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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 972363, 9 pages
Some Generalized Difference Sequence Spaces Defined by Ideal Convergence and Musielak-Orlicz Function
1Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, 11566 Cairo, Egypt
3Department of Mathematics, Faculty of Education, Alzaeim Alazhari University, P.O. Box 1432, 13311 Khartoum, Sudan
Received 25 February 2013; Accepted 8 May 2013
Academic Editor: Feyzi Başar
Copyright © 2013 Awad A. Bakery et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Throughout the paper , , , , and 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 , , 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 -normed linear spaces. By an ideal we mean a family of subsets of a nonempty set satisfying (i) ; (ii) imply ; (iii) , imply , while an admissible ideal of further satisfies for each . The notion of ideal convergence was introduced first by Kostyrko et al.  as a generalization of statistical convergence. The concept of 2-normed spaces was initially introduced by Gähler  in the 1960s, while that of -normed spaces can be found in , 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 . Later on, it was extended to -normed spaces by Gürdal and Şahiner . Given is a nontrivial ideal in , the sequence in a normed space is said to be -convergent to , if for each ,
A sequence in a normed space is said to be -bounded if there exists such that
A sequence in a normed space is said to be -Cauchy if for each , there exists a positive integer such that
An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and , as . If convexity of is replaced by , then it is called a modulus function, introduced by Nakano . Ruckle  and Maddox  used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function is said to satisfy -condition for all values of , if there exists a constant , such that . The -condition is equivalent to for all values of and for . Lindenstrauss and Tzafriri  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 is closely related to the space , which is an Orlicz sequence space with for .
A sequence of Orlicz functions for all is called a Musielak-Orlicz function, for a given Musielak-Orlicz function . Kızmaz  defined the difference sequence spaces , , and as follows: = , for , , and , where , for all . The above spaces are Banach spaces, normed by . The notion of difference sequence spaces was generalized by Et and Colak  as follows: , for , , and , where , and so that . Tripathy and Esi  introduced the following new type of difference sequence spaces.
, , , and , where , for all . Tripathy et al.  generalized the above notions and unified them as follows. Let , be nonnegative integers, then for a given sequence space we have where
Also let be nonnegative integers, then for a given sequence space we have where where , for
2. Definitions and Preliminaries
Let and be a linear space over the field of dimension , where and is the field of real or complex numbers. A real valued function on satisfies the following four conditions:(1) if and only if are linearly dependent in ;(2) is invariant under permutation;(3) for any ;(4); is called an -norm on and the pair is called an -normed space over the field . For example, we may take being equipped with the -norm the volume of the -dimensional parallelepiped spanned by the vectors which may be given explicitly by the formula
where for each .
Let be an -normed space of dimension and a linearly independent set in . Then, the function on defined by
defines an -norm on with respect to and this is known as the derived -norm. The standard -norm on , a real inner product space of dimension , is as follows:
where denotes the inner product on . If we take , then
For , this -norm is the usual norm .
Definition 1. A sequence in an -normed space is said to be convergent to if,
Definition 2. A sequence in an -normed space is called Cauchy (with respect to -norm) if, If every Cauchy sequence in converges to an , then is said to be complete (with respect to the -norm). A complete -normed space is called -Banach space.
Definition 3. A sequence in an -normed space is said to be -convergent to with respect to -norm, if for each , the set
Definition 4. A sequence in an -normed space is said to be -Cauchy if for each , there exists a positive integer such that the set Let be a sequence; then denotes the set of all permutations of the elements of ; that is, is a permutation of .
Definition 5. A sequence space is said to be symmetric if for all .
Definition 6. A sequence space is said to be normal (or solid) if , whenever and for all sequence of scalars with for all .
Definition 7. A sequence space is said to be a sequence algebra if then .
Lemma 8. Every -normed space is an -normed space for all . In particular, every -normed space is a normed space.
Lemma 9. On a standard -normed space , the derived -norm defined with respect to the orthogonal set is equivalent to the standard -norm . To be precise, one has
for all , where .
Let be a nondecreasing sequence of positive real numbers tending to infinity and let and . In summability theory, de La Vallée-Poussin mean was first used to define the -summability by Leindler . Also the -summable sequence spaces have been studied by many authors including [22, 23]. The generalized de La Vallée-Poussin’s mean of a sequence is defined as follows: , where for . We write
for some ,
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 for the spaces , , and reduce to the spaces , , and introduced by Maddox . The following new paranormed sequence space is defined in .
. If one takes for all ; the space reduced to normed space defined by . The details of the sequence spaces mentioned above can be found in .
For any bounded sequence of positive numbers, one has the following well-known inequality.
If and , then , for all and .
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 be an admissible ideal of , be a Musielak-Orlicz function, and an -normed space. Further, let be any bounded sequence of positive real numbers,
The above sequence spaces contain some unbounded sequences for . If , , for all and for all , then but .
Let us consider a few special cases of the above sets.(1)If = 2, = 1, and = , then the above classes of sequences are denoted by , , , , , , , , , , , , , , , and , , , , , respectively, which were defined and studied by Savaş .(2)If = , then the above classes of sequences are denoted by , , , , , , , , , , , , , , , and , , , , , respectively.(3)If , for all , then the above classes of sequences are denoted by , , , and , respectively.(4)If , for all , then we denote the above classes of sequences by , , , , , , , , , and , , , respectively.(5)If , , and for all , then the above classes of sequences are denoted by , , , and , respectively, which were defined and studied by Savaş .
Theorem 10. The spaces , , and are linear spaces.
Theorem 11. The spaces , , and are paranormed spaces (not totally paranormed) with respect to the paranorm defined by where .
Proof. Clearly and . Let and . Then, for we set Let , and , then we have Let where , and let as . We have to show that as . We set If and , by using nondecreasing and convexity of the Orlicz function for all that From the above inequality, it follows that and consequently Note that , for all . Hence, by our assumption, the right hand of (26) tends to 0 as , and the result follows. This completes the proof of the theorem.
Theorem 12. Let , , and be Musielak-Orlicz functions. Then, the following hold:
(a) , , , , , , , , , provided = be such that = > ,
(b) , , , , , , , , , + , , , .
(a) Let be given. Choose such that . Using the continuity of the Orlicz function , choose such that implies that . Let be any element in , put
Then, by definition of ideal convergent, we have the set . If then we have
Using the continuity of the Orlicz function for all and the relation (28), we have
Consequently, we get
This shows that
This proves the assertion.
(b) Let be any element in . Then, by the following inequality, the results follow:
Theorem 13. The inclusions are strict for in general where , , and .
Proof. We will give the proof for only. The others can be proved by similar arguments. Let . Then let be given; there exists such that Since for all is nondecreasing and convex, it follows that then we have Let for all , and for all . Consider a sequence . Then, but does not belong to , for . This shows that the inclusion is strict.
Theorem 14. Let for all , then .
Proof. Let , then there exists some such that This implies that for sufficiently large value of . Since for all is nondecreasing, we get Thus, . This completes the proof of the theorem.
(i) If < , then , , , , , , , .
(ii) If , then , , , , , , , .
Theorem 16. For any sequence of Orlicz functions which satisfies -condition, one has .
Theorem 17. Let and be bounded; then
Theorem 18. For any two sequences and of positive real numbers and for any two n-norms and on , the following holds: where , , , and .
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 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 we get the following sequence spaces:
Note. It is clear from definitions that .
Corollary 21. The sequence spaces , where , , and are paranormed spaces (not totally paranormed) with respect to the paranorm defined by
where and , , and . Also it is clear that the paranorm and are equivalent. We state the following theorem in view of Lemma 9. Let be a standard -normed space and an orthogonal set in . Then, the following hold:(a), , , , = , , , , ; (b), , , , = , , , , ;(c), , , , = , , , , ;(d), , , , = , , , , ,
where is the derived -norm defined with respect to the set and is the standard -norm on .
Theorem 22. The spaces and are equivalent as topological spaces, where , , and .
Proof. Consider the mapping ,, , , , , , defined by = for each = , , , , . Then, clearly is a linear homeomorphism and the proof follows.
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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