Applied Mathematics Letters 20 (2007) 964–970 www.elsevier.com/locate/aml
Double sequence spaces characterized by lacunary sequences Ekrem Savas¸ a,∗ , Richard F. Patterson b a Istanbul ˙ ¨ udar, Istanbul, ˙ Ticaret University, Department of Mathematics, Usk¨ Turkey b Department 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 |xk |, respectively. A sequence x ∈ l∞ is said to be almost convergent if and only if its Banach limits coincide. Let cˆ denote the set of almost convergent sequences. Lorentz in [4] proved that n o cˆ = x ∈ l∞ : lim tm,n (x) = exists, uniformly in, n m
where tm,n (x) =
xn + xn + · · · + xn+m . m+1
The space |c| ˆ of strongly almost convergent sequences was introduced by Moddox [5] and also independently by Freedman et al. [3] as follows: n o |c| ˆ = x ∈ l∞ : lim tm,n (|x − Le|) = 0, uniformly in n for some L m
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 kr − kr −1 → ∞ as r → ∞. The intervals determined by θ will be denoted by ∗ Corresponding author. Fax: +90 432 2251415.
E-mail addresses:
[email protected] (E. Savas¸),
[email protected] (R.F. Patterson). c 2007 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.09.008
E. Savas¸, R.F. Patterson / Applied Mathematics Letters 20 (2007) 964–970
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r Ir = (kr −1 , kr ] and h r = kr − kr −1 . The ratio krk−1 will be denoted by qr . The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al. [3] as follows: ) ( 1 X |xk − L| = 0 for some L . Nθ = x = xk : lim r →∞ h r k∈I r
There is a strong connection between Nθ and the space ( ) n 1X |(C, 1)| = x = xk : lim |xk − L| = 0 for some L . n→∞ n k=1 Recently, Das and Mishra [1] introduced the space ACθ of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences as follows: ) ( 1 X (xk+n − L) = 0, for some L , uniformly in n ≥ 0 , ACθ := x = xk : lim r hr k∈I r
and (
) 1 X |ACθ | := x = xk : lim |xk+n − L| = 0, for some L , uniformly in n ≥ 0 . r hr k∈I r
(2r ),
If we take θ = the above spaces ACθ and |ACθ | reduce to cˆ and |c|, ˆ respectively. A few years later, Das and Patel in [2] presented inclusion theorems for those spaces. Let ω00 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 00 . by l∞ 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 sequence θr,s = {(kr , ls )} is called double lacunary if there exist two increasing of integers such that k0 = 0,
h r = kr − kr −1 → ∞ as r → ∞
l0 = 0,
h s = ls − ls−1 → ∞ as s → ∞.
and
Notations: kr,s = kr ls , h r,s = h r h s , θr,s is determine by Ir,s = {(i, j) : kr −1 < i ≤ kr and ls−1 < j ≤ ls }. Also h¯ r,s = kr ls − kr −1ls−1 θ = θ¯r,s is determine by I¯r,s = {(i, j) : kr −1 < i ≤ kr ∪ ls−1 < j ≤ ls } \ (I 1 ∪ I 2 ) where kr ≤ i < kr −1 and I 1 = (i, j) : ls < j < ∞
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and ls−1 < j ≤ ls and I 2 = (i, j) : , kr < i < ∞
kr kr −1 , qs
ls ls−1 ,
and qr,s = qr qs . Throughout this paper, we will also use the following notations: β ≤ j n 10 for 0 ≤ q < q0 . Therefore (2.6) grants us the following: X 1 x − L (2.10) < . i, j p+m,q+n 2 |B p0 ,q0 | p+m,q+n B p0 ,q0 From (2.9) and (2.10) 1 X B 1 p+m,q+n xi, j − L ≤ | m,n + m,n |B p0 ,q0 2 |B m,n | |B | |B | p,q (i, j)∈B m,n p,q p,q p,q = o(1), for sufficiently large m and n. Hence the results follow.
00 . Theorem 2.2. (1) For some θ¯r,s , ACθ00 ; l∞ 00 ⇔ AC 00 . (2) For every θ¯r,s , ACθ00 ∩ l∞ 00 when k and l are even for r Proof. To establish the first part of this theorem, we only need to show that AC 00 ; l∞ r s and s. Let us consider the following double sequence (−1)k+l (kl)λ , if k = l; xk,l := 0, if otherwise
where, 0 < λ < 1. Thus the following series will contain an even number of terms X xi, j ≥ 0 r ,h s (i, j)∈B hp,q
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 00 . This completes the proof of Part (1). x ∈ ACθ00 where L = 0. However x 6∈ l∞ 00 then for > 0 there exist r , s , p , and q such that Let us establish part (2). Let x ∈ ACθ00 ∩ l∞ 0 0 0 0 1 X xi, j − L < (2.11) 2 h¯ r,s h r ,h s (i, j)∈B p,q for r ≥ r0 , s ≥ s0 , p ≥ p0 , and q ≥ q0 with 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 δ1 −1,δ X2 −1 X X 1 1 X + 1 x − L ≤ x − L |xi, j − L|. (2.12) i, j i, j m,n m,n m,n |B p,q | |B p,q | x,y=0,0 |B p,q | x,y x,y m,n (i, j)∈B m,n (i, j)∈A (i, j)∈B \A p,q p,q p,q p,q
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00 for all i and j, there exists B such that |x Since x ∈ l∞ i, j − L| < B. From (2.12) we have the following: ¯ 1 X ≤ 1 (δ1 )(δ2 ) 1 + B h r,s . x − L i, j m,n m,n m,n |B p,q | h¯ r,s 2 |B p,q | m,n |B p,q | (i, j)∈B p,q
Thus for m and n sufficiently large, we are granted the following: 1 X < for r ≥ r0 , s ≥ s0 , p ≥ p0 , and q ≥ q0 . x − L i, j m,n |B p,q | (i, j)∈B m,n p,q 00 ⇒ AC 00 . It is clear that AC 00 ⇒ AC 00 ∩ l 00 . This completes the proof. Thus, by Lemma 2.2, we have ACθ00 ∩ l∞ ∞ θ
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
G. Das, S. Mishra, Banach limits and lacunary strong almost convergent, J. Orissa Math. Soc. 2 (2) (1983) 61–70. G. Das, B.K. Patel, Lacunary distribution of sequences, Indian J. Pure Appl. Math. 26 (1) (1989) 64–74. A.R. Freedman, J.J. Sember, M. Raphael, Some Ces`aro type summability spaces, Proc. London Math. Soc. 37 (1978) 508–520. G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948) 167–190. I.J. Moddox, On strong almost convergent, Math. Proc. Cambridge Philos. Soc. 85 (2) (1979) 343–350. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (1) (2003) 223–231. Mursaleen, O.H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2004) 532–540. R.F. Patterson, Analogues of some fundamental theorems of summability theory, Int. J. Math. Math. Sci. 23 (1) (2000) 1–9. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900) 289–321.
Further reading [1] [2] [3] [4]
E. Savas¸, R.F. Patterson, On some double almost lacunary sequence spaces defined by Orlicz functions, Filomat 19 (2005) 35–44. E. Savas¸, V. Karaya, R.F. Patterson, Inclusion theorems for double lacunary sequence space, Acta Sci. Math. (Szeged) 20 (2005) 63–73. E. Savas¸, R.F. Patterson, Lacunary statistical convergence of multiple sequence, Appl. Math. Lett. 19 (6) (2006) 527–534. R.F. Patterson, E. Savas¸, Lacunary statistical convergence of double sequences, Math. Comm. 10 (1) (2005) 55–61.
