Some New Classes of Generalized Difference Strongly Summable -Normed Sequence Spaces Defined by Ideal Convergence and Orlicz Function
We study some new generalized difference strongly summable n-normed sequence spaces using ideal convergence and an Orlicz function in connection with de la Vallèe Poussin mean. We give some relations related to these sequence spaces also.
Let , , and be the Banach space of bounded, convergent, and null sequences , respectively, with the usual norm .
A sequence is said to be almost convergent if all of its Banach limits coincide.
Let denote the space of all almost convergent sequences.
Lorentz in  proved that where The following space of strongly almost convergent sequence was introduced by Maddox in : where .
Let be a one-to-one mapping from the set of positive integers into itself such that , , where denotes the th iterative of the mapping in .
A continuous linear functional on is said to be an invariant mean or a -mean, if and only if it satisfies the following conditions:(1), when the sequence is such that for all ;(2), where ;(3), for all .For a certain kind of mapping , we get that every invariant mean extends the functional limit on the space , such that for all . Consequently, we get that , where is the set of bounded sequences with equal -means.
Schaefer in  proved that where Thus we say that a bounded sequence is -convergent, if and only if such that for all , .
Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the -convergence leads naturally to the concept of strong -convergence.
A sequence is said to be strongly -convergent (Mursaleen ), if there exists a number such that We write to denote the set of all strong -convergent sequences and when (6) holds, we write .
Taking , we obtain . Then the strong -convergence generalizes the concept of strong almost convergence.
We also note that The notion of ideal convergence was first introduced by Kostyrko et al.  as a generalization of statistical convergence which was later studied by many other authors.
An Orlicz function is a function , which is continuous, nondecreasing, and convex with , , for , and , as .
Lindenstrauss and Tzafriri  used the idea of Orlicz function to construct the sequence space: The space with the norm becomes a Banach space which is called an Orlicz sequence space.
Kızmaz  studied the difference sequence spaces , , and of crisp sets. The notion is defined as follows: for and , where , for all .
The above spaces are Banach spaces, normed by Later the idea of Kızmaz  was applied to introduce different types of difference sequence spaces and study their different properties by many others later on.
The generalized difference notion is defined as follows. For and , This generalized difference has the following binomial representation: The concept of 2-normed space was initially introduced by Gähler , in the mid of 1960s, while that of -normed spaces can be found in Misiak . Since then, many other authors have used this concept and obtained various results. Recently, several various activities have been initiated to study summability, sequence spaces, and related topics in these spaces. The notion of ideal convergence in 2-normed spaces was initially introduced by Gurdal . Later on, it was extended to -normed spaces by Gurdal and Sahiner in .
2. Definitions and Preliminaries
Let and be a real vector space. A real valued function on satisfies the following four properties:(1) if and only if are linearly dependent;(2) is invariant under permutation;(3), for all ;(4)is called an -norm on and the pair is called an -normed space.
Let be a nonempty set. Then a family of sets (power sets of ) is said to be an if is additive that is and hereditary that is .
A sequence in a normed space is said to be -convergent to with respect to -norm, if for each , the set The generalized de la Vallée Poussin mean is defined by where for .
Then a sequence is said to be -summable to a number , if as , and we write for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by de la Vallée Poussin method.
Maddox introduced and studied the special case, where , for ; the sets , , and reduce to the sets , , and .
In this paper, we define some new sequence spaces in -normed spaces by using Orlicz function with notion of generalized de la Vallèe Poussin mean, generalized difference sequences, and ideals. We will also introduce and examine certain new sequence spaces using the above tools as also the -norm.
3. Main Results
Let be an admissible ideal of , let be an Orlicz function, and let be a -normed space. Further, let be a bounded sequence of positive real numbers. By , we denote the space of all sequences defined over .
In this paper, we have introduced the following sequence spaces: In particular, if we take , for all , we have Similarly, when , then , , and are reduced to In particular, if we put , for all , then we have the spaces Further when , for , the sets and are reduced to and , respectively.
Now, if we consider , then we can easily obtain If , with as uniformly in , then we write .
The following well-known inequality will be used later.
If and , then for all and .
Lemma 1. Let and . Then , if and only if , where .
Note that no other relation between and is needed in Lemma 1.
Theorem 2. Let . Then, implies . Let . If , then is unique.
Proof. Let .
By the definition of Orlicz function, we have, for all , Since , it follows that And consequently, .
Let . Suppose that , , and .
Now, from (22) and the definition of Orlicz, we have since Hence, Further, as , and therefore From (27) and (28), it follows that and by the definition of an Orlicz function, we have .
Hence, and this completes the proof.
Theorem 3. (i) Let . Then, (ii) Let . Then,
Theorem 4. Let stand for , , or and . Then the inclusion is strict. In general, for all and the inclusion is strict.
Proof. Let us take .
Let . Then for given , we have Since is nondecreasing and convex, it follows that Hence we have Since the set on the right hand side belongs to , so does the left hand side. The inclusion is strict as the sequence , for example, belongs to but does not belong to for and for all .
Theorem 5. and are complete linear topological spaces, with paranorm , where is defined by where .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The first author acknowledges that this research was part of the research project and partially supported by the Universiti Putra Malaysia under Grant no. ERGS 1-2013/5527179. The work of second author was carried out under the Postdoctoral Fellow under National Board of Higher Mathematics, DAE, Project no. NBHM/PDF.50/2011/64.
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