Double sequence spaces characterized by lacunary sequences

www.elsevier.com/locate/aml

Double sequence spaces characterized by lacunary sequences

Ekrem Savas¸

, Richard F. Patterson

a˙Istanbul Ticaret University, Department of Mathematics, ¨Usk¨udar, ˙Istanbul, Turkey

bDepartment of Mathematics and Statistics, University of North Florida, Building 14, Jacksonville, FL, 32224, USA Received 26 September 2006; accepted 26 September 2006

Abstract

In 1989, Das and Patel considered known sequence spaces to define two new sequence spaces called lacunary almost convergent and lacunary strongly almost convergent sequence spaces, and proved two inclusion theorems with respect to those spaces. In this paper, we shall extend those spaces to two new double sequence spaces and prove multidimensional analogues of Das and Patel’s results.

c

2007 Elsevier Ltd. All rights reserved.

Keywords:Lacunary double sequences; Almost lacunary sequences; P-convergent

1. Introduction and background

Let l∞and c be the Banach spaces of bounded and convergent sequences x =(xk) normed by ||x|| = supk|x_k|, respectively. A sequence x ∈ l∞is said to be almost convergent if and only if its Banach limits coincide. Let ˆcdenote the set of almost convergent sequences. Lorentz in [4] proved that

c =ˆ n

x ∈ l∞:lim

m tm,n(x) = exists, uniformly in, no where

tm,n(x) = x_n+x_n+ · · · +x_n+m

m +1 .

The space | ˆc|of strongly almost convergent sequences was introduced by Moddox [5] and also independently by Freedman et al. [3] as follows:

| ˆc| =n

x ∈ l∞:lim

m t_m,n(|x − Le|) = 0, uniformly in n for some Lo

where e = (1, 1, 1 . . .). By a lacunary θ = (kr); r = 0, 1, 2, . . . where k0 = 0, we shall mean an increasing sequence of non-negative integers with k_r−k_{r −1}→ ∞as r → ∞. The intervals determined byθ will be denoted by

∗Corresponding author. Fax: +90 432 2251415.

E-mail addresses:ekremsavas@yahoo.com(E. Savas¸),rpatters@unf.edu(R.F. Patterson).

0893-9659/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2006.09.008

I_r =(kr −1, kr]and h_r =k_r−kr −1. The ratio _k^kr

r −1 will be denoted by q_r. The space of lacunary strongly convergent sequences N_θ was defined by Freedman et al. [3] as follows:

N_θ = (

x = xk: lim

r →∞

1 h_r

X

k∈Ir

|xk−L| =0 for some L )

.

There is a strong connection between N_θand the space

|(C, 1)| = (

x = x_k : lim

n→∞

1 n

n

X

k=1

|x_k−L| =0 for some L )

.

Recently, Das and Mishra [1] introduced the space AC_θ of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences as follows:

AC_θ :=

(

x = x_k :lim

r

1 hr

X

k∈I_r

(xk+n−L) = 0, for some L, uniformly in n ≥ 0 )

,

and

|AC_θ| :=

(

x = x_k:lim

r

1 hr

X

k∈I_r

|x_k+n−L| =0, for some L, uniformly in n ≥ 0 )

.

If we takeθ = (2^r), the above spaces ACθ and | AC_θ|reduce to ˆcand | ˆc|, respectively. A few years later, Das and Patel in [2] presented inclusion theorems for those spaces.

Letω⁰⁰ denote the set of all double sequences of real numbers. By a bounded double sequence, we shall mean a positive number M exists such that |x_j,k|< M for all j and k. We will denote the set of all bounded double sequences by l⁰⁰_∞.

By the convergence of a double sequence we mean the convergence in the Pringsheim sense, that is, a double sequence x = (xk,l) has a Pringsheim limit L (denoted by P-lim x = L) provided that given  > 0 there exists N ∈N such that |xk,l−L|<  whenever k, l > N [9]. We shall describe such an x more briefly as “P-convergent”.

These notions were used to extend some known results from ordinary (i.e. single) sequences to double sequences by Mursaleen [6,7], Patterson [8], and others. The goal of this paper is to extend the notions of lacunary almost convergent and lacunary strongly almost convergent sequence spaces to double lacunary almost P-convergent and double lacunary strongly almost P-convergent sequence spaces. In addition, we shall also establish multidimensional analogues of Das and Patel’s results.

2. Main results

Definition 2.1. The double sequenceθr,s = {(kr, ls)} is called double lacunary if there exist two increasing of integers such that

k₀=0, hr =k_r−k_{r −1}→ ∞ as r → ∞ and

l₀=0, hs =l_s−l_s−1→ ∞ as s → ∞.

Notations: kr,s = krls, hr,s = hrhs,θr,s is determine by Ir,s = {(i, j) : kr −1 < i ≤ kr and ls−1 < j ≤ ls}. Also h¯r,s =krls−kr −1ls−1θ = ¯θr,sis determine by ¯Ir,s = {(i, j) : kr −1< i ≤ kr ∪ls−1< j ≤ ls} \(I¹∪I²) where

I¹=







kr ≤i < kr −1

(i, j) : and l_s < j < ∞







and

I²=







ls−1< j ≤ ls

(i, j) : and kr < i < ∞





 ,

with q_r = ^kr

kr −1, q_s = ^ls

ls−1, and q_r,s =q_rq_s. Throughout this paper, we will also use the following notations:

I_B¹ =







β ≤ j < β + n (i, j) : and

α + m < i < ∞





 ,

I_B² =







β + n < j < ∞ (i, j) : and

α ≤ i < α + m





 ,

B_α,β^m,n =







α ≤ i < α + m

(i, j) : or

β ≤ j < β + n







\(I_B¹∪I_B²),

I_A¹ =







β + (y + 1)hs < j < ∞

(i, j) : and

α + xhr ≤i ≤α + (x + 1)hr





 ,

I_A² =







β + yhs ≤ j ≤β + (y + 1)hs

(i, j) : and

α + (x + 1)hr < i < ∞





 ,

and

A^x_α,β^,y =







α + xhr ≤i < α + (x + 1)hr

(i, j) : or

β + yhs ≤i < β + (y + 1)hs







\(I¹_A∪I²_A).

Definition 2.2. The double sequence x is almost convergent if

AC⁰⁰:=

(

x ∈ω⁰⁰:P- lim

m,n

1 mn

m,n

X

i, j=1,1

(xi +r, j+s−L) = 0, for some L, uniformly in r, s ≥ 0 )

.

Definition 2.3. The double sequence x is strongly almost convergent if

|AC⁰⁰| :=

(

x ∈ω⁰⁰: P- lim

m,n

1 mn

m,n

X

i, j=1,1

|xi +r, j+s−L| =0, for some L, uniformly in r, s ≥ 0 )

.

We shall consider the following set.

|AC_θ^∗| :=







x ∈ω⁰⁰:P- lim

r,s

1 h_r,s

X

(i, j)∈Ir,s

|xi +r, j+s−L| =0, for some L, uniformly in r, s ≥ 0





 .

|AC_θ^∗| ⇔ |AC⁰⁰|is among the first conjectures we shall prove. Obviously, this conjecture is false in both directions.

Let us establish that | AC_θ^∗| ⇒ |AC⁰⁰|is false. Ifθ is such that {kr}and {ls}are both even integers and x is defined as follows:

x_k,l =

(kl)² if l = 1 and k = 1, 2, 3, . . . , 0 if otherwise

 .

It is clear that x ∈ | AC_θ^∗|and x 6∈ | AC⁰⁰|. Likewise it is also clear that | AC_θ^∗| 6⇐ |AC⁰⁰|. Now let us consider the following sequence spaces. Note that h_r,s and I_r,s are replaced with ¯h_r,sand ¯I_r,srespectively.

Definition 2.4. The double sequence x is lacunary almost convergent if

AC_θ⁰⁰:=







x ∈ω⁰⁰:P- lim

r,s

1 h¯r,s

X

(i, j)∈ ¯Ir,s

(xi +r, j+s−L) = 0, for some L uniformly in r, s ≥ 0





 .

Definition 2.5. The double sequence x is strongly lacunary almost convergent if

|AC_θ⁰⁰| :=







x ∈ω⁰⁰: P- lim

r,s

1 h¯_r,s

X

(i, j)∈ ¯Ir,s

|x_{i +r, j+s}−L| =0, for some L uniformly in r, s ≥ 0





 .

We shall now establish multidimensional analogues of Das and Patel’s results in [2].

Lemma 2.1. Suppose > 0 there exist m0, n₀, p₀, and q₀such that 1

|B^m,n_p,q| X

(i, j)∈B^m,np,q

|x_{i, j}−L|<  (2.1)

for m, n ≥ m0, n0and p, q ≥ p0, q0then x ∈ | AC⁰⁰|.

Proof. Let > 0 be given and choose m¹₀, n¹₀, p₀, and q₀such that 1

|B^m_p,q^,n| X

(i, j)∈B^m,np,q

|x_{i, j}−L|<

2 (2.2)

for all m ≥ m¹₀, n ≥ n¹₀, p ≥ p₀, and q ≥ q₀. We need only to show that given > 0 there exist m¹₀^,1and n¹₀^,1 such that

1

|B^m_p,q^,n| X

(i, j)∈B^m,np,q

|x_{i, j}−L|<  (2.3)

for m ≥ m¹₀^,1, n ≥ n¹₀^,1 and (p, q) ∈ D where D = {(i, j) : 0 ≤ i ≤ p0and 0 ≤ j ≤ q₀}. Since, taking m₀ =max{m¹₀, m¹₀^,1}and n₀ =max{n¹₀, n¹₀^,1},(2.3)will hold for m ≥ m₀, n ≥ n₀and for all p and q, which gives the results. Once p₀and q₀have been chosen, they are fixed. We have the following:

X

{(i, j):p≤i<p0∪q≤ j<q0}

|xi, j−L| = B (2.4)

is finite. Thus(p, q) ∈ D implies that p ≤ p0and q ≤ q₀. So for m ≥ p₀, and n ≥ q₀we obtain the following by (2.2):

1

|B^m,n_p,q| X

(i, j)∈B^m,np,q

|x_{i, j}−L| = 1

|B^m,n_p,q|

X

(i, j):p≤i<p0∪q≤ j<q0

|x_{i, j}−L| + 1

|B^m,n_p,q| X

(i, j)∈B^m,n_p0,q0

|x_{i, j}−L|

≤ B

|B^m,n_p,q| + 2.

Therefore taking m and n sufficiently large, we can make B

|B^m,n_p,q| + 2 < ε,

which gives(2.3)and hence result. 

Theorem 2.1. | AC_θ⁰⁰| ⇔ |AC⁰⁰|for every ¯θr,s.

Proof. Let x ∈ | AC_θ⁰⁰|; then given > 0 there exist r0, s₀, and L such that 1

h¯_r,s X

(i, j)∈B^hrp,q^,hs

|x_{i, j}−L|<  (2.5)

whenever r ≥ r₀, s ≥ s₀, p = k_{r −1}+1 +α, and q = ls−1+1 +β where α, β ≥ 0. Let m ≥ hr such that m =δ1h_r+θ1whereδ1is an integer. Also let n ≥ h_ssuch that n =δ2h_s+θ2whereδ2is an integer and 0 ≤θ1≤h_r and 0 ≤θ2≤h_s. Since m ≥ h_r forδ1≥1 and n ≥ h_s forδ2≥1 we have

1

|B^m_p,q^,n| X

(i, j)∈B^mp,q^,n

|x_{i, j}−L| ≤ 1

|B^m_p,q^,n|

X

(i, j)∈B(δ1+1)hr ,(δ2+1)hs p,q

|x_{i, j}−L|

= 1

|B^m_p,q^,n|

δ1,δ2

X

x,y=1,1

X

(i, j)∈A^xp^,y,q

|x_{i, j}−L|

≤ 1

|B^m_p,q^,n|

δ1,δ2

X

x,y=1,1

h¯_r,s

≤ (δ1+1)(δ2+1)

|B^mp,q^,n|

h¯r,s

=o(1), which gives the result.

Therefore byLemma 2.1, | AC⁰⁰_θ| ⇒ |AC⁰⁰|. It is clear that | AC⁰⁰| ⇒ |AC⁰⁰_θ|for every ¯θr,s. This completes the proof of this theorem. 

Lemma 2.2. Suppose > 0; then there exist m0, n₀, p₀, and q₀such that 1

|B^mp,q^,n|      

X

(i, j)∈B^mp,q^,n

xi, j−L      

< 

for all m, n ≥ m0, n0and p, q ≥ p0, q0, then x ∈ AC⁰⁰.

Proof. Let > 0 be given and choose m0, n₀, p₀, and q₀such that 1

|B^mp,q^,n|      

X

(i, j)∈B^mp,q^,n

xi, j−L      

< 

2 (2.6)

for all m, n ≥ m0, n0and p, q ≥ p0, q0. As inLemma 2.1it suffices to show that there exist m¹₀, n¹₀such that for m ≥ m¹₀implies

1

|B^m_p,q^,n|      

X

(i, j)∈B^mp,q^,n

x_{i, j}−L      

<  < 1, (2.7)

for all p and q with n ≥ n¹₀, 0 ≤ p ≤ p₀, and 0 ≤ q ≤ q₀. Since p₀and q₀are fixed, the following is finite X

{(i, j):0≤i<p0∪0≤ j<q0}

|xi, j−L| = B. (2.8)

Let 0 ≤ p ≤ p0, 0 ≤ q ≤ q0, and m, n > p0, q0and consider the following:

1

|B^mp,q^,n|      

X

(i, j)∈B^mp,q^,n

xi, j−L      

≤ 1

|B^mp,q^,n|

X

{(i, j):p≤i<p0∪q≤ j<q0}

|xi, j−L| + 1

|B^mp,q^,n|       

X

(i, j)∈B_p0,q0^p+m,q+n

xi, j−L       

= B

|B^m,n_p,q|+ 1

|B^m,n_p,q|       

X

B_p0,q0^p+m,q+n

x_{i, j}−L       

. (2.9)

Let m − p₀ ≥m¹₀, then m + p − p₀> m¹₀for 0 ≤ p< p0. Also, if we let n − q₀≥n¹₀, then n + q − q₀> n¹₀for 0 ≤ q< q0. Therefore(2.6)grants us the following:

1

|B_p^p+m_0,q0^,q+n|       

X

B_p0,q0^p+m,q+n

x_{i, j}−L       

< 

2. (2.10)

From(2.9)and(2.10) 1

|B^m_p,q^,n|      

X

(i, j)∈B^mp,q^,n

x_{i, j}−L      

≤ B

|B^m_p,q^,n|+ 1

|B^m_p,q^,n||B_p^p+m,q+n_0,q0 | 2

=o(1),

for sufficiently large m and n. Hence the results follow.  Theorem 2.2. (1) For some ¯θr,s, AC_θ⁰⁰; l∞⁰⁰ .

(2) For every ¯θr,s, AC⁰⁰_θ ∩l_∞⁰⁰ ⇔AC⁰⁰.

Proof. To establish the first part of this theorem, we only need to show that AC⁰⁰; l⁰⁰∞when k_r and l_s are even for r and s. Let us consider the following double sequence

x_k,l:=

(−1)^k+l(kl)^λ, if k = l;

0, if otherwise

where, 0< λ < 1. Thus the following series will contain an even number of terms X

(i, j)∈B^hrp,q^,hs

x_{i, j} ≥0

where p, q ≥ 0. Note that the sum of the terms is even in each block and is of order O(kl)^λ−1. It now follows that x ∈ AC_θ⁰⁰where L = 0. However x 6∈ l⁰⁰_∞. This completes the proof of Part (1).

Let us establish part (2). Let x ∈ AC_θ⁰⁰∩l⁰⁰_∞then for > 0 there exist r0, s₀, p₀, and q₀such that

1 h¯r,s

X

(i, j)∈B^hrp,q^,hs

xi, j−L       

< 

2 (2.11)

for r ≥ r0, s ≥ s0, p ≥ p0, and q ≥ q0with p = kr −1+1 +α where α ≥ 0, q = ls−1+1 +β and β ≥ 0. Let m ≥ h_r and n ≥ h_s where m and n are integers greater that or equal to 1; then

1

|B^m_p,q^,n|      

X

(i, j)∈B^m,np,q

x_{i, j}−L      

≤ 1

|B^m_p,q^,n|

δ1−1,δ2−1

X

x,y=0,0

X

(i, j)∈A^x,yp,q

x_{i, j}−L      

+ 1

|B^m_p,q^,n|

X

(i, j)∈B^mp,q^,n\A^x,yp,q

|x_{i, j}−L|. (2.12)

Since x ∈ l_∞⁰⁰ for all i and j , there exists B such that |x_{i, j}−L|< B. From (2.12) we have the following:

1

|B^mp,q^,n|      

X

(i, j)∈B^m,np,q

xi, j−L      

≤ 1

|B^mp,q^,n|(δ1)(δ2) 1 h¯_r,s



2 + B ¯h_r,s

|B^mp,q^,n|. Thus for m and n sufficiently large, we are granted the following:

1

|B^m_p,q^,n|      

X

(i, j)∈B^mp,q^,n

x_{i, j}−L      

<  for r ≥ r0, s ≥ s0, p ≥ p0, and q ≥ q0.

Thus, byLemma 2.2, we have AC_θ⁰⁰∩l_∞⁰⁰ ⇒AC⁰⁰. It is clear that AC⁰⁰⇒ AC_θ⁰⁰∩l_∞⁰⁰ . This completes the proof.  References

[1] G. Das, S. Mishra, Banach limits and lacunary strong almost convergent, J. Orissa Math. Soc. 2 (2) (1983) 61–70.

[2] G. Das, B.K. Patel, Lacunary distribution of sequences, Indian J. Pure Appl. Math. 26 (1) (1989) 64–74.

[3] A.R. Freedman, J.J. Sember, M. Raphael, Some Ces`aro type summability spaces, Proc. London Math. Soc. 37 (1978) 508–520.

[4] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948) 167–190.

[5] I.J. Moddox, On strong almost convergent, Math. Proc. Cambridge Philos. Soc. 85 (2) (1979) 343–350.

[6] Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (1) (2003) 223–231.

[7] Mursaleen, O.H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2004) 532–540.

[8] R.F. Patterson, Analogues of some fundamental theorems of summability theory, Int. J. Math. Math. Sci. 23 (1) (2000) 1–9.

[9] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900) 289–321.

Further reading

[1] E. Savas¸, R.F. Patterson, On some double almost lacunary sequence spaces defined by Orlicz functions, Filomat 19 (2005) 35–44.

[2] E. Savas¸, V. Karaya, R.F. Patterson, Inclusion theorems for double lacunary sequence space, Acta Sci. Math. (Szeged) 20 (2005) 63–73.

[3] E. Savas¸, R.F. Patterson, Lacunary statistical convergence of multiple sequence, Appl. Math. Lett. 19 (6) (2006) 527–534.

[4] R.F. Patterson, E. Savas¸, Lacunary statistical convergence of double sequences, Math. Comm. 10 (1) (2005) 55–61.