Abstract and Applied Analysis
Volume 2013, Issue 1 972363
Research Article
Open Access

Some Generalized Difference Sequence Spaces Defined by Ideal Convergence and Musielak-Orlicz Function

Awad A. Bakery, 

Corresponding Author

Awad A. Bakery

Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia kau.edu.sa

Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, 11566 Cairo, Egypt shams.edu.eg

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Elsayed Abdelbayen Elnour Mohamed, 

Elsayed Abdelbayen Elnour Mohamed

Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia kau.edu.sa

Department of Mathematics, Faculty of Education, Alzaeim Alazhari University, P.O. Box 1432, 13311 Khartoum, Sudan aau.edu.sd

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Mohamed Alamin Ahmed, 

Mohamed Alamin Ahmed

Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia kau.edu.sa

Department of Mathematics, Faculty of Education, Alzaeim Alazhari University, P.O. Box 1432, 13311 Khartoum, Sudan aau.edu.sd

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First published: 12 June 2013
https://doi.org/10.1155/2013/972363
Citations: 3
Academic Editor: Feyzi Başar

Abstract

In the present paper we introduced the ideal convergence of generalized difference sequence spaces combining de La Vallée-Poussin mean and Musielak-Orlicz function over n-normed spaces. We also study some topological properties and inclusion relation between these spaces.

1. Introduction

Throughout the paper ω, ℓ∞, c, c0, and ℓp denote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers. The sets of natural numbers and real numbers will be denoted by ℕ, ℝ, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using de La Vallée-Poussin mean and the notion of ideal, we aimed to introduce some new sequence spaces with respect to generalized difference operator and Musielak-Orlicz function in n-normed linear spaces. By an ideal we mean a family I ⊂ 2Y of subsets of a nonempty set Y satisfying (i) ϕ ∈ I; (ii) A, B ∈ I imply A ∪ B ∈ I; (iii) A ∈ I,  B ⊂ A imply B ∈ I, while an admissible ideal I of Y further satisfies {x} ∈ I for each x ∈ Y. The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence. The concept of 2-normed spaces was initially introduced by Gähler [2] in the 1960s, while that of n-normed spaces can be found in [3], and this concept has been studied by many authors; see for instance [4–7]. The notion of ideal convergence in 2-normed space was initially introduced by Gürdal [8]. Later on, it was extended to n-normed spaces by Gürdal and Şahiner [9]. Given I ⊂ 2ℕ is a nontrivial ideal in ℕ, the sequence (xn) n∈ℕ in a normed space (X; ∥·∥) is said to be I-convergent to x ∈ X, if for each ε > 0,
()
A sequence (xk) in a normed space (X, ∥·∥) is said to be I-bounded if there exists L > 0 such that
()
A sequence (xk) in a normed space (X, ∥·∥) is said to be I-Cauchy if for each ε > 0, there exists a positive integer m = m(ε) such that
()
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 M is replaced by M(x + y) ≤ M(x) + M(y), then it is called a modulus function, introduced by Nakano [10]. Ruckle [11] and Maddox [12] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function M is said to satisfy Δ2-condition for all values of x ≥ 0, if there exists a constant k > 0, such that M(2x) ≤ kM(x). The Δ2-condition is equivalent to M(lx) ≤ klM(x) for all values of x and for l > 1. Lindenstrauss and Tzafriri [13] used the idea of an Orlicz function to define the following sequence spaces:
()
which is a Banach space with the Luxemburg norm defined by
()

The space ℓM is closely related to the space ℓp, which is an Orlicz sequence space with M(x) = xp for 1 ≤ p < ∞.

Recently different classes of sequences have been introduced using Orlicz functions. See [7, 9, 14–16].

A sequence M = (Mk) of Orlicz functions Mk for all k ∈ ℕ is called a Musielak-Orlicz function, for a given Musielak-Orlicz function M. Kızmaz [17] defined the difference sequence spaces ℓ∞(Δ), c(Δ), and c0(Δ) as follows: Z(Δ)  =  {x = (xk):(Δxk) ∈ Z}, for Z = ℓ∞, c, and c0, where Δx = (xk − xk+1), for all k ∈ ℕ. The above spaces are Banach spaces, normed by ∥x∥ = |x1 | + sup k | Δxk|. The notion of difference sequence spaces was generalized by Et and Colak [18] as follows: Z(Δs) = {x = (xk):(Δsxk) ∈ Z}, for Z = ℓ∞,  c, and c0, where s ∈ ℕ, (Δsxk) = (Δs−1xk − Δs−1xk+1) and so that . Tripathy and Esi [19] introduced the following new type of difference sequence spaces.

Z(Δm) = {x = (xk):(Δmxk) ∈ Z}, Z = ℓ∞, c, and c0, where Δmxk = (xk − xk+m), for all k ∈ ℕ. Tripathy et al. [20] generalized the above notions and unified them as follows. Let m, s be nonnegative integers, then for Z a given sequence space we have
()
where
()
Also let m, s be nonnegative integers, then for Z a given sequence space we have
()
where
()
where xk = 0, for k < 0

2. Definitions and Preliminaries

Let n ∈ ℕ and X be a linear space over the field K of dimension d, where d ≥ n ≥ 2 and K is the field of real or complex numbers. A real valued function ∥·, …, ·∥ on Xn satisfies the following four conditions:
  • (1)

    ∥x1, x2, …, xn∥ = 0 if and only if x1, x2, …, xn are linearly dependent in X;

  • (2)

    ∥x1, x2, …, xn∥ is invariant under permutation;

  • (3)

    ∥αx1, x2, …, xn∥ = |α | ∥x1, x2, …, xn∥ for any α ∈ K;

  • (4)

    ∥x + x′, x2, …, xn∥ ≤ ∥x, x2, …, xn∥  +  ∥x′, x2, …, xn∥; is called an n-norm on X and the pair (X; ∥·, …, ·∥) is called an n-normed space over the field K. For example, we may take X = ℝn being equipped with the n-norm the volume of the n-dimensional parallelepiped spanned by the vectors x1, x2, …, xn which may be given explicitly by the formula

    ()

where xi = (xi1, xi2, …, xin) for each i ∈ ℕ.

Let (X, ∥·, …, ·∥) be an n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, a3, …, an} a linearly independent set in X. Then, the function ∥·,…,·∥∞ on Xn−1 defined by
()
defines an (n − 1)-norm on X with respect to a1, a2, a3, …, an and this is known as the derived (n − 1)-norm. The standard (n)-norm on X, a real inner product space of dimension d ≥ n, is as follows:
()
where 〈·, ·〉 denotes the inner product on X. If we take X = ℝn, then
()

For n = 1, this n-norm is the usual norm .

Definition 1. A sequence (xk) in an n-normed space is said to be convergent to x ∈ X if,

()

Definition 2. A sequence (xk) in an n-normed space is called Cauchy (with respect to n-norm) if,

()
If every Cauchy sequence in X converges to an x ∈ X, then X is said to be complete (with respect to the n-norm). A complete n-normed space is called n-Banach space.

Definition 3. A sequence (xk) in an n-normed space (X, ∥·, …, ·∥) is said to be I-convergent to x0 ∈ X with respect to n-norm, if for each ε > 0, the set

()

Definition 4. A sequence (xk) in an n-normed space (X, ∥·, …, ·∥) is said to be I-Cauchy if for each ε > 0, there exists a positive integer m = m(ε) such that the set

()
Let x = (xk) be a sequence; then S(x) denotes the set of all permutations of the elements of (xk); that is, S(x) = (xπ(n)) : π is a permutation of ℕ.

Definition 5. A sequence space E is said to be symmetric if S(x) ⊂ E for all x ∈ E.

Definition 6. A sequence space E is said to be normal (or solid) if (αkxk) ∈ E, whenever (xk) ∈ E and for all sequence (αk) of scalars with |αk | ≤ 1 for all k ∈ ℕ.

Definition 7. A sequence space E is said to be a sequence algebra if x, y ∈ E then x · y = (xkyk) ∈ E.

Lemma 8. Every n-normed space is an (n − r)-normed space for all r = 1,2, 3, …, n − 1. In particular, every n-normed space is a normed space.

Lemma 9. On a standard n-normed space X, the derived (n − 1)-norm ∥·,…,·∥∞ defined with respect to the orthogonal set {e1, e2, …, en} is equivalent to the standard (n − 1)-norm ∥·,…,·∥S. To be precise, one has

()

for all x1, x2, …, xn−1 ∈ X, where .

Let Λ = (λk) be a nondecreasing sequence of positive real numbers tending to infinity and let λ1 = 1 and λk+1 ≤ λk + 1. In summability theory, de La Vallée-Poussin mean was first used to define the (V, λ)-summability by Leindler [21]. Also the (V, λ)-summable sequence spaces have been studied by many authors including [22, 23]. The generalized de La Vallée-Poussin’s mean of a sequence x = (xk) is defined as follows: , where Ik = [k − λk + 1, k] for k ∈ ℕ. We write

,

for some l ∈ ℂ},

.

For the sequence spaces that are strongly summable to zero, strongly summable and strongly bounded by the de La Vallée-Poussin′s method, respectively. In the special case where λk = k for k ∈ ℕ the spaces [V, λ] 0, [V, λ], and [V, λ] ∞ reduce to the spaces v0, v, and v∞ introduced by Maddox [24]. The following new paranormed sequence space is defined in [22].

. If one takes pk = p for all k ∈ ℕ; the space V(λ, p) reduced to normed space Vp(λ) defined by . The details of the sequence spaces mentioned above can be found in [23].

For any bounded sequence (pn) of positive numbers, one has the following well-known inequality.

If 0 ≤ pk ≤ sup kpk = G and D = max (1, 2G−1), then , for all k and ak, bk ∈ ℂ.

3. Main Results

In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. We find out some relations related to these sequence spaces. Let I be an admissible ideal of ℕ, ℳ = (Mj) be a Musielak-Orlicz function, and (X, ∥·, …, ·∥) an n-normed space. Further, let p = (pk) be any bounded sequence of positive real numbers,
()

The above sequence spaces contain some unbounded sequences for s ≥ 1. If Mk(x) = x, m = 1, λk = k for all k ∈ ℕ and pk = 1 for all k ∈ ℕ, then but (ks) ∉ ℓ∞.

Let us consider a few special cases of the above sets.
  • (1)

    If n = 2,  m = 1, and Mk(x) = M(x), then the above classes of sequences are denoted by V[λ, M, ∥·, …, ·∥, p, Δs] I, V[λ, M, ∥, ·, ∥, p,  , V[λ, M, ∥, ·, ∥, p, Δs] ∞, and V[λ, M, ∥, ·, ∥, p, , respectively, which were defined and studied by Savaş [25].

  • (2)

    If Mk(x) = M(x), then the above classes of sequences are denoted by V[λ, M, ∥·, …, ·∥, p, , V[λ, M, ∥·, …, ·∥, p, , V[λ, M, ∥·, …, ·∥, p, , and V[λ, M, ∥·, …, ·∥, p, , respectively.

  • (3)

    If Mk(x) = x, for all k ∈ ℕ, then the above classes of sequences are denoted by , , , and , respectively.

  • (4)

    If pk = 1, for all k ∈ ℕ, then we denote the above classes of sequences by V[λ, ∥·, …, ·∥, , V[λ, ∥·, …, ·∥, , V[λ, ∥·, …, ·∥, , and V[λ, ∥·, …, ·∥, , respectively.

  • (5)

    If Mk(x) = M(x), m = 1, and λk = k for all k ∈ ℕ, then the above classes of sequences are denoted by V[M, ∥·, …, ·∥, p, Δs] I, , V[M, ∥·, …, ·∥, p, Δs] ∞, and , respectively, which were defined and studied by Savaş [7].

Theorem 10. The spaces , , and are linear spaces.

Theorem 11. The spaces , , and are paranormed spaces (not totally paranormed) with respect to the paranorm gΔ defined by

()
where H = max {1, sup kpk}.

Proof. Clearly gΔ(−x) = gΔ(x) and gΔ(θ) = 0. Let x = (xk) and  . Then, for ρ > 0 we set

()
Let ρ1 ∈ A1, ρ2 ∈ A2 and ρ = ρ1 + ρ2, then we have
()
Let λt → λ where λt, λ ∈ ℂ, and let gΔ(xt − x) → 0 as t → ∞. We have to show that gΔ(λtxt − λx) → 0 as t → ∞. We set
()
If ρt ∈ A3 and , by using nondecreasing and convexity of the Orlicz function Mj for all j ∈ ℕ that
()
From the above inequality, it follows that
()
and consequently
()
Note that gΔ(xt) ≤ gΔ(x) + gΔ(xt − x), for all t ∈ ℕ. Hence, by our assumption, the right hand of (26) tends to 0 as t → ∞, and the result follows. This completes the proof of the theorem.

Theorem 12. Let ℳ = (Mj), , and be Musielak-Orlicz functions. Then, the following hold:

(a)  V[λ, ℳ′, ∥·, …, ·∥, p, ⊆V[λ, ℳ · ℳ′, ∥·, …, ·∥, p, , provided p = (pk) be such that G0 = inf pk > 0,

(b)  V[λ, ℳ′, ∥·, …, ·∥, p, V[λ, ℳ′′, ∥·, …, ·∥, p, ⊆V[λ, ℳ′ + ℳ′′, ∥·, …, ·∥, p, .

Proof. (a) Let ε > 0 be given. Choose ε1 > 0 such that . Using the continuity of the Orlicz function M, choose 0 < δ < 1 such that 0 < t < δ implies that M(t) < ε1. Let x = (xk) be any element in , put

()
Then, by definition of ideal convergent, we have the set Aδ ∈ I. If n ∉ Aδ then we have
()
Using the continuity of the Orlicz function Mj for all j and the relation (28), we have
()
Consequently, we get
()
This shows that
()

This proves the assertion.

(b) Let x = (xk) be any element in . Then, by the following inequality, the results follow:

()

Theorem 13. The inclusions are strict for s, m ≥ 1 in general where Z = VI, , and .

Proof. We will give the proof for only. The others can be proved by similar arguments. Let . Then let ε > 0 be given; there exists ρ > 0 such that

()
Since Mj for all j ∈ ℕ is nondecreasing and convex, it follows that
()
then we have
()
Let Mk(x) = M(x) = x for all x ∈ [0, ∞[, k ∈ ℕ and λk = k for all k ∈ ℕ. Consider a sequence x = (xk) = (ks). Then, but does not belong to , for s = m = 1. This shows that the inclusion is strict.

Theorem 14. Let 0 < pk ≤ qk for all k ∈ ℕ, then .

Proof. Let , then there exists some ρ > 0 such that

()
This implies that
()
for sufficiently large value of j. Since Mj for all j ∈ ℕ is nondecreasing, we get
()
Thus, . This completes the proof of the theorem.

Theorem 15. (i) If 0<inf pk≤pk < 1, then  V[λ, ℳ, ∥·, …, ·∥, p, ⊆V[λ, ℳ, ∥·, …, ·∥, .

(ii) If 0<pk≤sup kpk<∞, then V[λ, ℳ, ∥·, …, ·∥, ⊆V[λ, ℳ, p, ∥·, …, ·∥, .

Theorem 16. For any sequence of Orlicz functions ℳ = (Mj) which satisfies Δ2-condition, one has .

Theorem 17. Let 0 < pn ≤ qn < 1 and (qn/pn) be bounded; then

()

Theorem 18. For any two sequences p = (pk) and q = (qk) of positive real numbers and for any two n-norms ∥·,…,·∥1 and ∥·,…,·∥2 on X, the following holds:

()
where Z = VI,  ,  , and V∞.

Proof. Proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.

Theorem 19. The sequence spaces , , , and are neither solid nor symmetric, nor sequence algebras for s, m ≥ 1 in general.

Proof. The proof is obtained by using the same techniques of Et [26, Theorems 3.6, 3.8, and 3.9].

Remark 20. If we replace the difference operator by , then for each ε > 0 we get the following sequence spaces:

()

Note. It is clear from definitions that .

Corollary 21. The sequence spaces , where , , and V∞ are paranormed spaces (not totally paranormed) with respect to the paranorm hΔ defined by

()

where H = max {1, sup kpk} and , , and V∞. Also it is clear that the paranorm gΔ and hΔ are equivalent. We state the following theorem in view of Lemma 9. Let X be a standard n-normed space and {e1, e2, …, en} an orthogonal set in X. Then, the following hold:

  • (a)

    V[λ, ℳ, ∥·,…,·∥∞, p, = V[λ, ℳ, ∥·,…,·∥n−1, p, ;

  • (b)

    V[λ, ℳ, ∥·,…,·∥∞, p, = V[λ, ℳ, ∥·,…,·∥n−1, p, ;

  • (c)

    V[λ, ℳ, ∥·,…,·∥∞, p, = V[λ, ℳ, ∥·,…,·∥n−1, p, ;

  • (d)

    V[λ, ℳ, ∥·,…,·∥∞, p, = V[λ, ℳ, ∥·,…,·∥n−1, p, ,

where ∥·,…,·∥∞ is the derived (n − 1)-norm defined with respect to the set {e1, e2, …, en} and ∥·,…,·∥n−1 is the standard (n − 1)-norm on X.

Theorem 22. The spaces and Z[λ, ℳ, ∥·,…,·∥∞, p] are equivalent as topological spaces, where , , and V∞.

Proof. Consider the mapping T : Z[λ,ℳ, ∥·,…,·∥∞, p, , ℳ, ∥·,…,·∥∞, p] defined by T(x) = for each x = (xk) ∈ Z[λ, ℳ, ∥·,…,·∥∞, p, . Then, clearly T is a linear homeomorphism and the proof follows.

Acknowledgments

The authors are most grateful to the editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper.

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