Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spaces
© Mohiuddine et al.; licensee Springer 2014
Received: 20 April 2014
Accepted: 5 August 2014
Published: 2 September 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
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.
defines an -norm on X with respect to .
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 .
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.
where η is known as the kernel of M, is a right differentiable for , , , η is nondecreasing and as .
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 .
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.
where is a double sequence of real numbers such that for j, k and , and is a double sequence of strictly positive real numbers.
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.
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.
Thus . This proves that is a linear space. Similarly we can prove that is also a linear space. □
where and .
- (iii)Let there exist positive numbers and such that
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.
and by continuity of , is a Cauchy sequence in ℝ for each fixed j and k.
Thus as . This proves that is a complete topological linear space.
Hence and this completes the proof. □
Theorem 3.4 Let be a Musielak-Orlicz function which satisfies the -condition. Then .
Since A is RH-regular and , we get . □
which implies that . This completes the proof. □
- (i)Let . Then
- (ii)Let . Then
- (ii)Let for each j and k and . Let . Then for each there exists a positive integer N such 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 .
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 .
where , so .
Hence . So is an ideal in for a Musielak-Orlicz function which satisfies the -condition.
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 .
The proof is complete. □
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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