A note on some strongly sequence spaces

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Volume 2011, Article ID 598393,8pages doi:10.1155/2011/598393

Research Article

A Note on Some Strongly Sequence Spaces

Ekrem Savas¸

¹

and Adem Kılıc¸man

²

1Istanbul Commerce University, 34840 Istanbul, Turkey

2Department of Mathematics and Institute for Mathematical Research, University of Putra Malaysia, Serdang, Selangor 43400, Malaysia

Correspondence should be addressed to Adem Kılıc¸man,akilicman@putra.upm.edu.my

Received 24 September 2010; Revised 4 April 2011; Accepted 30 May 2011 Academic Editor: Marcia Federson

Copyrightq 2011 E. Savas¸ and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce and study new sequence spaces which arise from the notions of generalized de la Vall´ee-Poussin means, invariant means, and modulus functions.

1. Introduction

Let w be the set of all real or complex sequences and let l_∞, c, and c0be the Banach spaces of bounded, convergent, and null sequences x xk, respectively, with the usual norm x sup_n|xn|.

A sequence x xk ∈ l∞is said to be almost convergent if its Banach limit coincides.

Letc denote the space of all almost convergent sequences. Lorentz 1 proved that

c

x ∈ l∞: lim

m tmnx exist uniformly in n

, 1.1

where

t_mnx x_n x_n1 · · · x_nm

m 1 . 1.2

The spacec of strongly almost convergent sequences was introduced by Maddox 2

as

c

x ∈ l_∞: lim

m t_mn|x − e| exist uniformly in n for some ∈ C

, 1.3

where e 1, 1, . . ..

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Let σ be a one-to-one mapping from the set of positive integers into itself such that σ^mn σ^m−1σn, m 1, 2, 3, . . ., where σ^mn denotes the mth iterate of the mapping σ in n, see 3. A continuous linear functional ϕ on l∞is said to be an invariant mean or a σ-mean, if and only if,

i ϕx ≥ 0, when the sequence x xn is such that xn≥ 0 for all n,

ii ϕe 1, where e 1, 1, . . .,

iii ϕxσn ϕx, for all x ∈ l∞.

For a certain kind of mapping σ, every invariant mean ϕ extends the functional limit on the space c, in the sense that ϕx lim x for all x ∈ c. Consequently, c ⊂ Vσ, where Vσis the set of bounded sequences with equal σ-means. Schaefer3 proved that

V_σ

x ∈ l_∞: lim

k t_kmx L uniformly in m for some L σ − lim x

, 1.4

where

tkmx xm xσm · · · xσ^km

k 1 , t_−1,m 0. 1.5

Thus we say that a bounded sequence x xk is σ-convergent, if and only if, x ∈ Vσsuch that σ^kn / n for all n ≥ 0, k ≥ 1. Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the σ-convergence leads naturally to the concept of strong σ-convergence.

A sequence x xk is said to be strongly σ-convergent see, Mursaleen 4, if there exists a number such that

1 k

k i1

xσⁱm− −→ 0, 1.6

as k → ∞ uniformly in m. We write Vσ to denote the set of all strong σ-convergent sequences and when 1.6 holds, we write Vσ − lim x . Taking σm m 1, we obtainVσ c. Then the strong σ-convergence generalizes the concept of strong almost convergence. We also note that

Vσ ⊂ Vσ ⊂ l_∞. 1.7

It is also well known that the concept of paranorm is closely related to linear metric spaces.

In fact, it is a generalization of absolute value. Let X be a linear space. A function p : X → R is called a paranorm, if

P:1 p0 ≥ 0,

P:2 px ≥ 0, for all x ∈ X,

P:3 p−x px, for all x ∈ X,

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P:4 px y ≤ px py, for all x, y ∈ X triangle inequality,

P:5 if λn is a sequence of scalars, with λn → λ n → ∞, and xn is a sequence of vectors with pxn− x → 0 n → ∞, then pλnxn− λx → 0 n → ∞ continuity of multiplication by scalars.

A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space X is said to be an FK space, if its metric is stronger than the metric of w on X, that is, convergence in the sequence space X implies coordinatewise convergencethe letters F and K stand for Fréchet and Koordinate, the German word for coordinate.

Note that, by Ruckle in5, a modulus function f is a function from 0, ∞ to 0, ∞

such that

i fx 0, if and only if, x 0,

ii fx y ≤ fx fy, for all x, y ≥ 0,

iii f increasing,

iv f is continuous from the right at zero.

Since|fx − fy| ≤ f|x − y|, it follows from condition iv that f is continuous on

0, ∞. Furthermore, from condition ii, we have fnx ≤ nfx for all n ∈ N, and thus

fx f

nx1

n

≤ nf

x n

, 1.8

hence

1

nfx ≤ f

x n

, ∀n ∈ N. 1.9

In5, Ruckle used the idea of a modulus function f in order to construct a class of FK spaces

L f

x xk :^∞

k1f|xk| < ∞

. 1.10

From the definition, we can easily see that the space Lf is closely related to the space l1, if we consider fx x for all real numbers x ≥ 0. Several authors study these types of spaces. For example, Maddox introduced and examined some properties of the sequence spaces w0f, wf and w∞f, defined by using a modulus f, which generalized the well-known spaces w0, w and w_∞of strongly summable sequences, see6. Similarly, Savas¸ in 7 generalized the concept of strong almost convergence by using a modulus f and examined some further properties of the corresponding new sequence spaces.

The generalized de la Vall´e-Poussin mean is defined by

tnx 1 λ_n

k∈In

xk, 1.11

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where In n − λn  1, n for n 1, 2, . . .. Then a sequence x xk is said to be V, λ- summable to a number Lsee 8, if tnx → L as n → ∞, and we write

V, λ₀

x : lim

n

1 λn

k∈In

|xk| 0

,

V, λ {x : x − e ∈ V, λ₀for some ∈ C},

V, λ_∞

x : sup

n

1 λ_n

k∈In

|xk| < ∞

,

1.12

for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by the de la Vall´e-Poussin method. In the special case where λn n, for n 1, 2, 3, . . ., the sets V, λ₀,V, λ, and V, λ_∞reduce to the sets w₀, w, and w_∞, which were introduced and studied by Maddox, see6.

We also note that the sets of sequence spaces such as strongly σ-summable to zero, strongly σ-summable, and strongly σ-bounded with respect to the modulus function were defined by Nuray and Savas¸ in9.

2. Main Results

Let p pk be a sequence of real numbers such that pk > 0 for all k, and sup_k p_k < ∞. This assumption is made throughout the rest of this paper. Then we now write

V_σ, λ, f, p

0

x : lim

n

1 λn

k∈In

f x_σ^k_m^p^k 0, uniformly in m

, Vσ, λ, f, p

x : x − e ∈

Vσ, λ, f, p

0 for some ∈ C ,

V_σ, λ, f, p

∞

x : sup

n,m

1 λn

k∈In

f x_σ^k_m^p^k < ∞

.

2.1

In particular, if we take pk 1 for all k, we have Vσ, λ, f

0

x : lim

n

1 λ_n

k∈In

f x_σ^k_m 0, uniformly in m , V_σ, λ, f

x : x − e ∈

V_σ, λ, f

Vσ, λ, f

∞

x : sup

n,m

1 λ_n

k∈In

f x_σ^k_m < ∞ .

2.2

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Similarly, when σm m 1, then Vσ, λ, f, p₀,Vσ, λ, f, p and Vσ, λ, f, p_∞are reduced to

V , λ, f, p

0

x : lim

n

1 λ_n

k∈In

f|xkm|pk 0, uniformly in m

,

V , λ, f, p

x : x − e ∈

V , λ, f, p

∞

x : sup

n,m

1 λ_n

k∈In

f|xkm|_p_k

< ∞

, respectively.

2.3

In particular, when p_k p for all k, then we have the spaces

V , λ, f, p

0 V , λ, f

0,

V , λ, f, p

V , λ, f

,

V , λ, f, p

∞ V , λ, f

∞, 2.4

which were introduced and studied by Malkowsky and Savas¸ in10. Further, when λn n, for n 1, 2, 3, . . ., the sets V , λ, f₀and V , λ, f are reduced to cf and c0f respectively, see7. Now, if we consider fx x, then one can easily obtain

Vσ, λ, p

0

x : lim

n

1 λn

k∈In

x_σ^k_m^p^k uniformly in m

, V_σ, λ, p

x : x − e ∈

V_σ, λ, p

0for some ∈ C , V_σ, λ, p

∞

x : sup

n,m

1 λn

k∈In

x_σ^k_m^p^k < ∞

.

2.5

If pk 1 for all k, then we can obtain the spaces Vσ, λ₀,Vσ, λ, and Vσ, λ_∞. Throughout this paper, we use the notation f|xk|^p^kinstead of{f|xk|}^p^k.

If p ∈ l∞, then it is clear thatVσ, λ, f, p₀, Vσ, λ, f, p, and V, σ, λ, f, p_∞ are linear spaces over the complex fieldC.

Lemma 2.1. Let f be any modulus. Then V_σ, λ, f

∞ _∞^σ f

x ∈ w :

f xσ^km ∞

. 2.6

Proof. Let x∈ Vσ, λ, f_∞. Then there is a constant M > 0 such that

1

λ₁f x_σ^k_m ≤ sup

m,n

1 λ_n

k∈In

f x_σ^k_m ≤ M, 2.7

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for all m, and sof|xσ^km| ∈ l∞. Let x ∈ _∞^σf. Then there is a constant M > 0 such that

f|xσ^km| ≤ M for all k and m, and so

1 λ_n

k∈In

f xσ^km ≤ M 1λ_n

k∈In

1≤ M, 2.8

for all m and n. Thus x∈ Vσ, λ, f_∞. This completes the proof.

If x∈ Vσ, λ, f, p, with 1/λn

k∈Inf|xσ^km− e|^p^k → 0 as n → ∞ uniformly in m, then we write x_k → lVσ, λ, f, p.

The following well-known inequality11, page 190 will be used later.

If 0≤ pk≤ sup pk H and C max1, 2^H−1, then

|ak bk|^p^k ≤ C

|ak|^p^k |bk|^p^k

, 2.9

for all k and ak, bk∈ C.

In the following theorem, we prove xk → implies xk → ∈ Vσ, λ, f, p and we also prove the uniqueness of the limit . To prove the theorem, we need the following lemma.

Lemma 2.2 see 2. Let pk> 0, q_k> 0. Then c_oq ⊂ c0p, if and only if, lim_{k → ∞}inf pk/q_k> 0, wherec₀p {x : |xk|^p^k → 0 as k → ∞}.

Note that no other relation betweenpk and qk is needed inLemma 2.2.

Theorem 2.3. Let limk → ∞inf pk > 0. Then xk → implies xk → ∈ Vσ, λ, f, p. Let lim_{k → ∞}pk r > 0. If xk → ∈ Vσ, λ, f, p, then is unique.

Proof. Let xk → . By the definition of modulus, we have f|xk − | → 0. Since lim_{k → ∞}inf p_k> 0, it follows from the above lemma that f|xk− |^p^k → 0 and consequently, xk → ∈ Vσ, f, p.

Let lim_{k → ∞}p_k r > 0. Suppose that xk → 1∈ Vσ, λ, f, p, xk → 2 ∈ Vσ, λ, f, p and

|1− 2|^p^k a > 0. Now, from 2.9 and the definition of modulus, we have

1 λ_n

k∈In

f|1− 2|^p^k ≤ C λ_n

k∈In

f xσ^km− 1^p^k

C λ_n

k∈In

f xσ^km− 2^p^k.

2.10

Hence,

1 λ_n

k∈In

f|1− 2|^p^k 0. 2.11

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Further, f|1− 2|^p^k → fa^ras k → ∞ and, therefore,

n → ∞lim 1 λ_n

k∈In

f|1− 2|^p^k fa^r. 2.12

From2.11 and 2.12, it follows that fa 0 and by the definition of modulus, we have a 0. Hence 1 2and this completes the proof.

Theorem 2.4. i Let 0 < infk pk≤ pk≤ 1. Then, V_σ, λ, f, p

⊂

V_σ, λ, f

. 2.13

ii Let 0 < pk≤ sup_k p_k< ∞. Then, V_σ, λ, f

⊂

V_σ, λ, f, p

. 2.14

Proof. i Let x ∈ Vσ, λ, f, p. Since 0 < infk p_k≤ 1, we get 1

λn

k∈In

f x_σ^k_m− e ≤ 1λn

k∈In

f x_σ^k_m− e^p^k, 2.15

and hence x∈ Vσ, λ, f.

ii Let p ≥ 1 for each k, and sup_k pk< ∞. Let x ∈ Vσ, λ, f. Then, for each k, 0 < ε < 1, there exists a positive integer N such that

1 λn

k∈In

f x_σ^k_m− e ≤ ε < 1, 2.16

for all m≥ N. This implies that 1

λ_n

k∈In

f xσ^km− e^p^k ≤ 1 λ_n

k∈In

f xσ^km− e. 2.17

Therefore, x∈ Vσ, λ, f, p. This completes the proof.

Finally, we conclude this paper by stating the following theorem. We omit the proof, since it involves routine verification and can be obtained by using standard techniques.

Theorem 2.5. Vσ, λ, f, p₀and Vσ, λ, f, p are complete linear topological spaces, with paranorm g, where g is defined by

gx sup

m,n

1 λ_n

k∈In

f x_σ^k_m^p^kM

, 2.18

whereM max1, {sup_k p_k}.

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Acknowledgment

The authors express their sincere thanks to the referees for careful reading of the paper and several helpful suggestions.

References

1 G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948.

2 I. J. Maddox, “Spaces of strongly summable sequences,” The Quarterly Journal of Mathematics, vol. 18, pp. 345–355, 1967.

3 P. Schaefer, “Infinite matrices and invariant means,” Proceedings of the American Mathematical Society, vol. 36, pp. 104–110, 1972.

4 M. Mursaleen, “Matrix transformations between some new sequence spaces,” Houston Journal of Mathematics, vol. 9, no. 4, pp. 505–509, 1993.

5 W. H. Ruckle, “FK spaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics, vol. 25, pp. 973–978, 1973.

6 I. J. Maddox, “Sequence spaces defined by a modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100, no. 1, pp. 161–166, 1986.

7 E. Savas¸, “On some generalized sequence spaces defined by a modulus,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 5, pp. 459–464, 1999.

8 L. Leindler, “ ¨Uber die verallgemeinerte de la Vall´ee-Poussinsche summierbarkeit allgemeiner orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965.

9 F. Nuray and E. Savas¸, “On strong almost A-summability with respect to a modulus and statistical convergence,” Indian Journal of Pure and Applied Mathematics, vol. 23, no. 3, pp. 217–222, 1992.

10 E. Malkowsky and E. Savas¸, “Some λ-sequence spaces defined by a modulus,” Archivum Mathematicum, vol. 36, no. 3, pp. 219–228, 2000.

11 I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, Cambridge, UK, 1970.

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A note on some strongly sequence spaces

Research Article