Ricerche mat. (2011) 60:299–311 DOI 10.1007/s11587-011-0106-0
On asymptotically equivalent difference sequences with respect to a modulus function Metin Ba¸sarir · Selma Altunda˘g
Received: 17 November 2009 / Revised: 28 May 2010 / Published online: 9 March 2011 © Università degli Studi di Napoli “Federico II” 2011
Abstract This paper presents new definitions which are a natural combination of the definition for asymptotically equivalence and m -lacunary strongly summable with respect to a modulus f . Using this definitions we have proved the ( f, m )-asymptotically equivalence and m -lacunary statistical asymptotically equivalence analogues of theorems of Tripathy and Et (Stud Univ Babe¸s-Bolyai Math (1):119–130, 2005) and Çolak’s theorems (Filomat 17:9–14, 2003). Keywords Difference sequence · Modulus function · Asymptotically equivalence · Lacunary sequence Mathematics Subject Classification (2000)
40D25 · 40B05 · 46A45
1 Introduction Let w be the set of all sequences of real or complex numbers and l∞ , c and c0 be, respectively, the Banach spaces of bounded, convergent and null sequences x = (xk ) with the usual norm x = sup|xk |. k
Communicated by Editor in Chief. M. Ba¸sarir (B) · S. Altunda˘g Department of Mathematics, Faculty of Science and Arts, Sakarya University, 54187 Sakarya, Turkey e-mail:
[email protected] S. Altunda˘g e-mail:
[email protected]
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Ruckle [2] used the idea of a modulus function f to construct a class of FK spaces L( f ) = x = (xk )w :
∞
f (|xk |) < ∞ .
k=1
The space L( f ) is closely related to the space l1 which is a L( f ) space with f (x) = x for all real x ≥ 0. Furthermore, modulus function has been discussed in [3–7] and many others. The difference sequence space X () was introduced by Kızmaz [8] as follows: X () = {x = (xk ) ∈ w : (xk ) ∈ X } for X = l∞ , c and c0 ; where xk = xk − xk+1 for all k ∈ N. The notion of difference sequence spaces was further generalized by Et and Çolak [9] as follows: X (m ) = x = (xk ) ∈ w : m xk ∈ X for X = l∞ , c and c0 ; where m xk = m−1 xk − m−1 xk+1 and 0 xk = xk for all k ∈ N. Taking X = l∞ ( p), c( p) and c0 ( p), these sequence spaces have been generalized by Et and Ba¸sarır [10]. The generalized difference has the following binomial representation:
m v m xk+v (−1) xk = v m
v=0
for all k ∈ N. Subsequently, difference sequence spaces have been discussed by several authors [11–14]. The idea of statistical convergence was introduced by Fast [16] and studied by various authors (see [17–19]). In [1], using lacunary sequence θ, the space of lacunary strongly convergent sequences Nθ was defined as follows: Nθ =
⎧ ⎨ ⎩
x = (xi )w : lim h r−1 r →∞
i∈Ir
|xi − s| = 0 for some s
⎫ ⎬ ⎭
.
In 1993, Marouf [20] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson [21] extended these concepts by presenting an asymptotically statistically equivalent analogues of these definitions and natural regularity conditions for nonnegative summability matrices. Furthermore, asymptotically equivalent sequences have been studied in [22–33].
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This paper presents the concept of m -asymptotically equivalence with respect to a modulus function f . In addition to this concept, some connections between m lacunary statistical asymptotically equivalence and ( f, m )-lacunary strongly asymptotically equivalence have also been presented. 2 Definitions and notations Now, we give some concepts which are used throughout the paper. Definition 1 [1] By a lacunary θ = (kr ); r = 0, 1, 2, . . . where k0 = 0, we shall mean an increasing sequence of nonnegative integers with kr − kr −1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr −1 , kr ] and h r = kr − kr −1 . The r will be denoted by qr . ratio krk−1 Definition 2 [2] A modulus function f is a function from [0, ∞) to [0, ∞) such that (i) (ii) (iii) (iv)
f (x) = 0 if and only if x = 0, f (x + y) ≤ f (x) + f (y) for all x, y ≥ 0, f increasing, f is continuous from at the right zero.
Since | f (x) − f (y)| ≤ f (|x − y|), it follows from condition (iv) that f is continuous on [0, ∞). Furthermore, we have f (nx) ≤ n f (x) for all n ∈ N, from condition (ii) and so
x 1 ≤ nf . f (x) = f nx n n Hence, for all n ∈ N x 1 f (x) ≤ f . n n A modulus may be bounded or unbounded. For example, f (x) = x p , for 0 < p ≤ 1 x is bounded. is unbounded, but f (x) = 1+x Definition 3 [16] A sequence x = (xk ) is said to be statistically convergent to the number L if for every ε > 0 1 μ({k ≤ n : |xk − L| ≥ ε}) = 0 n→∞ n lim
where μ({k ≤ n : |xk − L| ≥ ε}t) denotes the number of element belonging to {k ≤ n : |xk − L| ≥ ε}. In this case, we write S − lim x = L xk → L(S) and S denotes the set of all statistically convergent sequences. We give some definitions about asymptotically equivalent sequences.
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Definition 4 [20] Two nonnegative sequences x, y are said to be asymptotically equivalent if lim k
xk =1 yk
(denoted by x ∼ y). Definition 5 [24] Two nonnegative sequences x, y are said to be m -asymptotically equivalent if lim k
m x k =1 m yk
m
(denoted by x ∼ y). Definition 6 Let f be any modulus. The two nonnegative sequences x, y are said to be m -strongly asymptotically equivalent of multiple L with respect to a modulus f or ( f, m )-strongly asymptotically equivalent of multiple L provided that
m xk lim f m − L = 0 k yk (denoted by x L = 1.
( f L ,m )
∼
y) and simply ( f, m )-strongly asymptotically equivalent, if
m xk Since f is continuous and f (x) = 0 if and only if x = 0, then lim f m y − 1 = k k
m m xk xk m xk f lim − 1 = 0 if and only if lim m yk m yk − 1 = 0 i.e. lim m yk = 1. Therek
k
m
fore x ∼ y ⇐⇒ x
( f,m )
∼
k
y.
Definition 7 [24] Two nonnegative sequences x, y are m -strongly Cesaro asymptotically equivalent of multiple L provided that n 1 m xk lim m y − L = 0 n→∞ n k k=1
(denoted by x if L = 1.
|σ1 | L (m )
∼
y) and simply m -strongly Cesaro asymptotically equivalent,
Definition 8 [24] Let θ be a lacunary sequence. Two nonnegative sequences x, y are m -lacunary strongly asymptotically equivalent of multiple L provided that 1 m xk lim m y − L = 0 r →∞ h r k k∈Ir
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On asymptotically equivalent difference sequences
(denoted by x if L = 1.
NθL (m )
∼
303
y) and simply m -lacunary strongly asymptotically equivalent,
Definition 9 Two nonnegative sequences x, y are m -strongly almost asymptotically equivalent of multiple L provided that n 1 m xk+s =0 − L m n→∞ n yk+s lim
k=1
uniformly in s (denoted by x ically equivalent, if L = 1.
|AC| L (m )
∼
y) and simply m -strongly almost asymptot-
Definition 10 Let f be any modulus function. Two nonnegative sequences x, y are ( f, m )-strongly Cesaro asymptotically equivalent of multiple L provided that 1 lim f n→∞ n n
k=1
|σ1 | L ( f,m )
(denoted by x ∼ equivalent, if L = 1.
m xk m y − L = 0 k
y) and simply ( f, m )-strongly Cesaro asymptotically
Definition 11 Let f be any modulus function and θ be a lacunary sequence. Two nonnegative sequences x, y are ( f, m )-lacunary strongly asymptotically equivalent of multiple L provided that
m xk 1 f m − L = 0 lim r →∞ h r yk k∈Ir
NθL ( f,m )
(denoted by x ∼ alent, if L = 1.
y) and simply f, m )-lacunary strongly asymptotically equiv-
Definition 12 Let f be any modulus function. Two nonnegative sequences x, y are ( f, m )-strongly almost asymptotically equivalent of multiple L provided that 1 lim f n→∞ n k=1
m xk+s − L = 0 m y k+s
|AC| L ( f,m )
uniformly in s (denoted by x ∼ y) and simply ( f, m )-strongly almost asymptotically equivalent, if L = 1. In Definitions 10–12, if we take f (x) = x then we have Definitions 7–9, respectively.
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Definition 13 Let f be any modulus function, θ be a lacunary sequence and p = ( pk ) be any sequence of strictly positive real numbers. Two nonnegative sequences x, y are ( f, m , p)-lacunary strongly asymptotically equivalent of multiple L provided that pk m xk 1 f m − L lim =0 r →∞ h r yk k∈Ir
NθL ( f,m , p)
∼ (denoted by x equivalent, if L = 1.
y) and simply ( f, m , p)-lacunary strongly asymptotically
Definition 14 [21] Two nonnegative sequences x, y are said to be statistical asymptotically equivalent of multiple L provided that for every ε > 0 1 lim μ n→∞ n
k≤n
xk =0 : − L ≥ ε yk
SL
(denoted by x ∼ y) and simply statistical asymptotically equivalent, if L = 1. Definition 15 [24] Two nonnegative sequences x, y are m -statistical asymptotically equivalent of multiple L provided that for every ε > 0 1 μ n→∞ n
lim
(denoted by x
S L (m )
∼
m
xk =0 k ≤ n : m − L ≥ ε yk
y) and simply m -statistical asymptotically equivalent, if L = 1.
Definition 16 [24] Let θ be a lacunary sequence. Two nonnegative sequences x, y are m -lacunary statistical asymptotically equivalent of multiple L provided that for every ε > 0 1 μ lim r →∞ h r (denoted by x if L = 1.
SθL (m )
∼
k ∈ Ir
m
xk =0 : m − L ≥ ε yk
y) and simply m -lacunary statistical asymptotically equivalent,
3 Main results In [5, Lemma], if we take
x y
in place of x,we have the following lemma.
Lemma 1 Let f be a modulus function and let 0 < δ < 1. Then for y = 0 and each ( xy ) > δ we have f ( xy ) ≤ 2 f (1)δ −1 ( xy ).
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Theorem 1 Let f be a modulus function. Then x Proof Let x
NθL (m )
∼
τr =
NθL (m )
∼
y implies x
NθL ( f,m )
∼
y.
y. Then we have
h r−1
m xk m y − L → 0 as r → ∞, for some L. k
k∈Ir
Let ε > 0 and choose δ with 0 < δ < 1 such that f (u) < ε for u with 0 ≤ u ≤ δ. Then we can write h r−1
k∈Ir
m xk f m − L = h r−1 yk
m k∈Ir xk m y −L ≤δ k
+h r−1
m xk f m − L yk
f
m k∈Ir xk m y −L >δ
m xk − L m y k
k
≤ from Lemma 1. Therefore x
NθL ( f,m )
∼
h r−1 (h r ε) + h r−1 2 f (1)δ −1 h r τr
y. f (t) t→∞ t
Theorem 2 Let f be a modulus function. If lim
x
NθL ( f,m )
∼
y ⇐⇒ x
Proof By Theorem 1 we need only show that x β > 0 and x have h r−1
k∈Ir
f
NθL ( f,m )
∼
= β > 0, then
NθL (m )
∼
y.
NθL ( f,m )
∼
y implies x
NθL (m )
∼
y. Let
y. Since β > 0, we have f (t) ≥ βt for all t ≥ 0. Hence we
m m xk m xk xk ≥ h −1 = βh −1 . − L β − L − L r r m y m y m y k k k
Therefore we have x
k∈Ir
NθL (m )
∼
y.
k∈Ir
In Theorem 2, the condition β > 0 cannot be omitted. For this, consider the following example.
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Example 1 Let f (x) = ln(1 + x). Then β = 0. Define m xk to be h r + 1 at the (kr −1 + 1)th term in Ir for every r ≥ 1 and xi = 0 otherwise and yi =
α − (i − 1), (−1)m−1 (m−1)! (i − 2)(i − 3) . . . (i − m)α +
(−1)m (m)! (i
m =1 − 1)(i − 2) . . . (i − m), m > 1
where α ∈ C. Note that x is not mm -bounded. Then we have xk −1 −1 ln(1 + h ) → 0, as r → ∞ = h f − 1 hr r r k∈Ir m yk and so x
Nθ ( f,m )
∼
y, but
h r−1
m xk = h −1 h r → 1, as r → ∞ − 1 r m y k
k∈Ir
and so x
Nθ (m )
y.
Theorem 3 Let θ = (kr ) be a lacunary sequence. If 1 < lim inf qr ≤ lim supqr < ∞ r
r
then x
|σ1 | L ( f,m )
∼
y ⇐⇒ x
NθL ( f,m )
∼
y.
|σ1 | L ( f,m )
Proof Let x ∼ y and suppose that lim inf qr > 1. Then there exits a δ > 0 r such that qr = (kr /kr −1 ) ≥ 1 + δ for sufficiently large r. Since h r = kr − kr −1 , we have hkrr ≤ 1+δ δ . Now, we write kr−1
m kr m xi m xi xi hr f m − L ≥ kr−1 f m − L = h r−1 f m − L . yi yi kr yi i=1
i∈Ir
Hence x Let x
|σ1 | L ( f,m )
∼
NθL ( f,m )
∼
y ⇒ x
i∈Ir
NθL ( f,m )
∼
y for any modulus f .
y. Now suppose that lim supqr < ∞ and let ε > 0 be given. Then
there exits j0 such that for every j ≥ j0 Hj =
h −1 j
r
m xi f m − L < ε. yi
i∈I j
We can also find M > 0 such that H j ≤ M for all j. If lim supqr < ∞ then there exists B > 0 such that qr < B for every r .
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Now let n be any integer with kr −1 < n ≤ kr . Then 1 f n n
i=1
m m kr xi xi −1 f m − L m y − L ≤ kr −1 yi i i=1 ⎧ ⎫
⎬ ⎨ m x m xi i −1 f m − L + · · · + f m − L = kr −1 ⎩ ⎭ yi yi i∈I1 i∈Ir ⎧ ⎫
⎬ j0 m r ⎨ m xi xi −1 f m − L + f m − L = kr −1 ⎩ ⎭ yi yi j= j0 +1i∈I j
j=1i∈I j
m xi + ε kr − k j k −1 f − L ≤ kr−1 0 −1 r −1 m y i j=1i∈I j −1 = kr−1 −1 h 1 H1 + h 2 H2 + · · · + h j0 H j0 + ε kr − k j0 kr −1 −1 −1 ≤ kr−1 −1 ( sup Hi )k j0 + ε kr − k j0 kr −1 < Mkr −1 k j0 + ε B j0
1≤i≤ j0
which yields that x
|σ1 | L ( f,m )
∼
y.
If we take f (x) = x, then we have the following corollary. Corollary 1 Let θ = (kr ) be a lacunary sequence. If 1 < lim inf qr ≤ lim sup qr < ∞ r then x
|σ1 | L (m )
∼
y ⇐⇒ x
NθL (m )
∼
y.
Theorem 4 For every lacunary θ = (kr ) x
|AC| L ( f,m )
∼
y ⇒ x
NθL ( f,m )
∼
y.
|AC| L ( f,m )
y. Then for ε > 0 there exists N > 0 and L such that
m xk+s 1 f m − L < ε for n > N , s = 0, 1, 2, . . . . n yk+s
Proof Let x
∼
k=1
Since θ is lacunary, we can choose R > 0 such that r ≥ R implies h r > N and NθL ( f,m ) m xk consequently τr = h r−1 f m yk − L < ε. Thus x ∼ y.
k∈Ir
If we take f (x) = x, then we have the following corollary. Corollary 2 For every lacunary θ = (kr ) x
|AC| L (m )
∼
y ⇒ x
NθL (m )
∼
y.
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Theorem 5 Let 0 < pk ≤ tk and
x
NθL ( f,m ,t)
∼
tk pk
be bounded. Then
y ⇒ x
NθL ( f,m , p)
∼
y.
tk m xk Proof If we take wk = f − L for all k, using the technique applied for m yk establishing Theorem 5 of Maddox [15], we can easily prove the theorem.
Theorem 6 Let f be a modulus function and sup pk = H . Then k
x Proof Let x
NθL ( f,m , p)
∼
NθL ( f,m , p)
∼
y ⇒ x
SθL (m )
∼
y.
y and ε > 0 be given. Then
pk m xk 1 1 f m − L = hr yk hr k∈Ir
k∈Ir m xk m y −L ≥ε k
1 + hr
k∈Ir m xk m y −L 0 be given. Then pk m xk 1 1 f m − L = hr yk hr k∈Ir
k∈Ir m xk m y −L ≥ε k
1 + hr
pk m xk f m − L yk
k∈Ir m xk m y −L 1
m √ xk n ≤ : m − 1 ≥ ε yk n
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M. Ba¸sarir, S. Altunda˘g Sθ (m )
for all n and so x ∼
This contradicts to x
y, but x
SθL (m )
∼
y
Nθ ( f,m , p)
y for θ = (kr ) and pk = 1 for all k ∈ N.
⇐⇒
x
NθL ( f,m , p)
∼
y .
Acknowledgments We would like to express our gratitude to the reviewer for his/her careful reading and valuable suggestions which is improved the presentation of the paper.
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