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Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spaces
Journal of Inequalities and Applicationsvolume 2014, Article number: 332 (2014)
The aim of this paper is to introduce some generalized spaces of double sequences with the help of the Musielak-Orlicz function and four-dimensional bounded-regular (shortly, RH-regular) matrices over n-normed spaces. Some topological properties and inclusion relations between these spaces are investigated.
1 Introduction, notations, and preliminaries
The concept of 2-normed spaces was first introduced by Gähler  in the mid-1960s, while that of n-normed spaces one can find in Misiak . Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi . Let and X be a linear space over the field of real numbers ℝ of dimension d, where . A real valued function on satisfying the following four conditions:
if and only if are linearly dependent in X,
is invariant under permutation,
for any , and
is called an n-norm on X, and the pair is called a n-normed space over the field ℝ.
For example, we may take being equipped with the n-norm = the volume of the n-dimensional parallelepiped spanned by the vectors which may be given explicitly by the formula
where for each . Let be an n-normed space of dimension and be a linearly independent set in X. Then the function on defined by
defines an -norm on X with respect to .
A sequence in a n-normed space is said to converge to some if
A sequence in a n-normed space is said to be Cauchy if
If every Cauchy sequence in X converges to some , then X is said to be complete with respect to the n-norm. A complete n-normed space is called n-Banach space.
Of the definitions of convergence commonly employed for double series, only that due to Pringsheim permits a series to converge conditionally. Therefore, in spite of any disadvantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series. Chief among the reasons why the theory of double sequences, under the Pringsheim definition of convergence, presents difficulties not encountered in the theory of simple sequences is the fact that a double sequence may converge without being a bounded function of i and j. Thus it is not surprising that many authors in dealing with the convergence of double sequences should have restricted themselves to the class of bounded sequences or, in dealing with the summability of double series, to the class of series for which the function whose limit is the sum of the series is a bounded function of i and j. Without such a restriction, peculiar things may sometimes happen; for example, a double power series may converge with partial sum unbounded at a place exterior to its associated circles of convergence. Nevertheless there are problems in the theory of double sequences and series where this restriction of boundedness as it has been applied is considerably more stringent than need be. In , Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences has also been done by Bromwich . Later on, it was studied by various authors, e.g. Móricz , Móricz and Rhoades , Başarır and Sonalcan , Mursaleen and Mohiuddine [11, 12], and many others. Mursaleen  has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see ). Altay and Başar  have recently introduced the double sequence spaces , , , , , and consisting of all double series whose sequence of partial sums are in the spaces , , , , , and , respectively. Başar and Sever  extended the well known space from single sequence to double sequences, denoted by , and established its interesting properties. The authors of  defined some convex and paranormed sequences spaces and presented some interesting characterization. Most recently, Mohiuddine and Alotaibi  introduced some new double sequences spaces for σ-convergence of double sequences and invariant mean, and also determined some inclusion results for these spaces. For more details on these concepts, one is referred to [19–21].
The notion of difference sequence spaces was introduced by Kızmaz , who studied the difference sequence spaces , , and . The notion was further generalized by Et and Çolak  by introducing the spaces , , and .
Let w be the space of all complex or real sequences and let r and s be two nonnegative integers. Then for , we have the following sequence spaces:
where and for all , which is equivalent to the following binomial representation:
We remark that for and , we obtain the sequence spaces which were introduced and studied by Et and Çolak  and Kızmaz , respectively. For more details as regards sequence spaces, see [24–31] and references therein.
An Orlicz function is a continuous, nondecreasing, and convex such that , for and as . If convexity of the Orlicz function is replaced by , then this function is called modulus function. Lindenstrauss and Tzafriri  used the idea of Orlicz function to define the sequence space
known as an Orlicz sequence space. The space is a Banach space with the norm
Also it was shown in  that every Orlicz sequence space contains a subspace isomorphic to . An Orlicz function M can always be represented in the integral form
where η is known as the kernel of M, is a right differentiable for , , , η is nondecreasing and as .
is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space and its subspace are defined as follows:
where is a convex modular defined by
We consider equipped with the Luxemburg norm
or equipped with the Orlicz norm
A Musielak-Orlicz function is said to satisfy the -condition if there exist constants , and a sequence (the positive cone of ) such that the inequality
holds for all and , whenever .
A double sequence is said to be bounded if . We denote by , the space of all bounded double sequences.
By the convergence of double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence is said to converge to the limit L in Pringsheim sense (denoted by ) provided that given there exists such that whenever (see ). We shall write more briefly as P-convergent. If, in addition, , then x is said to be boundedly P-convergent to L. We shall denote the space of all bounded convergent double sequences (or, boundedly P-convergent) by .
Let and let be given. By , we denote the characteristic function of the set .
Let be a four-dimensional infinite matrix of scalars. For all , where , the sum
is called the A-means of the double sequence . A double sequence is said to be A-summable to the limit L if the A-means exist for all m, n in the sense of Pringsheim’s convergence:
A four-dimensional matrix A is said to be bounded-regular (or RH-regular) if every bounded P-convergent sequence is A-summable to the same limit and the A-means are also bounded.
The following is a four-dimensional analog of the well-known Silverman-Toeplitz theorem .
The four-dimensional matrix A is RH-regular if and only if
(RH1) for each j and k,
(RH3) for each k,
(RH4) for each j,
(RH5) for all .
2 Some spaces of double sequences over n-normed spaces
Recently, Yurdakadim and Tas  defined the spaces of double sequences for RH-regular four-dimensional matrices and Orlicz functions and also established some interesting results. Quite recently, Mohiuddine et al.  defined and studied some paranormed double difference sequence spaces for four-dimensional bounded-regular matrices and Musielak-Orlicz functions.
Recall that a linear topological space X over the real field ℝ (the set of real numbers) is said to be a paranormed space if there is a subadditive function such that , and scalar multiplication is continuous, i.e., and imply for all α’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.
Let be a n-normed space and denotes the space of X-valued sequences. Let be a Musielak-Orlicz function, that is, ℳ is a sequence of Orlicz functions and let be a nonnegative four-dimensional bounded-regular matrix. Then we define the following double difference sequence spaces over n-normed spaces:
where is a double sequence of real numbers such that for j, k and , and is a double sequence of strictly positive real numbers.
We obtain the following sequence spaces from the above sequence spaces: and by giving particular values to ℳ, p, u, and A.
If , then we write and instead of and , respectively.
If for all j, k, then we write and instead of and , respectively.
If for all j, k, then we write and instead of and , respectively.
If , then we write and instead of and , respectively, where denotes the nm th Cesàro mean of double sequence .
If and , then we write and instead of and , respectively.
Throughout the paper, we shall use the following inequality: Let and be two double sequences. Then
where and (see ).
3 Main results
Theorem 3.1 Let be a Musielak-Orlicz function, be a nonnegative four-dimensional RH-regular matrix, be a bounded sequence of positive real numbers and be a sequence of strictly positive real numbers. Then and are linear spaces over the field ℝ of reals.
Proof Suppose and and . Then there exist positive numbers , such that
Let = . Since is a nondecreasing and convex so by using inequality (2.1), we have
Thus . This proves that is a linear space. Similarly we can prove that is also a linear space. □
Theorem 3.2 Let be a Musielak-Orlicz function, be a nonnegative four-dimensional RH-regular matrix, be a bounded sequence of positive real numbers and be a sequence of strictly positive real numbers. Then and are paranormed spaces with the paranorm
where and .
Proof (i) Clearly for . Since , we get .
Let there exist positive numbers and such that
Let . Then by using Minkowski’s inequality, we have
Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition,
where . Since , we have
So, the fact that the scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem. □
Theorem 3.3 Let be a Musielak-Orlicz function, be a nonnegative four-dimensional RH-regular matrix, be a bounded sequence of positive real numbers and be a sequence of strictly positive real numbers. Then and are complete topological linear spaces.
Proof Let be a Cauchy sequence in , that is, as . Then we have
Thus for each fixed j and k as , since is nonnegative, we are granted that
and by continuity of , is a Cauchy sequence in ℝ for each fixed j and k.
Since ℝ is complete as , we have for each . For , there exists a natural number N such that
Since for any fixed natural number M, we have
and by letting in the above expression we obtain
Since M is arbitrary, by letting we obtain
Thus as . This proves that is a complete topological linear space.
Now we shall show that is a complete topological linear space. For this, since is also a sequence in by definition of , for each q there exists with
whence, from the fact that and from the definition of a Musielak-Orlicz function, we have as and so converges to L. Thus
Hence and this completes the proof. □
Theorem 3.4 Let be a Musielak-Orlicz function which satisfies the -condition. Then .
Proof Let , that is,
Let and choose δ with such that for . Write and consider
For , we use the fact that . Hence
Since ℳ satisfies the -condition, we have
Since A is RH-regular and , we get . □
Theorem 3.5 Let be a Musielak-Orlicz function and let be a nonnegative four-dimensional RH-regular matrix. Suppose that . Then
Proof In order to prove that . It is sufficient to show that . Now, let . By definition of β, we have for all . Since , we have for all . Let . Thus, we have
which implies that . This completes the proof. □
Let . Then
Let . Then
Proof (i) Let . Then since , we obtain the following:
Let for each j and k and . Let . Then for each there exists a positive integer N such that
This implies that
Therefore . This completes the proof. □
Lemma 3.7 Let G be an ideal in and let . Then x is in the closure of G in if and only if for all .
Proof It is easy to prove so we omit the proof. □
Lemma 3.8 Let be a Musielak-Orlicz function which satisfies the -condition and let be a nonnegative four-dimensional RH-regular matrix. Then is an ideal in .
Proof Let and . We need to show that . Since , there exists such that . In this case for all j, k. Since ℳ is nondecreasing and satisfies -condition, we have
for all j, k, and . Therefore . Thus, . This completes the proof. □
Lemma 3.9 If A is a nonnegative four-dimensional RH-regular matrix, then is a closed ideal in .
Proof We have and it is clear that . For , we get . Now, we have
by the -condition and the convexity of M. Since
where , so .
Let and . Thus, there exists a positive integer K, so that for every j, k, we have . Therefore
Hence . So is an ideal in for a Musielak-Orlicz function which satisfies the -condition.
Now, we have to show that is closed. Let there exists such that . For every there exists such that for all , . Now, for , we have
Since and A is RH-regular, we get
so . This completes the proof. □
Theorem 3.10 Let be a bounded sequence, be a Musielak-Orlicz function which satisfies the -condition and A be a nonnegative four-dimensional RH-regular matrix. Then .
Proof Without loss of generality we may take and establish
Since , therefore . We need to show that . Notice that if , then
for all n, m. Observe that whenever by Lemma 3.7 and Lemma 3.8, so
The proof is complete. □
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The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.