Abstract and Applied Analysis
Volume 2013, Issue 1 123798
Research Article
Open Access

Generalized Difference λ-Sequence Spaces Defined by Ideal Convergence and the 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, Cairo 11566, Egypt shams.edu.eg

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First published: 22 December 2013
https://doi.org/10.1155/2013/123798
Citations: 1
Academic Editor: Abdelghani Bellouquid

Abstract

We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect to λ-sequences and the Musielak-Orlicz function over n-normed spaces. We also studied some topological properties and inclusion relations 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 and ℝ, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using an infinite matrix of complex numbers and the notion of ideal, we aimed to introduce some new sequence spaces with respect to generalized difference operatoron  λ-sequences and the 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 the following: (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]; this concept has been studied by many authors; see for instance [4–7]. The notion of ideal convergence in a 2-normed space was initially introduced by Gürdal [8]. Later on, it was extended to  n-normed spaces by Gürdal and Şahiner [9]. Given that  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
()
In paper [10], the notion of  λ-convergent and bounded sequences is introduced as follows: letbe a strictly increasing sequence of positive real numbers tending to infinity; that is,
()
We say that a sequence x = (xj) ∈ ω is λ-convergent to the number  l ∈ ℂ, called the  λ-limit of  x, if  Λj(x) → l  as  j → ∞, where
()

The class of all sequences  (λj)  satisfying this property is denoted by  Λ.

In particular, we say that x is a λ-null sequence if  Λj(x) → 0  as  j → ∞. Further, we say that  x  is λ-bounded if  sup j|Λj(x)| < ∞. Here and in the sequel, we will use the convention that any term with a zero subscript is equal to naught; for example,  λ0 = 0  and  x0 = 0. Now, it is well known [10] that if  lim j xj = a  in the ordinary sense of convergence, then
()
This implies that
()
which yields that lim jΛj(x) = a and hence x is λ-convergent to  a. We therefore deduce that the ordinary convergence implies the  λ-convergence to the same limit.
An Orlicz function is a function  M : [0, ∞)→[0, ∞)  which is continuous, nondecreasing, and convex with  M(0) = 0 and  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 [11]. Ruckle [12] and Maddox [13] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function  M  is said to satisfy the 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. Lindentrauss and Tzafriri [14] 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, 15–17].

A sequence  M = (Mk)  of Orlicz functions  Mk  for all  k ∈ ℕ is called a Musielak-Orlicz function.

Kizmaz [18] defined the difference sequences 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 [19] 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 [20] 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. [21], generalized the previous notions and unified them as follows.

Let m and s be nonnegative integers, then for  Z  a given sequence space we have
()
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, …, and  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∥ + ∥x1, x2, …, xn∥, which 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-normthe volume of the  n-dimensional parallelepiped spanned by the vectors  x1, x2, …, and  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, …, and  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 an 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 sequences (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.

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) a Musielak-Orlicz function, and the forward generalized difference operator on the class of all sequences  (λj)  satisfying the property  Λ  and an  n-normed space  (X, ∥·…·∥). Further, let  p = (pk)  be any bounded sequence of positive real numbers; we will define the following sequence spaces:
()

Let us consider a few special cases of the aforementioned sets.

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

(2) If pk = 1  for all  k ∈ ℕ then the previous classes of sequences are denoted by , , , and , respectively.

(3) If  Mk(x) = x, for all  k ∈ ℕ and  x ∈ [0, ∞[, then the previous classes of sequences are denoted by , , , and , respectively.

(4) If we take  Mk(x) = M(x), for all  k ∈ ℕand  A = (akj)  as
()
then we denote the previous classes of sequences by, , , and, respectively.
(5) If we take  Mk(x) = M(x)  and  A = (akj)  as
()
where  (ϕk)  is a nondecreasing sequence of positive numbers tending to ∞,  ϕ1 = 1, and ϕk+1 ≤ ϕk + 1, then we denote the previous classes of sequences by,, , and .

(6) If  A = (akj)  as in (22), then we denote the previous classes of sequences by, , , and.

And if λj = j  for all  j ∈ ℕ, then the previous classes of sequences are denoted by, , , andand they are a generalization of the sequence spaces defined by Bakery et al. [22].

(7) By a lacunary  θ = (jr), r = 0,1, 2, …, where  j0 = 0, we will mean an increasing sequence of nonnegative integers with  jr − jr−1 → ∞  as  r → ∞. The interval determined by  θ  will be denoted by  Ir = ]jr−1, jr]  and  hr = jr − jr−1  and let  A = (akj)  as
()

Then we denote the previous classes of sequences by, , , and, respectively.

(8) If  Mk(x) = M(x), for all  k ∈ ℕ, A = I, and  λj = j, then the previous classes of sequences are denoted by, , , and.

(9) If  s = 1, then the previous classes of sequences are denoted by  W[A, ℳ, Δm, Λ, p, ∥·…·∥] I, , W[M, Δm, C, p, ∥·…·∥] ∞, and .

(10) If  m = 1, then the previous classes of sequences are denoted by  [A, ℳ, Δs, Λ, p, ∥·…·∥] I, , W[M, Δs, C, p, ∥·…·∥] ∞, and.

Theorem 10. The spaces , and are linear spaces.

Proof. We will prove the assertion for; the others can be proved similarly. Assume that  x = (xk), , and  α, β ∈ ℂ. Then, there exist  ρ1  and  ρ2  such that the sets

()
()

Since  (X, ∥·…·∥)  is an  n-norm,and  Λj  are linear, and the Orlicz function  Mj  is convex for all  j ∈ ℕ, the following inequality holds:

()
where  L = max {|α|ρ1/(|α|ρ1 + |β|ρ2), |β|ρ2/(|α|ρ1 + |β|ρ2)}. On the other hand from the above inequality we get
()

Since the two sets on the right hand side belong to  I, this completes the proof.

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 , then by using non-decreasing and convexity of the Orlicz function  Mj  for all j ∈ ℕ we get

()

From the previous 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 (34) tends to 0 as  t → ∞, and the result follows. This completes the proof of the theorem.

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

  • (a)

    , provided  p = (pk) such that  G0 = inf pk > 0,

  • (b)

    .

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 and put

()

Then, by the 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 (36), 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, 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 exist ρ > 0 such that

()

Since Mj for all j ∈ ℕ is non-decreasing and convex, it follows that

()
and 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 a sufficiently large value of j. Since Mj for all j ∈ ℕ is non-decreasing, we get
()

Thus, . This completes the proof of the theorem.

Theorem 15. (i) If  0 < inf pk ≤ pk < 1, then

()

(ii) If 1 < pk ≤ sup kpk < ∞, then .

Proof. (i) Let ; since 0 < inf kpk ≤ pk < 1, then we have

()
and hence .

(ii) Let 1 < pk ≤ sup kpk < ∞ and . Then for each 0 < ε < 1 there exists a positive integer j0 such that

()
for all j ≥ j0. This implies that
()
Thus and this completes the proof.

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

Proof. Let and ε > 0  be given. Then, there exist  ρ > 0  such that the set

()

By taking , let ε > 0 and choose δ wit 0 < δ < 1 such that Mj(t) < ε for all j ∈ ℕ; for 0 ≤ t ≤ δ, consider that

()
since  Mj  is continuous for all  n ∈ ℕ.

and for  yj > δ, we use the fact that  yj < yj/δ < 1 + yj/δ. Since ℳ = (Mj)  is non-decreasing and convex, it follows that

()

Since ℳ = (Mj)  satisfies the Δ2-condition, then

()

Hence

()
and then we have
()

This proves that .

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

()

Proof. Let  x = (xj) ∈ W[A, ℳ, Λ, q, ∥·…·∥] ∞  and we put

()

Then  0 < βj ≤ 1, for all  j ∈ ℕ. Let it be such that  0 < β ≤ βj  for all  j ∈ ℕ. Define the sequences  (aj)  and  (bj)  as follows: for  yj ≥ 1, let  aj = yj  and  bj = 0; for  yj < 1, let  aj = 0  and  bj = yj. Then clearly, for all  j ∈ ℕ we have  yj = aj + bj,, , and. Therefore, we have

()

Hence.

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, and  W∞.

Proof. The 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 [23] and Theorems 15, 17, and 18.

Note 1. It is clear from definitions that

()

Theorem 20. The spaces and Z[A, ℳ, p, ∥·…·∥] are equivalent as topological spaces, where , and W∞.

Proof. Consider the mapping

()
defined by for each . Then, clearly T is a linear homeomorphism and the proof follows.

Remark 21. If we replace the difference operator by , then for each ε > 0 we get the following sequence spaces:

()

Corollary 22. The sequence spaces, where, and  W∞, are paranormed spaces (not totally paranormed) with respect to the paranorm  hΔ  defined by

()
where  H = max {1, sup kpk} and, and  W∞. Also it is clear that the paranorms  gΔ  and  hΔ  are equivalent.

We state the following theorem in view of Lemma 9.

Theorem 23. Let X be a standard n-normed space and {e1, e2, …, en} an orthogonal set in X. Then, the following hold:

  • (a)

    ,

  • (b)

    ,

  • (c)

    ,

  • (d)

    ,

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.

Acknowledgment

The author is 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 it.

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