Double almost lacunary statistical convergence of order α

Savas Advances in Difference Equations 2013, 2013:254http://www.advancesindifferenceequations.com/content/2013/1/254

RESEARCH Open Access

Double almost lacunary statisticalconvergence of order αEkrem Savas*

*Correspondence:[email protected] of Mathematics,Istanbul Commerce University,Üsküdar, Istanbul, Turkey

AbstractIn this paper, we define and study lacunary double almost statistical convergence oforder α. Further, some inclusion relations have been examined. We also introduce anew sequence space by combining lacunary double almost statistical convergenceand Orlicz function.MSC: Primary 40B05; secondary 40C05

Keywords: statistical convergence; Orlicz function; double statistical convergence oforder α; lacunary statistical convergence; double almost statistical convergence

1 IntroductionThe notion of convergence of a real sequence was extended to a statistical convergence byFast [] (see also Schoenberg []) as follows. If N denotes the set of natural numbers andK ⊂ N, then K(m,n) denotes the cardinality of the set K ∩ [m,n]. The upper and lowernatural density of the subset K is defined by

d(K) = limn→∞ sup

K(,n)n

and d(K) = limn→∞ inf

K(,n)n

.

If d(K) = d(K), then we say that the natural density of K exists, and it is denoted simply byd(K). Clearly d(K) = limn→∞ K (,n)

n .A sequence x = (xk) of real numbers is said to be statistically convergent to L if for arbi-

trary ε > , the set K(ε) = {k ∈N : |xk – L| ≥ ε} has a natural density zero.Statistical convergence turned out to be one of the most active areas of research in

summability theory after the works of Fridy [] and Šalát []. For some very interestinginvestigations concerning statistical convergence, one may consult the papers of Cakalli[], Miller [], Maddox [] and many others, where more references on this importantsummability method can be found.On the other hand, in [, ], a different direction was given to the study of statistical

convergence, where the notion of statistical convergence of order α, < α < was intro-duced by replacing n by nα in the denominator in the definition of statistical convergence.It was observed in [] that the behaviour of this new convergence was not exactly parallelto that of statistical convergence, and some basic properties were obtained. One can alsosee [] for related works.In this paper, we define and study lacunary double almost statistical convergence of or-

der α. Also some inclusion relations have been examined.

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Letw be the set of all real or complex double sequences. By the convergence of a doublesequence, we mean the convergence on the Pringsheim sense, that is, double sequencex = (xij) has a Pringsheim limit L, denoted by P-limx = L, provided that given ε > , andthere exists N ∈ N such that |xij – L| < ε whenever i, j ≥ N . We shall describe such anx more briefly as ‘P-convergent’ (see, []). We denote by c the space of P-convergentsequences. A double sequence x = (xij) is bounded if ‖x‖ = supi,j≥ |xij| < ∞. Let l∞ andc∞ be the set of all real or complex bounded double sequences and the set bounded andconvergent double sequences, respectively. Moricz and Rhoades [] defined the almostconvergence of double sequence as follows: x = (xij) is said to be almost convergent to anumber L if

P- limp,q→∞ sup

m,n

∣∣∣∣∣ (p + )(q + )

m+p∑i=m

n+q∑j=n

xij – L∣∣∣∣∣ = ,

that is, the average value of (xij) taken over any rectangle

D ={(i, j) :m ≤ i≤ m + p,n ≤ j ≤ n + q

},

tends to L as both p and q tend to ∞, and this convergence is uniform in m and n. Wedenote the space of almost convergent double sequence by c, as

c ={x = (xij) : lim

k,l→∞∣∣tklpq(x) – L

∣∣ = , uniformly in p,q},

where

tklpq(x) =

(k + )(l + )

k+p∑i=p

l+q∑j=q

xij.

The notion of almost convergence for single sequences was introduced by Lorentz []and some others.A double sequence x is called strongly double almost convergent to a number L if

P- limk,l→∞

(k + )(l + )

k+p∑i=p

l+q∑j=q

|xij – L| = , uniformly in p,q.

By [c], we denote the space of strongly almost convergent double sequences.The notion of strong almost convergence for single sequences has been introduced by

Maddox [].The idea of statistical convergence was extended to double sequences byMursaleen and

Edely []. More recent developments on double sequences can be found in [, –]. Forthe single sequences; statistical convergence of orderα and strongly p-Cesàro summabilityof order α introduced by Çolak []. Quite recently, in [], Çolak and Bektaş generalizedthis notion by using de la Valée-Poussin mean.LetK ⊆N×N be a two-dimensional set of positive integers, and letKm,n be the numbers

of (i, j) in K such that i≤ n and j ≤ m.



Then the lower asymptotic density of K is defined as

P-lim infm,n

Km,n

mn= δ(K).

In the case when the sequence (Km,nmn )∞,∞

m,n=, has a limit, we say that K has a natural densityand is defined as

P-limm,n

Km,n

mn= δ(K).

For example, let K = {(i, j) : i, j ∈N}, where N is the set of natural numbers. Then

δ(K) = P-limm,n

Km,n

mn≤ P-lim

m,n

√m

√n

mn=

(i.e., the set K has a double natural density zero).Mursaleen and Edely [] presented the notion of a statistical convergence for the dou-

ble sequence x = (xij) as follows: A real double sequence x = (xij) is said to be statisticallyconvergent to L, provided that for each ε >

P-limm,n

mn

∣∣{(i, j) : i ≤ m and j ≤ n, |xij – L| ≥ ε}∣∣ = .

We now write the following definition.The double statistical convergence of order α is defined as follows. Let < α ≤ be

given. The sequence (xij) is said to be statistically convergent of order α if there is a realnumber L such that

P- limmn→∞

(mn)α

∣∣{i≤ m and j ≤ n : |xij – L| ≥ ε}∣∣ =

for every ε > , in this, case we say that x is double statistically convergent of order α to L.In this case, we write Sα

-limxij = L. The set of all double statistically convergent sequencesof order α will be denoted by Sα

. If we take α = in this definition , we can have the previousdefinition.By a lacunary θ = (kr); r = , , , . . . , where k = , we shall mean an increasing sequence

of nonnegative integers with kr – kr– → ∞ as r → ∞. The intervals determined by θ willbe denoted by Ir = (kr–,kr] and hr = kr – kr–. The ratio kr

kr– will be denoted by qr .Fridy and Orhan [] introduced the idea of lacunary statistical convergence for single

sequence as follows.The number sequence x = (xi) is said to be lacunary statistically convergent to the num-

ber � if for each ε > ,

limn

hr

∣∣{k ∈ Ir : |xi – L| ≥ ε}∣∣ = .

In this case, we write Sθ -limi xi = �, and we denote the set of all lacunary statistically con-vergent sequences by Sθ .



Definition . By a double lacunary θrs = {(krls)}, r, s = , , , . . . , where k = and l = ,we shall mean two increasing sequences of nonnegative integers with

hr = kr – kk– → ∞ as r → ∞

and

hs = ls – ls– → ∞ as s → ∞.

Let us denote krs = krls, hrs = hrhs and the intervals determined by θrs will be denoted byIrs = {(k, l) : kr– < k ≤ kr and ls– < l ≤ ls}, qr = kr

kr– , qs =lsls– , and qrs = qrqs. We will denote

the set of all double lacunary sequences by Nθrs .Let K ⊆N ×N have double lacunary density δθ

(K) if

P-limrs

hrs

∣∣{(k, l) ∈ Irs : (k, l) ∈ K}∣∣

exists.

Example Let θ = {(r – , s – )} and K = {(k, l) : k, l ∈ N ×N}. Then δθ (K) = . But it

is obvious that δ(K) = /.

In , Patterson and Savaş [] studied double lacunary statistical convergence bygiving the definition for complex sequences as follows.

Definition . Let θrs be a double lacunary sequence; the double number sequence x isSθ -convergent to L, provided that for every ε > ,

P-limrs

hrs

∣∣{(k, l) ∈ Irs : |xkl – L| ≥ ε}∣∣ = .

In this case, write Sθ -limx = L or xklP→ L(Sθ ).

More investigation in this direction and more applications of double lacunary and dou-ble sequences can be found in [–] and [].

2 Main resultsIn this section, we define lacunary double almost statistically convergent sequences oforder α. Also we shall prove some inclusion theorems.We now have the following.

Definition . Let < α ≤ be given. The sequence x = (xij) ∈ w is said to be Sαθrs -

statistical convergence of order α if there is a real number L such that

P-limrs

hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣ = , uniformly in p,q,

where hαrs denote the αth power (hrs)α of hrs. In case x = (xij) is Sα

θrs -statistically convergentof order α to L, wewrite Sα

θrs-limxij = L.We denote the set of all Sαθrs -statistically convergent

sequences of order α by Sαθrs .



We know that the Sαθrs -statistical convergence of order α is well defined for < α ≤ ,

but it is not well defined for α > in general. It is easy to see by taking x = (xij) asfixed.

Definition . Let < α ≤ be any real number, and let t be a positive real number.A sequence x is said to be strongly wα

θrs (t)-summable of order α, if there is a real numberL such that

P-limrs

hαrs

∑(k,l)∈Irs

∣∣tklpq(x) – L∣∣t = , uniformly in p,q.

If we take α = , the strong wαθrs (t)-summability of order α reduces to the strong wθrs (t)-

summability.We denote the set of all strongly wα

θrs (t)-summable sequence of order α by wαθrs (t).

We now state the following theorem.

Theorem . If < α ≤ β ≤ , then Sαθrs ⊂ Sβ

θrs .

Proof Let < α ≤ β ≤ . Then

hβrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣ ≤

hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

for every ε > , and finally, we have that Sαθrs ⊂ Sβ

θrs . This proves the result. �

Theorem . For any lacunary sequences θ , Sα ⊆ Sα

θrs , if lim infqr > and lim inf qs > .

Proof Suppose that lim infqαr > and lim infqα

s > , lim infqαr = α and lim infqα

s = α, say.Write β = (α – )/ and β = (α – )/. Then there exist a positive integer r and s suchthat qα

r ≥ + β for r ≥ r and qs ≥ + β for s ≥ s. Hence for r ≥ r, and s ≥ s,

hαrs

(krls)α

= –(kαr–kαr

)× –

(lαs–lαs

)

=( –

qαr

)×

( –

qαs

)

≥ –

( + β)× –

( + β)

=β

+ β× β

+ β.

Take any (xkl) ∈ Sα , and Sα

-lim(k,l)→∞ xkl = L, say. We prove that Sαθrs-lim(k,l)→∞ xkl = L.

Then for r ≥ r and s≥ s, we have

(krls)α

∣∣{k ≤ kr , l ≤ ls :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

≥ (krls)α

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣



= hαrs

(krls)α

hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

≥ β

+ β× β

+ β

hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣.

Therefore, Sαθrs-lim(k,l)→∞ x(k, l) = L. �

Remark . The converse of this result is true for α = . However, for α < it is not clear,and we leave it as an open problem.

Theorem . For any double lacunary sequence θrs, Sαθrs ⊆ Sα

if lim supr qαr < ∞ and

lim sups qαs <∞.

Proof Suppose that lim supr qαr < ∞ and lim sups qα

s <∞. Then there existsH > such thatqαr <H and qα

s <H for all r and s. Suppose that xkl → L(Sθαrs ) and

Nrs =∣∣{(k, l) ∈ Irs :

∣∣tklpq(x) – L∣∣ ≥ ε

}∣∣by the definition of xkl → L(Sθrs ) given ε > , there exists r, s ∈N such that Nrs

hαrs< ε for all

r > r and s > s. Let

M := max{Nrs : ≤ r ≤ r and ≤ s ≤ s}.

Let n andm be such that kr– <m≤ kr and ls– < n≤ ls. Therefore, we obtain the following:

(mn)α

∣∣{k ≤ m and l ≤ n :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

≤ (kr–ls–)α

∣∣{k ≤ kr and l ≤ ls :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

=

(kr–ls–)α

{ r,s∑i,j=,

Ni,j

}

≤ Mrs(kr–ls–)α

+

(kr–ls–)α

{ r,s∑i,j=r+,r+

Ni,j

}


+

(kr–ls–)α

{ r,s∑i,j=r+,r+

Ni,jhαi,j

hαi,j

}

≤ Mrskr–ls–

+

(kr–ls–)α

(sup

i,j≥r,r

Ni,j

hαi,j

){ r,s∑i,j=r+,r+

hαi,j

}


+ ε

{ r,s∑i,j=r+,r+

hαi,j

}


+ εH.

This completes the proof of the theorem. �

Theorem . Let < α ≤ β ≤ and t be a positive real number, then wαθrs (t)⊆ wβ

θrs (t).



Proof Let x = (xij) ∈ wαθrs (t). Then given α and β such that < α ≤ β ≤ and a positive real

number t we write

hβrs

∑(k,l)∈Irs

∣∣tklpq(x) – L∣∣t ≤

hαrs

∑(k,l)∈Irs

∣∣tklpq(x) – L∣∣t ,

and we get that wαθrs (t) ⊆ wβ

θrs (t). �

As a consequence of Theorem ., we have the following.

Corollary . Let < α ≤ β ≤ and t be a positive real number. Then:(i) If α = β , then wα

θrs (t) = wβ

θrs (t).(ii) wα

θrs (t) ⊆ wθrs (t) for each α ∈ (, ] and < t < ∞.

Theorem. Let α and β be fixed real numbers such that < α ≤ β ≤ and < t <∞. If asequence is a strongly wα

θrs (t)-summable sequence of order α, to L, then it is Sβ

θrs -statisticallyconvergent of order β , to L, i.e., wα

θrs (t) ⊂ Sβ

θrs .

Proof For any sequence x = (xij) and ε > , we write

∑(k,l)∈Irs

∣∣tklpq(x) – L∣∣t = ∑

(k,l)∈Irs|tklpq(x)–L|≥ε

∣∣tklpq(x) – L∣∣t + ∑

(k,l)∈Irs|tklpq(x)–L|<ε

∣∣tklpq(x) – L∣∣t

≥∑

(k,l)∈Irs|tklpq(x)–L|≥ε

∣∣tklpq(x) – L∣∣t ≥ ∣∣{(k, l) ∈ Irs :

∣∣tklpq(x) – L∣∣ ≥ ε

}∣∣ · εt

and so that

hαrs

∑(k,l)∈Irs

∣∣tklpq(x) – L∣∣t ≥

hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣ · εt

≥ hβrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣ · εt ,

this shows that if x = (xij) is strongly wαθrs (t)-summable sequence of order α to L, then it is

Sβ

θrs -statistically convergent of order β to L. This completes the proof. �

We have the following.

Corollary . Let α be fixed real numbers such that < α ≤ and < t < ∞.(i) If a sequence is strongly wα

θrs (t)-summable sequence of order α to L, then it isSα

θrs -statistically convergent of order α to L, i.e., wαθrs (t)⊂ Sα

θrs .(ii) wα

θrs (t) ⊂ Sθrs , for < α ≤ .

3 New sequence spaceIn this section, we study the inclusion relations between the set of Sα

θrs -statistical conver-gent sequences of order α and strongly wα

θrs [M, t]-summable sequences of order α withrespect to an Orlicz functionM.



The study of Orlicz sequence spaces was initiated with a certain specific purpose inBanach space theory. Lindenstrauss and Tzafriri [] investigated Orlicz sequence spacesin more detail, and they proved that every Orlicz sequence space lM contains a subspaceisomorphic to lp ( ≤ p < ∞). The Orlicz sequence spaces are the special cases of Orliczspaces studied in []. Orlicz spaces find a number of useful applications in the theory ofnonlinear integral equations. Whereas the Orlicz sequence spaces are the generalizationof lp spaces, the lp-spaces find themselves enveloped in Orlicz spaces [].Recall in [] that an Orlicz function M : [,∞) → [,∞) is continuous, convex, non-

decreasing function such thatM() = andM(x) > for x > , andM(x)→ ∞ as x → ∞.An Orlicz function M is said to satisfy �-condition for all values of u, if there exists

K > such thatM(u) ≤ KM(u), u≥ .In the later stage different classes of Orlicz sequence spaces were introduced and studied

by Parashar and Choudhary [], Savaş [–] and many others.

Definition . LetM be an Orlicz function, t = (tkl) be a sequence of strictly positive realnumbers, and let α ∈ (, ] be any real number. Now, we write

wαθrs [M, t] =

{x = (xkl) : P-limrs

hαrs

∑(k,l)∈Irs

[M(|tklpq(x) – L|)ρ

]tkl= ,

uniformly in p,q, for some L and ρ > }.

If x ∈ wαθrs [M, t], then we say that x is strongly double almost lacunary summable of order

α with respect to the Orlicz functionM.

If we consider various assignments ofM, θrs and t in the sequence spaces above, we aregranted the following:() IfM(x) = x, θ = rs, and tk,l = for all (k, l) then wα

θrs [M, t] = [wα].() If tk,l = for all (k, l), then wα

θrs [M, t] = wαθrs [M].

() If tk,l = for all (k, l) and θ = rs, then wαθrs[M, t] = wα[M].

() If θ = rs, then wαθ [M, t] = wα[M, t].

In the followings theorems, we shall assume that t = (tkl) is bounded and < h =infkl tkl ≤ tkl ≤ supkl tkl =H < ∞.

Theorem . Let α,β ∈ (, ] be real numbers such that α ≤ β , and let M be an Orliczfunction, then wα

θrs[M, t] ⊂ Sβ

θrs.

Proof Let x ∈ wαθ [M, t], ε > be given and

∑ and

∑ denote the sums over (k, l) ∈ Irs,

|tklpq(x) – L| ≥ ε and (k, l) ∈ Irs, |tklpq(x) – L| < ε, respectively. Since hαrs ≤ hβ

rs for each r, swe write

hαrs

∑(k,l)∈Irs


]tkl

=hαrs

[∑


]tkl+

∑


]tkl]

≥ hβrs

[∑


]tkl+

∑


]tkl]



≥ hβrs

[∑

[M(ε/ρ)

]]tkl

≥ hβrs

∑

min([M(ε)

]h, [M(ε)]H)

, ε =ε

ρ

≥ hβrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣min

([M(ε)

]h, [M(ε)]H)

.

Since x ∈ wαθrs[M, t], the left hand side of the inequality above tends to zero as r, s → ∞

uniformly in p, q. Hence the right hand side tends to zero as r, s → ∞ uniformly in p, q,and, therefore, x ∈ Sβ

θrs . This proves the result. �

Corollary . Let α ∈ (, ] and M be an Orlicz function, then wαθrs [M, t] ⊂ Sα

θrs .

We finally prove the following theorem.

Theorem . Let M be an Orlicz function, and let x = (xij) be a bounded sequence, thenSα

θrs ⊂ wαθrs [M, t].

Proof Suppose that x ∈ �∞ and Sα

θrs-limxij = L. Since x ∈ �∞ , then there is a constant K >

such that |tklpq(x)| ≤ K . Given ε > , we write for all p, q

hαrs

∑(k,l)∈Irs

[M

( |tklpq(x) – L|ρ

)]rkl=

hαrs

∑

[M


)]rkl

+hαrs

∑

[M


)]rkl

≤ hαrs

∑

max

{[M

(Kρ

)]h,[M

(Kρ

)]H}

+hαrs

∑

[M

(ε

ρ

)]tkl

≤ max{[M(T)

]h, [M(T)]H}

× hαrs

∣∣{(k, l) ∈ Irs :∣∣tklpq(x) – L

∣∣ ≥ ε}∣∣

+ max{[M(ε)

]h, [M(ε)]H}

,Kρ

= T , ε

ρ= ε.

Therefore, x ∈ x ∈ wαθrs [M, t]. This proves the result. �

Competing interestsThe author declares that they have no competing interests.

AcknowledgementsThe author would like to thank the referees for their help and useful discussions.

Received: 27 April 2013 Accepted: 22 July 2013 Published: 21 August 2013



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doi:10.1186/1687-1847-2013-254Cite this article as: Savas: Double almost lacunary statistical convergence of order α. Advances in Difference Equations2013 2013:254.


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Double almost lacunary statistical convergence of order α