Proyecciones Journal of Mathematics Vol. 29, No 1, pp. 1-8, May 2010. Universidad Cat´olica del Norte Antofagasta - Chile
SOME DIFFERENCE SEQUENCES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS AHMAD H. A. BATAINEH and IBRAHIM M. A. SULAIMAN AL AL-BAYT UNIVERSITY, JORDAN Received : November 2009. Accepted : January 2010
Abstract The idea of difference sequence spaces was introduced by Kizmaz [6], and this concept was generalized by Bektas and Colak [1]. In this m paper, we define the sequence spaces c0 (F, ∆m u x) and l∞ (F, ∆u x), where F = (fk ) is a sequence of modulus functions, and examine some inclusion relations and properties of these spaces. Subjclass [2000] : 40A05, 40C05,46A45. Keywords : Difference sequence spaces, A sequence of modulus functions.
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Ahmad H. A. Bataineh and M. A. Sulaiman
1. Definitions and notations Let w denote the set of all complex sequences x = (xk ), and l∞ , c, and c0 be the linear spaces of bounded, convergent, and null sequences with complex terms, respectively, normed by k x k= supk | xk |, where k ∈ N, the set of positive integers. Kizmaz [6] defined the sequence spaces l∞ (∆) = {x ∈ w : ∆x ∈ l∞ }, c (∆) = {x ∈ w : ∆x ∈ c}, and c0 (∆) = {x ∈ w : ∆x ∈ c0 }, where for any sequence x = (xk ), the difference sequence ∆x is defined by ∞ ∆x = (∆xk )∞ k=1 = (xk − xk+1 )k=1 , 0 0 ∞ ∞ ∆ x = (∆ xk )k=1 = (xk )k=1 . Kizmaz [6] proved that these are Banach spaces with the norm k x k=| x1 | + k x k∞ . Also, he showed that E ⊂ E(∆), where E = {l∞ , c, c0 }, since there exists a sequence x = (xk ) such that xk = k, for each k,that is , x = (1, 2, 3, · · ·) for which ∆x = (−1, −1, −1, · · ·), so that although x is not convergent, but it is ∆−convergent. If m is a nonnegative integer and x = (xk ) is any sequence, then the difference sequences ∆x, ∆2 x, · · · , ∆m x are defined by ∞ ∆x = (∆xk )∞ k=1 = (xk − xk+1 )k=1 , ∆2 x = ∆(∆x) = ∆xk − ∆xk+1 , .. . ∆m x = ∆(∆m−1 x) = ∆m−1 xk − ∆m−1 xk+1 , so that m ¡ ¢ P ∆m xk = (−1)r m r xk+r . r=0
Let U be the set of all sequences u = (uk ) such that uk 6= 0 for each k. Then Gnanaseelan and Srivastava [4] defined and studied the sequence spaces E(u,∆) = {x ∈ w : u∆x ∈ E}, where E = {l∞ , c, c0 } and u∆x = uk xk − uk+1 xk+1 . After then, Et. and Colak [2] defined the sequence spaces E(∆m ) = {x ∈ w : ∆m x ∈ E}, where E = {l∞ , c, c0 }. They proved that these are Banach spaces with the norm k x k=
m P
i=1
|
xi | + k ∆m x k∞ . We recall that a modulus function f is a function from [0, ∞) to [0, ∞) such that
Some difference sequences defined by a sequence of modulus functions 3 (i) f (x) = 0 if and only if x = 0, (ii) f (x + y) ≤ f (x) + f (y) for all x, y ≥ 0, (iii) f is increasing, (iv) f is continuous from the right at 0. Since | f (x) − f (y) |≤ f (| x − y |), it follows from conditions (ii) and (iv) that f is continuous everywhere on [0, ∞). A modulus function may be bounded or unbounded. For example, t is bounded but f (t) = tp (0 < p ≤ 1) is unbounded. Furf (t) = t+1 thermore, we have f (nx) ≤ nf (x), from condition (ii), and so f (x) = f (nx n1 ) ≤ nf ( nx ), hence n1 f (x) ≤ f ( nx ), for all n ∈ N. Ruckle [13] used the concept of modulus function to define the sequence space L(f)={x∈ w :
∞ P
f (| xk |) < ∞}.
∞ P
| xk |p < ∞}.
∞ P
| xk |< ∞}.
k=1
If we take f (x) = xp , then L(f ) reduces to the familiar space lp which is given by lp = {x ∈ w :
k=1
In particular, if f (x) = x, then L(f ) = l1 which is given by l1 = {x ∈ w :
k=1
Several authors including Maddox ([8]-[11]), Ozturk and Bilgin [12], and some others studied some sequence spaces defined by a modulus function. Let X be a sequence space. Then the sequence space X(f ) is defined by X(f)={x∈ w : f (| xk |) ∈ X}. Kolk [7] gave an extension of X(f ) by considering a sequence of modulus functions F = (fk ) and defined the space X(F)={x∈ w : fk (| xk |) ∈ X}. Gaur and Mursaleen [3] defined and studied the following sequence spaces l∞ (F, ∆) = {x ∈ w : ∆x ∈ l∞ (F )}, and c0 (F, ∆) = {x ∈ w : ∆x ∈ c0 (F )}. For a nonnegative integer m, Bektas and Colak [1] extended the above mentioned spaces to l∞ (F, ∆m ) = {x ∈ w : ∆m x ∈ l∞ (F )}, and c0 (F, ∆m ) = {x ∈ w : ∆m x ∈ c0 (F )}. We further give an extension of the spaces of Bektas and Colak [1] and define the sequence spaces
4
.. .
Ahmad H. A. Bataineh and M. A. Sulaiman m l∞ (F, ∆m u ) = {x ∈ w : ∆u x ∈ l∞ (F )}, and m c0 (F, ∆m u ) = {x ∈ w : ∆u x ∈ c0 (F )}, where u = (uk ) is any sequence such that uk 6= 0 for each k,and ∆0u x = uk xk , ∆1u x = uk xk − uk+1 xk+1 , ∆2u x = ∆(∆1u x), m−1 x), ∆m u x = ∆(∆u so that m ¡ ¢ m x = P (−1)r m u ∆m x = ∆ k k+r xk+r . u uk r r=0
If u = e = (1, 1, 1, · · ·), then these spaces will give the spaces of Bektas and Colak [1] as special cases.
2. Main results For a sequence F = (fk ) of modulus functions, we will give the necessary m and sufficient conditions for the inclusion between X(∆m u ) and Y (F, ∆u ), where X, Y = l∞ or c0 . We need the following Lemmas ( see Kolk [7] ) : Lemma 2.1. The condition supk fk (t) < ∞, t > 0 holds if and only if there exists a point t0 > 0 such that supk fk (t0 ) < ∞ .
Lemma 2.2. The condition inf k fk (t) > 0, t > 0 holds if and only if there exists a point t0 > 0 such that inf k fk (t0 ) > 0. Now, we prove the following theorems Theorem 2.3. For a sequence F = (fk ) of modulus functions, the following statements are equivalent : m (i) l∞ (∆m u ) ⊆ l∞ (F, ∆u ) m (ii) c0 (∆m u ) ⊆ l∞ (F, ∆u ) (iii) supk fk (t) < ∞, t > 0
Some difference sequences defined by a sequence of modulus functions 5 Proof. (i) implies (ii) is obvious m (ii) implies (iii) : Let c0 (∆m u ) ⊆ l∞ (F, ∆u ). Suppose that (iii) is not true. Then by Lemma 2.1 , we see that supk fk (t) = ∞, for all t > 0, and therefore there is a sequence (ki ) of positive integers such that 1 fki ( ) > i for i = 1, 2, 3, · · · i Define(x = (xk ) as follows 1 k = ki i, xk = 0, otherwise Then x ∈ c0 (∆m / l∞ (F, ∆m u ), but by (2.1), x ∈ u ) which contradicts (ii). Hence (iii) must hold. (iii) implies (i) : Let (iii) be satisfied and x∈ l∞ (∆m u ). If we suppose m x |) = ∞ for ∆m x ∈ l . Now ), then sup f (| ∆ that x ∈ / l∞ (F, ∆m ∞ k k k u u u x |, then sup f (t) = ∞ which contradicts (iii). Hence take t =| ∆m k k u m m l∞ (∆u ) ⊆ l∞ (F, ∆u ). This completes the proof of the theorem. 2 (2.1)
Theorem 2.4. For a sequence F = (fk ) of modulus functions, the following statements are equivalent : m (i) c0 (F, ∆m u ) ⊆ c0 (∆u ) m (ii) c0 (F, ∆u ) ⊆ l∞ (∆m u) (iii) inf k fk (t) > 0, (t > 0)
Proof. (i) implies (ii) is obvious m (ii) implies (iii) : Let c0 (F, ∆m u ) ⊆ l∞ (∆u ). Suppose that (iii) does not hold. Then by Lemma 2.2, we see that (2.2)
inf fk (t) = ∞, (t > 0) k
and therefore there is a sequence (ki ) of positive integers such that fki (i2 ) < 1i for i = 1, 2, 3, · · · . Define(x = (xk ) as follows i2 , k = ki for i = 1, 2, 3, · · · . xk = 0, otherwise Then by (2.2), x ∈ c0 (F, ∆m / l∞ (∆m u ) but x ∈ u ) which contradicts (ii). Hence (iii) must hold. (iii) implies (i) : Let (iii) be satisfied and x∈ c0 (F, ∆m u ) that is limk fk (| m m ∆u xk |) = 0. Suppose that x ∈ / c0 (∆u ), then for some number ε0 > 0 and positive integer k0 , we have | ∆m u xk | ε0 > 0 for k ≥ k0 . Therefore fk (ε0 ) ≤
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Ahmad H. A. Bataineh and M. A. Sulaiman
fk (| ∆m u xk |) for k ≥ k0 and hence limk fk (ε0 ) = 0 which contradicts (iii). m Thus c0 (F, ∆m u ) ⊆ c0 (∆u ) and this completes the proof of the theorem. 2 m Theorem 2.5. The inclusion l∞ (F, ∆m u ) ⊆ c0 (∆u ) holds if and only if
(2.3)
lim fk (t) = ∞ for t > 0 k
m Proof. Let l∞ (F, ∆m u ) ⊆ c0 (∆u ) and suppose that (2.3) does not hold . Then there is a number t0 > 0 and a sequence (ki ) of positive integers such that
(2.4)
fki (t0 ) ≤ L < ∞
Define(the sequence x = (xk ) by t0 , k = ki for i = 1, 2, 3, · · · xk = 0, otherwise Then x ∈ l∞ (F, ∆m / c0 (∆m u ) by (2.4) but x ∈ u ) so that (2.4) must hold m m if l∞ (F, ∆u ) ⊆ c0 (∆u ). Conversely, Suppose that (2.3) is satisfied. If x ∈ l∞ (F, ∆m u ), then m fk (| ∆u xk |) ≤ L < ∞ for k = 1, 2, 3, · · · Now suppose that x ∈ / c0 (∆m u ). Then for some ε0 > 0 and positive m integer k0 , we have | ∆u xk |> ε0 for k ≥ k0 . Therefore fk (ε0 ) ≤ fk (| m ∆m u xk |) ≤ L for k ≥ k0 which contradicts (2.3). Hence x ∈ c0 (∆u ). 2 m Theorem 2.6. The inclusion l∞ (∆m u ) ⊆ c0 (F, ∆u ) holds if and only if
(2.5) Proof. Then (2.6)
lim fk (t) = 0 for t > 0 k
m Let l∞ (∆m u ) ⊆ c0 (F, ∆u ) and suppose that (2.5) does not hold .
lim fk (t) = l 6= 0 k
for some t0 > 0. Define the sequence x = (xk ) by
Some difference sequences defined by a sequence of modulus functions 7
xk = t0
m P
(−1)m
r=0
¡m+k−u−1¢ k−u
for k = 1, 2, 3, · · · .
Then x ∈ / c0 (F, ∆m u ) by (2.6). Hence (2.5) must hold. Conversely, Suppose that (2.5) is satisfied. If x ∈ l∞ (∆m u ), then | m ∆u xk |≤ L < ∞ for k = 1, 2, 3, · · · . m Therefore fk (| ∆m u xk |) ≤ fk (L) for k = 1, 2, 3, · · · and limk fk (| ∆u xk | m ) ≤ limk fk (L) = 0, by (2.5). Hence x ∈ c0 (F, ∆u ). 2
References [1] Bektas, Cigdem, A. and Colak, R., Generalized difference sequence spaces defined by a sequence of modulli, Soochow J. Math., 29(2), pp. 215-220, (2003). [2] Et., M. and Colak, R., On some generalized difference sequence spaces, Soochow J. Math., 21, pp. 377-386, (1995). [3] Gaur, A. K. and Mursaleen, Difference sequence spaces, Internat. J. Math. & Math. Sci., 21(4), pp. 701-706, (1998). [4] Gnanaseelan, C. and Srivastava, P. D., The α−, β−, γ−duals of some generalized difference sequence spaces, Indain J. Math., 38(2), pp. 111120, (1996). [5] Kamthan, P. K. and Gupta, M., Sequence spaces and series, Lecture Notes in Pure and Applied Mathematics, vol. 65 , Marcel Dekker, New York, (1981). [6] Kizmaz, H., On certain sequence spaces, Canad. Math. Bull. 24, pp. 169-176, (1981). [7] Kolk, E., Inclusion theorems for some sequence spaces defined by a sequence of modulli, Acta Comment. Univ. Tartu, 970, pp. 65-72, (1994). [8] Maddox, I. J., Some properties of paranormed sequence spaces, J. London Math. Soc. 1, pp. 316-322, (1969). [9] Maddox, I. J. , Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Phil. Soc., 64, pp. 431-435, (1969).
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[10] Maddox, I. J., Elements of Functional Analysis, 2nd Edition, Cambridge University Press, (1970). [11] Maddox, I. J., Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc. 100, pp. 161-166, (1986). ¯ urk, E. and Bilgen, T., Strongly summable sequence spaces defined [12] Ozt¨ by a modulus, Indian J. Pure Appl. Math. 25(6), pp. 621-625, (1994). [13] Ruckle, W. H., FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25, pp. 973-978, (1973). Ahmad H. A. Bataineh Department of Mathematics Al al-Bayt University P. O. Box 130095 Mafraq Jordan e-mail :
[email protected] and
Ibrahim M. A. Sulaiman Department of Mathematics Al al-Bayt University P. O. Box 130095 Mafraq Jordan e-mail :
[email protected] 
