Research Article | Open Access
Some Paranormed Double Difference Sequence Spaces for Orlicz Functions and Bounded-Regular Matrices
The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz function and four-dimensional bounded-regular (shortly, RH-regular) matrices . We also make an effort to study some topological properties and inclusion relations between these double difference sequence spaces.
1. Introduction, Notations, and Preliminaries
In , Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences is also found by Bromwich . Later on, it was studied by various authors, for example, Móricz , Móricz and Rhoades , Başarır and Sonalcan , Mursaleen and Mohiuddine [6–8], 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 [10, 11]). 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 -convergence of double sequences and invariant mean and also determined some inclusion results for these spaces. For more details on these concepts, one can be referred to [16–18].
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 be the space of all complex or real sequences and let and 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 about sequence spaces see [21–27] and references therein.
An Orlicz function is continuous, nondecreasing, and convex such that , for and as . If convexity of Orlicz function is replaced by , then this function is called modulus function. Lindenstrauss and Tzafriri  used the idea of Orlicz function to define the following sequence space: which is 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 can always be represented in the following integral form: where is known as the kernel of , is a right differentiable for is nondecreasing, and as .
A sequence of Orlicz functions is said to be a Musielak-Orlicz function (see [29, 30]). A sequence is defined by which is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , 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 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; that is, a double sequence is said to converge to the limit in Pringsheim sense (denoted by, ) provided that given there exists such that whenever (see ). We will write more briefly as -convergent. If, in addition, , then is said to be boundedly P-convergent to . We will denote the space of all bounded convergent double sequences (or boundedly -convergent) by .
Let and let be given. By , we denote the characteristic function of the set .
Let be a four-dimensional infinite matrix of scalers. For all , where , the sum is called the - of the double sequence . A double sequence is said to be - to the limit if the -means exist for all in the sense of Pringsheim’s convergence:
A four-dimensional matrix is said to be bounded-regular (or RH-regular) if every bounded -convergent sequence is -summable to the same limit and the -means are also bounded.
The following is a four-dimensional analogue of the well-known Silverman-Toeplitz theorem .
2. The Double Difference Sequence Spaces
In this section, we define some new paranormed double difference sequence spaces with the help of Musielak-Orlicz functions and four-dimensional bounded-regular matrices. Before proceeding further, first we recall the notion of paranormed space as follows.
A linear topological space over the real field (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; that is, and imply for all ’s in and all ’s in , where is the zero vector in the linear space .
Let be a Musielak-Orlicz function; that is, is a sequence of Orlicz functions and let be a nonnegative four-dimensional bounded-regular matrix. Then, we define the following: where is a double sequence of real numbers such that for , , and is a double sequence of strictly positive real numbers.
Remark 2. If we take in and , then we have the following spaces:
Remark 3. Let for all . Then and are reduced to respectively.
Remark 4. Let for all . Then, the spaces and are reduced to respectively.
Remark 5. If we take in and , then we have the following spaces:
Remark 6. If we take and in and , then we have the following spaces:
Remark 7. Let for all . If, in addition, and , then the spaces and are reduced to and which were introduced and studied by Yurdakadim and Tas  as below:
Throughout the paper, we will use the following inequality: let and be two double sequences. Then where and (see ). We will also assume throughout this paper that the symbol will denote the sublinear Musielak-Orlicz function.
3. Main Results
Theorem 8. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and a sequence of strictly positive real numbers. Then and are linear spaces over the complex field .
Proof. Let and . Then there exist integers and such that and .
Since is a nondecreasing function, so by inequality (21), we have
Thus . This proves that is a linear space. Similarly we can prove that is also a linear space.
Theorem 9. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and a sequence of strictly positive real numbers. Then and are paranormed spaces with the paranorm where and .
Proof. We will prove the result for . Let . Then for each , exists. Also it is clear that , and .
We now show that the scalar multiplication is continuous. First observe the following: where denotes the integer part of . It is also clear that if and implies . For fixed , if , then . We need to show that for fixed implies . Let . Thus
Then, for there exists such that for . Also, for each with , since there exists an integer such that
Let . We have for each with
Also from (26), for , we have
Thus is an integer independent of such that
Since , therefore
For each and by the continuity of as , we have the following: in Pringsheim’s sense. Now choose such that implies
In the same manner, we have
It follows from (31), (34), (35), and (36) that
Thus as . Therefore is a paranormed space. Similarly, we can prove that is a paranormed space. This completes the proof.
Theorem 10. Let be a sublinear Musielak-Orlicz function, a nonnegative four-dimensional -regular matrix, a bounded sequence of positive real numbers, and 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 and as , since is nonnegative, we are granted that and by continuity of , is a Cauchy sequence in for each fixed and .
Since is complete as , we have for each . Now from (36), we have that, for , there exists a natural number such that
Since for any fixed natural number , from (38) we have By letting in the above expression we obtain
Since is arbitrary, by letting we obtain
Thus as . This proves that is a complete topological linear space.
Now we will show that is a complete topological linear space. For this, since is also a sequence in by definition of , for each , there exists with whence from the fact that and from the definition of Musielak-Orlicz function, we have as and so converges to . Thus
Hence and this completes the proof.
Theorem 11. Let be a sublinear Musielak-Orlicz function which satisfies the -condition. Then .
Proof. Let ; that is,
Let and choose with such that for . Write and consider
For , we use the fact that . Hence
Since satisfies the -condition, we have and hence
Since is -regular and , we get .
Theorem 12. Let be a sublinear 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 , we have for all . Since , we have for all . Let . Thus, we have which implies that . This completes the proof.
(i) Let . Then
(ii) Let . Then
(i) Let . Then since , we obtain the following:
(ii) Let for each and and . Let . Then for each there exists a positive integer such that
This implies that
Therefore . This completes the proof.
Lemma 14. Let be a sublinear Musielak-Orlicz function which satisfies the -condition and let be a nonnegative four-dimensional -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 . Since is nondecreasing and satisfies -condition, we have for all and . Therefore . Thus . This completes the proof.
Lemma 15. Let be an ideal in and let . Then is in the closure of in if and only if for all .
Proof. Let be in the closure of and let be given. Suppose that such that and observe that . Define a double sequence by
Clearly . Since and , hence .
Conversely, if then . It follows that for all ; then is in the closure of .
Lemma 16. If is a nonnegative four-dimensional -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 . Since