# On Generalized Statistical Convergence of Orderα of Difference Sequences

Authors: Mikail Et; Hıfsı Altınok; Yavuz Altın
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce the concept of statistical convergence of orderα of difference sequences, and we give some relations between the set of statistical convergence of order α of difference sequences and strong wλα(Δm,p)-summability of order α. Furthermore some relations between the spaces wλα(Δm,(p),M) and Sλα(Δm) are examined.


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## Body

## 1. Introduction

The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor [5], Fridy [6], Khan and Orhan [7], Miller and Orhan [8], Mursaleen [9], Rath and Tripathy [10], Šalát [11], Savaş [12], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.Letw be the set of all sequences of real or complex numbers, and let ℓ∞, c, and c0 be, respectively, the Banach spaces of bounded, convergent, and null sequences x=(xk) with the usual norm ∥x∥=sup|xk|, where k∈ℕ={1,2,…}, the set of positive integers. Also by bs, cs, ℓ1, and ℓp, we denote the spaces of all bounded, convergent, absolutely, and p-absolutely convergent series, respectively.Letλ=(λn) be a nondecreasing sequence of positive numbers tending to ∞ such that λn+1≤λn+1, λ1=1. The generalized de la Vallée-Poussin mean is defined by tn(x)=(1/λn)∑k∈Inxk, where In=[n-λn+1,n] for n=1,2,…. A sequence x=(xk) is said to be (V,λ)-summable to a number ℓ if tn(x)→ℓ as n→∞ [13]. If λn=n, then (V,λ)-summability is reduced to the Cesàro summability.The notion of difference sequence spaces was introduced by Kızmaz [14], and it was generalized by Et and Çolak [15] such as
(1)Δm(X)={x=(xk):(Δmxk)∈X}
for X=ℓ∞, c, or c0, where m∈ℕ, Δ0x=(xk), Δmx=(Δm-1xk-Δm-1xk+1), and so Δmxk=∑i=0m(-1)i(mi)xk+i. The sequence spaces Δm(X) are the Banach spaces normed by
(2)∥x∥Δ=∑i=1m|xi|+∥Δmxk∥∞
for X=ℓ∞, c or c0. Recently the difference sequence spaces have been studied in [16–21].The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan [22] and after that statistical convergence of order α and strongly p-Cesàro summability of order α were studied by Çolak [23] and generalized by Çolak and Bektaş [24].Let0<α≤1 be given. The sequence (xk) is said to be statistically convergent of order α if there is a real number L such that
(3)limn→∞1nα|{k≤n:|xk-L|≥ε}|=0
for every ε>0. In this case we write Sα-limxk=L. The set of all statistically convergent sequences of order α will be denoted by Sα.LetX be a sequence space. Then X is called(i)
solid (or normal) if(αkxk)∈X whenever (xk)∈X for all sequences (αk) of scalars with |αk|≤1,(ii)
symmetric if(xk)∈X implies (xπ(k))∈X, where π(k) is a permutation of ℕ,(iii)
sequence algebra ifX is closed under multiplication.


## 2. Main Results

In this section we give the main results of the paper. In Theorem5 we give the inclusion relations between the sets of Sλα(Δm)-statistically convergent sequences of order α for different α’s. In Theorem 9 we give the relationship between the strong wλα(Δm,p)-summability of order α and the strong wλβ(Δm,p)-summability of order β. In Theorem 12 we give the relationship between the strong wλα(Δm,p)-summability of order α and the Sλβ(Δm)-statistical convergence of order β.Definition 1.
Let the sequenceλ=(λn) of real numbers be defined as above, and let 0<α≤1 be given. The sequence x=(xk)∈w is said to be Sλα(Δm)-statistically convergent of order α if there is a real number L such that
(4)limn→∞1λnα|{k∈In:|Δmxk-L|≥ε}|=0,
where In=[n-λn+1,n] and λnα denote the αth power (λn)α of λn; that is λα=(λnα)=(λ1α,λ2α,…,λnα,…). In this case we write Sλα(Δm)-limxk=L. The set of all Sλα(Δm)-statistically convergent sequences of order α will be denoted by Sλα(Δm). For λn=n for all n∈ℕ, we will write Sα(Δm) instead of Sλα(Δm) and in the special case α=1, we will write Sλ(Δm) instead of Sλα(Δm), and also in the special case α=1 and λn=n for all n∈ℕ we will write S(Δm) instead of Sλα(Δm), which were defined and studied by Et and Nuray [25].TheSλα(Δm)-statistical convergence of order α is well defined for 0<α≤1, but it is not well defined for α>1 in general. For this let λn=n for all n∈ℕ and x=(xk) be defined as follows:
(5)xk={-1,ifkisodd1,ifkiseven,
and so
(6)Δmxk={-2m,ifkisodd2m,ifkiseven;
then both
(7)limn→∞1λnα|{k∈In:|Δmxk-(-2m)|≥ε}|=limn→∞λn2λnα=0,limn→∞1λnα|{k∈In:|Δmxk-2m|≥ε}|=limn→∞λn2λnα=0
for α>1, and so Sλα(Δm)-statistically converges of order α, both to -2m and 2m; that is, Sλα(Δm)-limxk=2m and Sλα(Δm)-limxk=-2m. But this is impossible.Theorem 2.
Let0<α≤1 and x=(xk),y=(yk) be sequences of real numbers; then(i)
ifSλα(Δm)-limxk=x0 and c∈ℝ, then Sλα(Δm)-lim(cxk)=cx0,(ii)
ifSλα(Δm)-limxk=x0 and Sλα(Δm)-limyk=y0, then Sλα(Δm)-lim(xk+yk)=x0+y0.Proof.
Omitted.Definition 3.
Let the sequenceλ=(λn) be as above, α∈(0,1] any real number, and p a positive real number. A sequence x is said to be strongly wλα(Δm,p)-summable of order α if there is a real number L such that
(8)limn→∞1λnα∑k∈In‍|Δmxk-L|p=0,
where In=[n-λn+1,n]. In this case we write wλα(Δm,p)-limxk=L. The strong wλα(Δm,p)-summability of order α reduces to the strong wλ(Δm,p)-summability for α=1. The set of all strongly wλα(Δm,p)-summable sequences of order α will be denoted by wλα(Δm,p).Theorem 4.
Let the sequenceλ=(λn) be as above, αo∈(0,1], and po a positive real number. The sequence space wλαo(Δm,po) is a Banach space for 1≤po<∞ normed by
(9)∥x∥Δ=∑i=1m‍|xi|+supn(1λnαo∑k∈In‍|Δmxk|po)1/po
and a complete p-normed space for 0<po<1 by
(10)∥x∥Δ′=∑i=1m‍|xi|po+supn1λnαo∑k∈In‍|Δmxk|po.Proof.
It is a routine verification thatwλαo(Δm,po) is a normed space normed by (9). Let (xs) be a Cauchy sequence such that xs=(xks)k=(x1s,x2s,…)∈wλαo(Δm,po) for each s∈ℕ. Then we have
(11)∥xs-xt∥Δ=∑i=1m‍|xis-xit|+supn(1λnαo∑k∈In‍|Δmxks-Δmxkt|po)1/po⟶0
as s,t→∞ and so that
(12)∑i=1m‍|xis-xit|⟶0,(1λnαo∑k∈In‍|Δmxks-Δmxkt|po)1/po⟶0,∀n∈ℕ
as s,t→∞. Now for k≤m we get |xis-xis|→0 as s,t→∞, and for all k∈ℕ we get |Δmxks-Δmxkt|→0 as s,t→∞. Since the inequality
(13)|Δmxk+ms-Δmxk+mt|≤|Δmxks-Δmxkt|+(m0)|xks-xkt|+⋯+(mm-1)×|Δmxk+m-1s-Δmxk+m-1t|
holds for all k∈ℕ, we get |xis-xis|→0 as s,t→∞ and for each k∈ℕ. This implies that the sequence (xks)k=(xk1,xk2,…) is a Cauchy in ℝ for each k∈ℕ. Also, it is convergent, because ℝ is complete. Assume that limsxks=xk for each k∈ℕ. Since (xs) is a Cauchy sequence, then for every ε>0 there exists a number n0=n0(ε) such that
(14)∥xs-xt∥Δ<ε
for all s,t≥n0. Hence for all n∈ℕ and all s,t≥n0 we get
(15)∑i=1m‍|xis-xt|<ε,1λnαo∑k∈In‍|Δmxks-Δmxkt|po<εpo,∀n∈ℕ.
Taking limit as t→∞ in the last inequalities, we get
(16)limt∑i=1m‍|xis-xit|=∑i=1m‍|xis-xi|<ε,limt1λnαo∑k∈In‍|Δmxks-Δmxkt|po=1λnαo∑k∈In‍|Δmxks-Δmxk|po<εpo
for s≥n0 and all n∈ℕ. This implies that
(17)∥xs-x∥Δ=∑i=1m‍|xis-xi|+supn(1λnα∑k∈In‍|Δmxks-Δmxk|po)1/po⟶0,
and hence xs→x as s→∞. Since
(18)1λnαo∑k∈In‍|Δmxk-L|po≤2po{1λnαo∑k∈In‍|Δmxkn0-L|po+1λnαo∑k∈In‍|Δmxkn0-Δmxk|po},
we get x∈wλαo(Δm,po).
It can be shown thatwλαo(Δm,po) is a complete p-normed space for 0<po<1 by (10).Theorem 5.
If0<α≤β≤1, then Sλα(Δm)⊆Sλβ(Δm) and the inclusion is strict.Proof.
Proof follows from the inequality(19)1λnβ|{k∈In:|Δmxk-L|≥ε}|≤1λnα|{k∈In:|Δmxk-L|≥ε}|.
To show that the inclusion is strict, define x=(xk) such that
(20)Δmxk={k,ifkissquare0,otherwise.
Then x∈Sλβ(Δm) for 1/2<β≤1, but x∉Sλα(Δm) for 0<α≤1/2.From Theorem5 we have the following.Corollary 6.
If a sequence isSλα(Δm)-statistically convergent of order α, to L, then it is Sλ(Δm)-statistically convergent to L.Theorem 7.
S
α
(
Δ
m
)
⊆
S
λ
α
(
Δ
m
) if
(21)limn→∞infλnαnα>0.Proof.
For givenε>0 we have
(22){k≤n:|Δmxk-L|≥ε}⊃{k∈In:|Δmxk-L|≥ε}
and so
(23)1nα|{k≤n:|Δmxk-L|≥ε}|≥λnαnα1λnα|{k∈In:|Δmxk-L|≥ε}|.
Taking the limit as n→∞ and using (21), we get Sλα(Δm)-limxk=L.Theorem 8.
S
(
Δ
m
)
=
S
λ
α
(
Δ
m
) if and only if
(24)limn→∞infλnαn>0.Proof.
From Theorem7 we have S(Δm)⊆Sλα(Δm).
Conversely, suppose thatlimn→∞infλnα/n=0. We can choose a subsequence (n(j))j=1∞ such that λnα(j)/n(j)<1/j. If we consider a sequence x=(xk) defined by
(25)Δmxi={1,ifi∈In(j)forsomej=1,2,3,…0,otherwise,
then x∈S(Δm) but x∉Sλ(Δm). From Corollary 6 we have x∉Sλα(Δm). This completes the proof.Theorem 9.
Let0<α≤β≤1 and p a positive real number; then wλα(Δm,p)⊆wλβ(Δm,p) and the inclusion is strict.Proof.
Letx=(xk)∈wλα(Δm,p). Then given α and β such that 0<α≤β≤1 and a positive real number p, we may write
(26)1λnβ∑k∈In‍|Δmxk-L|p≤1λnα∑k∈In‍|Δmxk-L|p
and this gives that wλα(Δm,p)⊆wλβ(Δm,p).
To show that the inclusion is strict definex=(xk) such that
(27)Δmxk={1,ifkissquare0,otherwise.
Then x∈wλβ(Δm,p) for 1/2<β≤1 but x∉wλα(Δm,p) for 0<α≤1/2, for λ=(n).The following result is a consequence of Theorem9.Corollary 10.
Let0<α≤β≤1 and p a positive real number. Then(i)
ifα=β, then wλα(Δm,p)=wλβ(Δm,p),(ii)
wλα(Δm,p)⊆wλ(Δm,p) for each α∈(0,1] and 0<p<∞.Theorem 11.
Let0<α≤1 and 0<p<q<∞. Then wλα(Δm,q)⊂wλα(Δm,p).Proof is seen from the Hölder inequality.Theorem 12.
Letα and β be fixed real numbers such that 0<α≤β≤1 and 0<p<∞. If a sequence is strongly wλα(Δm,p)-summable of order α, to L, then it is Sλβ(Δm)-statistically convergent of order β, to L.Proof.
For any sequencex=(xk) and ε>0, we have
(28)∑k∈In‍|Δmxk-L|p=∑k∈In|Δmxk-L|≥ε‍|Δmxk-L|p+∑k∈In|Δmxk-L|<ε‍|Δmxk-L|p≥∑k∈In|Δmxk-L|≥ε‍|Δmxk-L|p≥|{k∈In:|Δmxk-L|≥ε}|εp
and so
(29)1λnα∑k∈In‍|Δmxk-L|p≥1λnα|{k∈In:|Δmxk-L|≥ε}|εp≥1λnβ|{k∈In:|Δmxk-L|≥ε}|εp.
From this it follows that if x=(xk) is strongly wλα(Δm,p)-summable of order α, to L, then it is Sλβ(Δm)-statistically convergent of order β, to L.Corollary 13.
Letα be a fixed real number such that 0<α≤1 and 0<p<∞.(i)
If a sequence is stronglywλα(Δm,p)-summable of order α, to L, then it is Sλα(Δm)-statistically convergent of order α, to L.(ii)
If a sequence is stronglywλα(Δm,p)-summable of order α, to L, then it is Sλ(Δm)-statistically convergent to L.


## 3. Results Related to the Orlicz Function

In this section we give the inclusion relations between the sets ofSλα(Δm)-statistically convergent sequences of order α and strongly wλα(Δm,(p),M)-summable sequences of order α with respect to the Orlicz function M.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→∞. An Orlicz function M can always be represented in the following integral form:
(30)M(x)=∫0xq(t)dt,
where q known as the kernel of M is right differentiable for t≥0, q(0)=0, q(t)>0 for t>0, q is nondecreasing, and q(t)→∞ as t→∞.Let(Ω,Σ,μ) be a finite measure space. We denote by E(μ) the space of all (equivalence classes of) Σ-measurable functions x from Ω into [0,∞). Given an Orlicz function M, we define on E(μ) a convex functional IM by
(31)IM(x)=∫Ω‍M(x(t))dμ,
and an Orlicz space LM(μ) by LM(μ)={x∈E(μ):IM(λx)<+∞forsomeλ>0} (for detail see [26]).The study of Orlicz sequence spaces was initiated with a certain specific purpose in the Banach space theory. Indeed, Lindberg [27] got interested in Orlicz spaces in connection with finding the Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c0 or ℓp(1≤p<∞). Subsequently, Lindenstrauss and Tzafriri [28] used the idea of the Orlicz function to construct the sequence space
(32)ℓM={x∈w:∑k=1∞‍M(|xk|ρ)<∞forsomeρ>0}.
The space ℓM with the norm
(33)∥x∥=inf{ρ>0:∑k=1∞‍M(|xk|ρ)≤1}
becomes a Banach space, called an Orlicz sequence space. The space ℓM is closely related to the space ℓp which is an Orlicz sequence space with M(x)=|x|p for 1≤p<∞. Lindenstrauss and Tzafriri [28] proved that every Orlicz sequence space ℓM contains a subspace isomorphic to ℓp  (1≤p<∞). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [26].It is well known that ifM is a convex function and M(0)=0, then M(λx)≤λM(x) for all λ with 0<λ<1.Recently, the Orlicz sequence spaces have been studied by Altin et al. [29], Bhardwaj and Singh [30], Et et al. [31, 32], Mursaleen et al. [33], Savaş and Rhoades [34], and many others.Definition 14.
LetM an Orlicz function, p=(pk) a sequence of strictly positive real numbers, and α∈(0,1] any real number. Now we define
(34)wλα(Δm,(p),M)={∑k∈In‍[M(|Δmxk-L|ρ)]pkx=(xk):limn→∞1λnα×∑k∈In‍[M(|Δmxk-L|ρ)]pk=0,forsomeLandρ>0∑k∈In‍[M(|Δmxk-L|ρ)]pk}.
If x∈wλα(Δm,(p),M), then we say that x is strongly wλα(Δm,(p),M)-summable of order α with respect to the Orlicz function M. For pk=p for all k∈ℕ and M(x)=x we will write wλα(Δm,p) instead of wλα(Δm,(p),M) and in the special case M(x)=x we will write wλα(Δm,(p)) instead of wλα(Δm,(p),M).In the following two theorems we will assume that the sequencep=(pk) is bounded and 0<h=infkpk≤pk≤supkpk=H<∞.Theorem 15.
Letα,β∈(0,1] be real numbers such that α≤β and M an Orlicz function; thenwλα(Δm,(p),M)⊂Sλβ(Δm).Proof.
Letx∈wλα(Δm,(p),M), ε>0 be given, and let ∑1‍ and ∑2‍ denote the sums over k∈In, |Δmxk-L|≥ε and k∈In, |Δmxk-L|<ε, respectively. Since λnα≤λnβ for each n we may write
(35)1λnα∑k∈In‍[M(|Δmxk-L|ρ)]pk=1λnα[∑1‍[M(|Δmxk-L|ρ)]pk+∑2‍[M(|Δmxk-L|ρ)]pk]≥1λnβ[∑1‍[M(|Δmxk-L|ρ)]pk+∑2‍[M(|Δmxk-L|ρ)]pk]≥1λnβ∑1‍[f(ε)]pk≥1λnβ∑1‍min([M(ε1)]h,[M(ε1)]H),ε1=ερ≥1λnβ|{k∈In:|Δmxk-L|≥ε}|×min([f(ε1)]h,[f(ε1)]H).
Since x∈wλα(Δm,(p),M), the left-hand side of the above inequality tends to zero as n→∞. Therefore the right-hand side tends to zero as n→∞ and hence x∈Sλβ(Δm), because min([f(ε1)]h,[f(ε1)]H)>0.Corollary 16.
Let0<α≤1 and M an Orlicz function; then wλα(Δm,(p),M)⊂Sλα(Δm).Theorem 17.
LetM be an Orlicz function and x=(xk)Δm-bounded sequence; then Sλα(Δm)⊂wλα(Δm,(p),M).Proof.
Suppose thatx=(xk) a Δm-bounded sequence; that is, x∈ℓ∞(Δm) and Sλα(Δm)-limxk=L. Since x∈ℓ∞(Δm), then there is a constant T>0 such that |Δmxk|≤T. Given ε>0 we have
(36)1λnα∑k∈In‍[M(|Δmxk-L|ρ)]pk=1λnα∑1‍[M(|Δmxk-L|ρ)]pk+1λnα∑2‍[M(|Δmxk-L|ρ)]pk≤1λnα∑1‍max{[M(Tρ)]h,[M(Tρ)]H}+1λnα∑2‍[M(ερ)]pk≤max{[M(K)]h,[M(K)]H}×1λnα|{k∈In:|Δmxk-L|≥ε}|+max{[M(ε1)]h,[M(ε1)]H},dddddddddTρ=K,ερ=ε1.
Hence x∈wλα(Δm,(p),M).Theorem 18.
The sequence spaceswλα(Δm,p), Sλα(Δm), and wλα(Δm,(p),M) are neither solid nor symmetric, nor sequence algebras for m≥1.Proof.
Letλk=k for all k∈ℕ; then (xk)=(km-1)∈wλα(Δm,p) but (αkxk)∉wλα(Δm,p) when αk=(-1)k for all k∈ℕ. Hence wλα(Δm,p) is not solid. The other cases can be proved on considering similar examples.Theorem 19.
Iflimpk>0 and x=(xk) is strongly wλα(Δm,(p),M)-summable of order α, to L with respect to the Orlicz function M, then wλα(Δm,(p),M)-limxk=L uniquely.Proof.
Letlimpk=s>0. Suppose that wλα(Δm,(p),M)-limxk=L and wλα(Δm,(p),M)-limxk=L1. Then (37)limn→∞1λnα∑k∈In‍[M(|Δmxk-L|ρ1)]pk=0,forsomeρ1>0,limn→∞1λnα∑k∈In‍[M(|Δmxk-L1|ρ2)]pk=0,forsomeρ2>0.
Defineρ=max(2ρ1,2ρ2). Since M is nondecreasing and convex, we have
(38)1λnα∑k∈In‍[M(|L-L1|ρ)]pk≤Dλnα∑k∈In‍12pk([M(|Δmxk-L|ρ1)]pk+[M(|Δmxk-L1|ρ2)]pk)≤Dλnα∑k∈In‍[M(|Δmxk-L|ρ1)]pk+Dλnα∑k∈In‍[M(|Δmxk-L1|ρ2)]pk⟶0,(n⟶∞),
where supkpk=H and D=max(1,2H-1). Hence
(39)limn→∞1λnα∑k∈In‍[M(|L-L1|ρ)]pk=0.
Since limk→∞pk=s we have
(40)limk→∞[M(|L-L1|ρ)]pk=[M(|L-L1|ρ)]s
and so L=L1. Thus the limit is unique.



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