The main aim of this paper is to present an extension of the modular sequence spaces by means of Cesàro mean of order one, to investigate several relevant algebraic and topological properties, and derive some other spaces in the sequel.

1. Introduction

Throughout the paper w(X) will represent the spaces of all X valued sequences spaces, where (X, q) is a seminormed space, seminormed by q. For X = C, the space of complex numbers, these represent the corresponding scalar valued sequence spaces. The zero sequence is denoted by , where θ is the zero element of X.

An Orlicz function is a function M : [0, ∞)→[0, ∞), which is continuous, nondecreasing, and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) → ∞, as x → ∞.

If convexity of Orlicz function M is replaced by M(x + y) ≤ M(x) + M(y), then this function is called a modulus function introduced by Nakano [1].

Lindenstrauss and Tzafriri [2] used the idea of Orlicz function to construct sequence space

()

The space ℓ_M becomes a Banach space, with the norm

()

which is called an Orlicz space. The space ℓ_M is closely related to the space ℓ_p which is an Orlicz sequence space with M(x) = x^p for 1 ≤ p < ∞.

Another generalization of Orlicz sequence spaces is due to Woo [3]. Let {M_k} be a sequence of Orlicz functions. Define the vector space ℓ{M_k} by

()

and equip this space with the norm

()

Then ℓ{M_k} becomes a Banach space and is called a modular sequence space. The space ℓ{M_k} also generalizes the concept of modular sequence space introduced earlier by Nakano [4], who considered the space ℓ{M_k} when , where 1 ≤ α_k < ∞ for k ≥ 1.

An Orlicz function M is said to satisfy the Δ₂-condition for all values of u, if there exists a constant K > 0, such that M(2u) ≤ kM(u) (u ≥ 0). The Δ₂-condition is equivalent to the satisfaction of inequality M(lu) ≤ kluM(u) for all values of u and for l > 1 (see [5]).

The previous Δ₂-condition implies , for all u > 0, l > 1.

Bektaş and Altin [6], Parasar and Choudhary [7], Mursaleen et al. [8], Dutta and Başar [9], Dutta and Bilgin [10], Karakaya and Dutta [11], Tripathy and Dutta [12], Jebril [13], and many others have studied different summable spaces and other sequence spaces using Orlicz functions.

A BK-space (introduced by Zeller [14]) (X, ∥·∥) is a Banach space of complex sequences x = (x_k) in which the coordinate maps are continuous; that is, , whenever ∥xⁿ − x∥ → 0 as n → ∞, where , for all n ∈ N and x = (x_k).

Let A denote the set of all complex sequences which have only a finite number of nonzero coordinates, and let λ denote a BK-space of sequences x = (x_k) which contains A. An element x = (x_k) of λ will be called sectionally convergent if

()

where e_k = (δ_ki), where δ_kk = 1, δ_ki = 0 for k ≠ i.

λ will be called AK-space if and only if each of its elements is sectionally convergent.

Let M = (M_k) be a sequence of Orlicz functions, let X be a seminormed space with seminorm q, let p = (p_k) be sequence of positive real numbers, and let

be the Cesàro matrix of order one with c_nk = 1/(n + 1) if 0 ≤ k ≤ n and c_nk = 0, otherwise. Then for nonnegative real numbers s we define

()

The following inequality will be used throughout the paper. Let p = (p_k) be a positive sequence of real numbers with 0 < p_k ≤ sup p_k = G, D = max {1, 2^G−1 }. Then for all a_k, b_k ∈ C for all k ∈ N, we have

()

2. Main Results

In this section we give the theorems that characterize the structure of the class of sequences ℓ{M_k, p, q, s, C} and some other spaces which can be derived from this space.

Theorem 1. Let p = (p_k) be bounded sequence of positive reals; then ℓ{M_k, p, q, s, C} is a linear space over the field of complex numbers.

Proof. Let x, y ∈ ℓ{M_k, p, q, s, C} and α, β ∈ C. Then there exist some ρ₁ > 0 and ρ₂ > 0 such that

()

We consider ρ₃ = max (2|α|ρ₁, 2|β|ρ₂). Since each M_k is non-decreasing and convex, and since q is a seminorm,

()

This completes the proof.

Theorem 2. ℓ{M_k, p, q, s, C} is a paranormed space (need not total paranorm) space with paranorm g, defined as follows

()

where H = max {1, sup _kp_k}.

Proof. Clearly g(x) = g(−x). Since M_k(0) = 0, for all k ∈ N we get for x = θ.

Now let x, y ∈ ℓ{M_k, p, q, s, C}, and let us choose ρ₁ > 0 and ρ₂ > 0 such that

()

Let ρ = ρ₁ + ρ₂. Then we have

()

Hence g(x + y) ≤ g(x) + g(y).

Finally let λ be a given non-zero scalar; then the continuity of the scalar multiplication follows from the following equality

()

This completes the proof.

The proof of the following theorem is easy, so omitted.

Theorem 3. Let M = (M_k) and T = (T_k) be sequences of Orlicz functions. For any two sequences p = (p_k) and t = (t_k) of bounded positive real numbers and for any two seminorms q₁ and q₂ one has

(i)
if q₁ is stronger than q₂, then ℓ{M_k, p, q₁, s, C} ⊂ ℓ{M_k, p, q₂, s, C},
(ii)
ℓ{M_k, p, q₁, s, C}∩ℓ{M_k, p, q₂, s, C} ⊂ ℓ{M_k, p, q₁ + q₂, s, C},
(iii)
ℓ{M_k, p, q, s, C}∩ℓ{T_k, p, q, s, C} ⊂ ℓ{M_k + T_k, p, q, s, C},
(iv)
ℓ{M_k, p, q₁, s, C}∩ℓ{M_k, t, q₂, s, C} ≠ φ,
(v)
If s₁ ≤ s₂, then ℓ{M_k, p, q, s₁, C} ⊂ ℓ{M_k, p, q, s₂, C}.

Theorem 4. Let M = (M_k) and T = (T_k) be sequences of Orlicz functions which satisfy Δ₂-condition and s > 1, then

()

Proof. Let x ∈ ℓ{M_k, p, q, s, C} and ε > 0. We choose 0 < δ < 1 such that each M_k(u) < ε for 0 ≤ u ≤ δ. We write and consider

()

where the first summation is over y_k ≤ δ and the second is over y_k > δ. Now we have

()

For y_k > δ, we use the fact that

()

Since each T_k is non-decreasing and convex, it follows that

()

Since each T_k satisfies Δ₂-condition, we have

()

Hence

()

Thus

()

Hence x ∈ ℓ{T_k∘M_k, p, q, s, C}.

This completes the proof.

Taking M_k(x) = x, for all x ∈ [0, ∞) and k in N, in Theorem 4, we get the next corollary.

Corollary 5. Let M = (M_k) be any sequence of Orlicz functions which satisfy Δ₂-condition and s > 1, then

()

We will write f ≈ g for non-negative functions fand g whenever C₁f ≤ g ≤ C₂f for some C_j > 0, j = 1, 2.

Theorem 6. Let M = (M_k) and T = (T_k) be sequences of Orlicz functions. If M_k ≈ T_k for each k ∈ N, then ℓ{M_k, p, q, s, C} = ℓ{T_k, p, q, s, C}.

Proof. The proof is obvious.

Theorem 7. Let M = (M_k) be a sequence of Orlicz functions. If lim _t→0 (M_k(t)/t) > 0 and lim _t→0 (M_k(t)/t) < ∞, for each k ∈ N, then ℓ{M_k, p, q, s, C} = ℓ{p, q, s, C}.

Proof. If the given conditions are satisfied, we have M_k(t) ≈ t for each k, and the proof follows from Theorem 6.

If we take s = 0, the sequence space ℓ{M_k, p, q, s, C} reduces to the following sequence space:

()

Theorem 8. Let p = (p_k) be bounded sequence of positive reals, and let (X, q) be a complete seminormed space, then ℓ{M_k, p, q, C} is a complete paranormed space paranormed by h, defined by

()

where H = max {1, sup _k p_k}.

Proof. Let (xⁱ) be a Cauchy sequence in ℓ{M_k, p, q, C}. Let δ > 0 be fixed, and let r > 0 be such that for a given 0 < ε < 1, ε/rδ > 0, and rδ ≥ 1. Then there exists a positive integer n₀ such that

()

Hence we have

()

It follows that

()

For r > 0 with M_k(rδ/2) ≥ 1, we have

()

Since M_k is non-decreasing for each k ∈ N, we have

()

Hence it follows that

is a Cauchy sequence in (X, q) for each k ∈ N. But (X, q) is complete, and so

is convergent in (X, q) for each k ∈ N.

Let exists for each k ∈ N.

Now we have for all s, t ≥ n₀,

()

Then we have

()

Using the continuity of Orlicz functions, we have

()

This implies

()

It follows that (x^s − x) ∈ ℓ{M_k, p, q, C}.

Since (x^s) ∈ ℓ{M_k, p, q, C} and ℓ{M_k, p, q, C} is a linear space, so we have x = x^s − (x^s − x) ∈ ℓ{M_k, p, q, C}.

This completes the proof.

If we take s = 0 and p_k = l, a constant the sequence space ℓ{M_k, p, q, s, C} reduces to the following sequence space:

()

Theorem 9. Let (X, q) be a complete normed space; then ℓ{M_k, q, C} is a Banach space normed by ∥·∥, defined by

()

Proof. We prove that ∥·∥ is a norm on ℓ{M_k, q, C}. The completeness part can be proved using similar arguments as applied to prove the previous theorem.

If x = θ, then it is obvious that ∥x∥ = 0. Conversely assume ∥x∥ = 0. Then using the definition of norm, we have

()

This implies that for a given ε > 0, there exists some ρ_ε (0 < ρ_ε < ε) such that

()

It follows that

()

Thus

()

Suppose

, for some i. Let ε → 0 then

It follows that as ε → 0 for some n_i ∈ N. This is a contradiction.

Therefore .

It follows that x_k = 0 for all k ≥ 1. Hence x = θ.

Again proof of the properties ∥x + y∥≤∥x∥+∥y∥ and for any scalar α, ∥αx∥ = | α | ∥x∥ are similar to that of Theorem 2.

It is easy to see that ∥xⁱ∥→0 implies that for each i ≥ 1. Hence we have the following proposition.

Proposition 10. The space ℓ{M_k, q, C} is a BK-space.

Now we study the AK-characteristic of the space ℓ{M_k, q, s, C}. Before that we give a new definition and prove some results which will be required.

Definition 11. For any sequence of Orlicz functions M = (M_k), we define

()

Clearly h{M_k, q, C} is a subspace of ℓ{M_k, q, C}. The topology of h{M_k, q, C} is the one it inherits from ∥·∥.

Proposition 12. Let M = (M_k) be a sequence of Orlicz functions which satisfy Δ₂-condition. Then

()

Proof. It is enough to prove that ℓ{M_k, q, C}⊆h{M_k, q, C}.

Let x ∈ ℓ{M_k, q, C}, then for some ρ > 0,

()

Choose an arbitrary η > 0. If ρ ≤ η then

()

Let now η < ρ and put l = ρ/η > 1.

Since each M_k satisfies the Δ₂-condition, there exist constants K_k such that

()

Let

. Then for every η > 0

()

This completes the proof.

Proposition 13. Let (X, q) be a complete normed space, then h{M_k, q, C} is an AK-space.

Proof. Let x ∈ h{M_k, q, C}. Then for each ε, 0 < ε < 1, we can find an s₀ such that

()

Hence for s ≥ s₀,

()

Thus we can conclude that h{M_k, q, C} is an AK space.

Combining Proposition 10 and Proposition 12, we have the following theorem.

Theorem 14. Let M = (M_k) be a sequence of Orlicz functions which satisfy Δ₂-condition, then ℓ{M_k, q, C} is an AK-space.

Proposition 15. The space h{M_k, q, C} is a closed subspace of ℓ{M_k, q, C}.

Proof. Let {x^s} be a sequence in h{M_k, q, C} such that ∥x^s − x∥ → 0, where x ∈ ℓ{M_k, q, C}.

To complete the proof we need to show that x ∈ h{M_k, q, C}; that is,

()

To ρ > 0 there corresponds an l such that ∥x^l − x∥ ≤ ρ/2. Then using convexity of each M_k,

()

Now from Theorem 9, using the definition of norm ∥·∥, we have

()

It follows that

()

Thus x ∈ h{M_k, q, C}.

Hence we have the following corollary.

Corollary 16. The space h{M_k, q, C} is a BK-space.

Acknowledgment

This research is supported by Deanship of Scientific Research, Taibah University, Almadinah Almunawwarah, Saudi Arabia.

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An Extension of Modular Sequence Spaces

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