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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 123798, 13 pages
Generalized Difference λ-Sequence Spaces Defined by Ideal Convergence and the 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, Cairo 11566, Egypt
Received 2 June 2013; Accepted 22 September 2013
Academic Editor: Abdelghani Bellouquid
Copyright © 2013 Awad A. Bakery. 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.
We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect to -sequences and the Musielak-Orlicz function over -normed spaces. We also studied some topological properties and inclusion relations between these spaces.
Throughout the paper,,,, anddenote 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 and , 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-sequences and the Musielak-Orlicz function in-normed linear spaces. By an ideal we mean a familyof subsets of a nonempty setsatisfying the following: (i); (ii)imply; (iii), imply, while an admissible idealoffurther satisfiesfor 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 ; 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 . Later on, it was extended to-normed spaces by Gürdal and Şahiner . Given thatis a nontrivial ideal in , the sequencein a normed spaceis said to be-convergent to, if, for each,
A sequencein a normed spaceis said to be-bounded if there existssuch that
A sequencein a normed spaceis said to be-Cauchy if, for each, there exists a positive integersuch that
In paper , the notion of-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 is -convergent to the number, called the-limit of, ifas, where
The class of all sequencessatisfying this property is denoted by.
In particular, we say that is a -null sequence ifas. Further, we say thatis -bounded if. Here and in the sequel, we will use the convention that any term with a zero subscript is equal to naught; for example,and. Now, it is well known  that ifin the ordinary sense of convergence, then
This implies that which yields that and hence is -convergent to. We therefore deduce that the ordinary convergence implies the-convergence to the same limit.
An Orlicz function is a functionwhich is continuous, nondecreasing, and convex with andforand, as. If convexity ofis 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 functionis said to satisfy the -condition for all values of, if there exists a constant, such that. The-condition is equivalent tofor all values ofand for. Lindentrauss 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
A sequenceof Orlicz functionsfor all is called a Musielak-Orlicz function.
Kizmaz  defined the difference sequences , , 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 previous notions and unified them as follows.
Let and be nonnegative integers, then fora given sequence space we have where, for.
2. Definitions and Preliminaries
Let andbe a linear space over the fieldof dimension, whereandis the field of real or complex numbers. A real valued function onsatisfies the following four conditions:(1)if and only ifare linearly dependent in;(2)is invariant under permutation;(3)for any;(4), which is called an-norm onand the pairis called an-normed space over the field. For example, we may takebeing equipped with the-normthe volume of the-dimensional parallelepiped spanned by the vectorswhich may be given explicitly by the formula where for each.
Letbe an-normed space of dimensionanda linearly independent set in. Then, the functionondefined 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 sequencein an-normed space is said to be convergent toif
Definition 2. A sequencein an-normed space is called Cauchy (with respect to-norm) if
If every Cauchy sequence inconverges to an, thenis said to be complete (with respect to the-norm). A complete-normed space is called an -Banach space.
Definition 3. A sequencein an-normed space is said to be-convergent towith respect to-norm, if, for each , the set
Definition 4. A sequencein an-normed spaceis said to be-Cauchy if, for each , there exists a positive integersuch that the set
Letbe a sequence; thendenotes the set of all permutations of the elements of; that is,is a permutation of .
Definition 5. A sequence spaceis said to be symmetric iffor all.
Definition 6. A sequence spaceis said to be normal (or solid) if, wheneverand for all sequences of scalars withfor all.
Definition 7. A sequence spaceis 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-normdefined with respect to the orthogonal setis equivalent to the standard-norm. To be precise, one has
for all, where.
For any bounded sequenceof positive numbers, one has the following well known inequality: ifand, then, for alland.
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 , a Musielak-Orlicz function, and the forward generalized difference operator on the class of all sequencessatisfying the propertyand an-normed space. Further, letbe 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, for all then the previous classes of sequences are denoted by, , , and , respectively.
(2) If for all then the previous classes of sequences are denoted by , , , and , respectively.
(3) If, for all and, then the previous classes of sequences are denoted by , , , and , respectively.
(4) If we take, for allandas then we denote the previous classes of sequences by, , , and, respectively.
(5) If we takeandas whereis a nondecreasing sequence of positive numbers tending to ,, and , then we denote the previous classes of sequences by,, , and .
(6) Ifas in (22), then we denote the previous classes of sequences by, , , and.
And if for all, then the previous classes of sequences are denoted by, , , andand they are a generalization of the sequence spaces defined by Bakery et al. .
(7) By a lacunary, , where, we will mean an increasing sequence of nonnegative integers withas. The interval determined bywill be denoted byandand letas
Then we denote the previous classes of sequences by, , , and, respectively.
(8) If, for all, , and, then the previous classes of sequences are denoted by, , , and.
(9) If, then the previous classes of sequences are denoted by, , , and .
(10) If, then the previous classes of sequences are denoted by, , , and.
Theorem 10. The spaces , and are linear spaces.
Proof. We will prove the assertion for; the others can be proved similarly. Assume that, , and. Then, there existandsuch that the sets
Sinceis an-norm,andare linear, and the Orlicz functionis convex for all, the following inequality holds: where. On the other hand from the above inequality we get
Since the two sets on the right hand side belong to, this completes the proof.
Theorem 11. The spaces , , and are paranormed spaces (not totally paranormed) with respect to the paranorm defined by where.
Proof. Clearlyand. Letand. Then, forwe set
Let, , and; then we have
Letwhere, and let as . We have to show that as . We set
If and , then by using non-decreasing and convexity of the Orlicz functionfor all we get
From the previous inequality, it follows that and consequently
Note that, for all. Hence, by our assumption, the right hand of (34) tends to 0 as, and the result follows. This completes the proof of the theorem.
Theorem 12. Let , , and be the Musielak-Orlicz functions. Then, the following hold:(a), provided such that,(b).
Proof. (a) Letbe given. Choosesuch that. Using the continuity of the Orlicz function, choosesuch thatimplies that.
Let be any element in and put
Then, by the definition of ideal convergent, we have the set. If, then we have
Using the continuity of the Orlicz functionfor alland the relation (36), 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 exist such that
Since for all is non-decreasing and convex, it follows that and 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 a sufficiently large value of . Since for all is non-decreasing, we get
Thus, . This completes the proof of the theorem.
Theorem 15. (i) If, then
(ii) If , then .
Proof. (i) Let ; since , then we have
and hence .
(ii) Let and . Then for each there exists a positive integer such that for all . This implies that Thus and this completes the proof.
Theorem 16. For any sequence of the Orlicz functions which satisfies the -condition, we have .
Proof. Let and be given. Then, there existsuch that the set
By taking , let and choose wit such that for all ; for , consider that sinceis continuous for all.
and for, we use the fact that. Since is non-decreasing and convex, it follows that
Since satisfies the -condition, then
Hence and then we have
This proves that