the concepts of (∆m, I) −lacunary statistical conver-gence of order α and lacunary strong ∆mp −summability of order α of sequences and give some relations about these concepts.
Definition 2.1. Let θ = (kr) be a lacunary sequence and α ∈ (0, 1] be a fixed real number. We say that the sequence x = (xk) is Sθα(∆m, I) −convergent (or (∆m, I) −lacunary statistically convergent sequences of order α) if there is a real number L such that
r ∈ N : h1α r |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≥ δ ∈ I, where Ir = (kr−1, kr] and hαr denote the αth power (hr)
α
of hr, that is hα = (hα
r) = (hα1, hα2, ..., hαr, ...) . In this case we write Sθα(∆
m, I) − lim x
k = L or xk → L(Sα
θ (∆
m, I)). We will denote the set of all Sα θ (∆
m, I) −convergent sequences by Sα
θ (∆
m, I). If θ = (2r) , then we will write Sα(∆m, I) in the place of Sα θ (∆
m, I) and if α = 1 and θ = (2r) , then we will write S (∆m, I) in the place of Sα
θ (∆m, I) . Definition 2.2. Let θ = (kr) be a lacunary sequence and α ∈ (0, 1] be a fixed real number. We say that the sequence x = (xk) is Nθα(∆
m, I) −summable to L (or lacunary strongly ∆m−summable sequence of order α) if, for any ε > 0,
( r ∈ N : h1α r X k∈Ir |∆mx k− L| ≥ ε ) ∈ I.
In this case we write xk → L (Nθα(∆m, I)) and we will denote the set of all Nα
θ (∆m, I) −summable sequences by Nθα(∆m, I). It can be shown that Sα
θ (∆
m, I) − convergence is well defined for 0 < α ≤ 1, but it is not well defined for α > 1 in general.
The inclusion parts of the following three theorems are straightforward, so we omit these parts of their proofs.
Theorem 2.1. If xk → L(Sθα(∆
m, I)), then x
k→ L(Sθβ(∆
m, I)) and the inclusion is proper.
Proof. Define a sequence x = (xk) by ∆mxk = k, if k is square 0, otherwise . (2.1) Then x ∈ Sθβ(∆m, I) for 1 2 < β ≤ 1 but x /∈ S α θ(∆ m, I) for 0 < α ≤ 1 2. Theorem 2.2. If xk → L(Nθα(∆ m, I)), then x k → L(N β θ (∆
m, I)) and the inclu-sion is proper.
Proof. Define a sequence x = (xk) by
∆mxk =
1, if k is square
0, otherwise . (2.2)
Then x ∈ Nθβ(∆m, I) for 12 < β ≤ 1 but x /∈ Nα θ (∆ m, I) for 0 < α ≤ 1 2. Theorem 2.3. If xk→ L(Nθα(∆ m, I)), then x k→ L(Sθα(∆
m, I)) and the inclusion is proper.
Proof. Define a sequence x = (xk) by ∆mxk =
√hr , k = 1, 2, 3, ...,√hr
0, otherwise . (2.3)
Then we have for every ε > 0 and 12 < α ≤ 1, 1 hα r |{k ∈ Ir: |∆mxk− 0| ≥ ε}| ≤ √hr hα r , and for any δ > 0 we get
r ∈ N :h1α r |{k ∈ Ir: |∆mxk− 0| ≥ ε}| ≥ δ ⊆ ( r ∈ N :√hr hα r ≥ δ )
and so xk→ 0 (Sθα(∆m, I)) for 1
2 < α ≤ 1. On the other hand, for 0 < α ≤ 1, 1 hα r X k∈Ir |∆mx k− 0| = √hr √hr hα r → ∞ and for α = 1, √hr √hr hα r → 1. Then we can write
( r ∈ N : h1α r X k∈Ir |∆mxk− 0| ≥ 1 ) = ( r ∈ N :√hr √hr hα r ≥ 1 ) = {a, a + 1, a + 2, ...} ∈ F (I) for some a ∈ N, since I is admissible. Thus xk 9 0 (Nθα(∆
m, I)) .
The proof of each of the following results is obvious, so we do not give the proof of theorems.
Theorem 2.4. If lim infrqr> 1, then xk → L(Sα(∆m, I)) implies xk→ L(Sθα(∆ m, I)).
ON (∆ , I) − LACUNARY STATISTICAL CONVERGENCE OF ORDER α 81
Theorem 2.5. If limr→∞inf hα
r kr
> 0, then xk → L(S (∆m, I)) implies xk → L(Sα
θ (∆ m, I)).
Theorem 2.6. If lim suprqr< ∞, then xk → L(Sθ(∆m, I)) implies xk→ L(S (∆m, I)). Theorem 2.7. Sθα(∆m, I) ∩ `∞(∆m) is a closed subset of `∞(∆m) for 0 < α ≤ 1.
Now let θ = (kr) and θ0 = (sr) be two lacunary sequences such that Ir ⊆ Jr for all r ∈ N, and let α, β ∈ (0, 1] such that 0 < α ≤ β ≤ 1. Now we research inclusion connections between the sets of Sα
θ (∆m, I) −convergent sequences and Nα
θ (∆
m, I) −summable sequences for different α’s and θ’s.
Theorem 2.8. Let θ = (kr) and θ0 = (sr) be two lacunary sequences and let α, β ∈ (0, 1] such that 0 < α ≤ β ≤ 1. (i) If lim r→∞inf hαr `βr > 0 (2.4) then Sθβ0(∆m, I) ⊆ Sθα(∆m, I) , (ii) If lim r→∞ `r hβr = 1 (2.5) then Sα θ (∆ m, I) ⊆ Sβ θ0(∆ m, I) , where I r = (kr−1, kr] , Jr = (sr−1, sr] , hr = kr− kr−1, `r= sr− sr−1.
Proof. (i) Assume that Ir⊂ Jr for all r ∈ N and let (2.4) be satisfied. For given ε > 0 we have {k ∈ Jr: |∆mxk− L| ≥ ε} ⊇ {k ∈ Ir: |∆mxk− L| ≥ ε} , 1 `βr |{k ∈ Jr: |∆mxk− L| ≥ ε}| ≥ hα r `βr 1 hα r |{k ∈ Ir: |∆mxk− L| ≥ ε}| and so r ∈ N : h1α r |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≥ δ ⊆ r ∈ N : 1 `βr |{k ∈ Jr: |∆mxk− L| ≥ ε}| ≥ δ hα r `βr ∈ I
for all r ∈ N. Now taking the limit as r → ∞ in the last inequality and using (2.4) we obtain Sβ
θ0(∆
m, I) ⊆ Sα
(ii) Let x ∈ Sθα(∆m, I) and (2.5) be satisfied. Since Ir ⊂ Jr, for ε > 0 we may write 1 `βr |{k ∈ Jr: |∆mxk− L| ≥ ε}| = 1 `βr |{sr−1< k ≤ kr−1: |∆mxk− L| ≥ ε}| +1 `βr |{kr< k ≤ sr: |∆mxk− L| ≥ ε}| +1 `βr |{kr−1< k ≤ kr: |∆mxk− L| ≥ ε}| ≤ kr−1− sr−1 `βr +sr− kr `βr + 1 `βr |{k ∈ Ir: |∆mxk− L| ≥ ε}| = `r− hr `βr + 1 `βr |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≤ `r− h β r hβr + 1 hβr |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≤ `r hβr − 1 + 1 hα r |{k ∈ Ir: |∆mxk− L| ≥ ε}| and r ∈ N : 1 `βr |{k ∈ Jr: |∆mxk− L| ≥ ε}| ≥ δ ⊆ r ∈ N : h1α r |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≥ δ ∈ I
for all r ∈ N. Thus Sα
θ (∆m, I) ⊆ S β θ0 (∆
m, I) .
Theorem 2.9. Let θ = (kr) and θ0 = (sr) be two lacunary sequences such that Ir⊆ Jr for all r ∈ N, α and β be fixed real numbers such that 0 < α ≤ β ≤ 1. Then we have
(i) If (2.4) holds then Nβ θ0(∆
m, I) ⊂ Nα
θ (∆m, I) , (ii) If (2.5) holds and x ∈ `∞(∆m) then Nθα(∆
m, I) ⊂ Nβ θ0(∆
m, I) .
Proof. Omitted.
Theorem 2.10. Let θ = (kr) and θ0 = (sr) be two lacunary sequences such that Ir⊆ Jr for all r ∈ N, and let α, β ∈ (0, 1] such that 0 < α ≤ β ≤ 1. Then
(i) Let (2.4) holds, if xk→ L(Nθβ0(∆m, I)), then xk → L(Sθα(∆m, I)),
(ii) Let (2.5) holds and x = (xk) be a ∆m−bounded sequence, if xk → L( Sθα(∆m, I)), then xk→ L(Nθβ0(∆m, I)).
ON (∆ , I) − LACUNARY STATISTICAL CONVERGENCE OF ORDER α 83
(ii) Assume that Sθα(∆m, I) − lim xk= L and x ∈ ∆m(`∞) . Then we may write 1 `βr X k∈Jr |∆mx k− L| = 1 `βr X k∈Jr−Ir |∆mx k− L| + 1 `βr X k∈Ir |∆mx k− L| ≤ `r− hr `βr M + 1 `βr X k∈Ir |∆mx k− L| ≤ `r− h β r `βr M + 1 `βr X k∈Ir |∆mx k− L| ≤ `r hβr − 1 M + 1 hβr X k∈Ir |xk−L|≥ε |∆mx k− L| + 1 hβr X k∈Ir |xk−L|<ε |∆mx k− L| ≤ `r hβr − 1 M +M hα r |{k ∈ Ir: |∆mxk− L| ≥ ε}| + `r hβr ε and so ( r ∈ N : 1 `βr X k∈Jr |∆mx k− L| ≥ δ ) ⊆ r ∈ N :h1α r |{k ∈ Ir: |∆mxk− L| ≥ ε}| ≥ δ M ∈ I, for all r ∈ N. Using (2.5) we obtain that Nθβ0(∆
m, I) − lim x
k = L, whenever Sα
θ (∆m, I) − lim xk= L.
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Mikail Et
Department of Mathematics, Fırat University 23119, Elazig, TURKEY E-mail address: mikailet68@gmail.com
Hacer S¸eng¨ul
Department of Mathematics, Siirt University 56100, Siirt, TURKEY E-mail address: hacer.sengul@hotmail.com
