Generalized difference strongly summable sequence spaces of fuzzy real numbers defined by ideal convergence and Orlicz function
© Kılıçman and Borgohain; licensee Springer. 2013
Received: 25 June 2013
Accepted: 10 September 2013
Published: 7 November 2013
We study some new generalized difference strongly summable sequence spaces of fuzzy real numbers 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.
MSC:40A05, 40A25, 40A30, 40C05.
KeywordsOrlicz function difference operator ideal convergence de la Vallèe Poussin mean
Let , c 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.
Let σ be a one-to-one mapping from the set of positive integers into itself such that , , where denotes the m th iterative of the mapping σ in n, see .
Thus, we say that a bounded sequence is σ-convergent if and only if such that for all , .
We write to denote the set of all strong σ-convergent sequences, and when (1) holds, we write .
Taking , we obtain . Then the strong σ-convergence generalizes the concept of strong almost convergence.
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, non-decreasing and convex with , , for and , as .
becomes a Banach space, which is called an Orlicz sequence space.
for , where , for all .
The generalized difference is defined as follows:
The concept of fuzzy set theory was introduced by Zadeh in the year 1965. It has been applied for the studies in almost all the branches of science, where mathematics is used. Workers on sequence spaces have also applied the notion and introduced sequences of fuzzy real numbers and studied their different properties.
2 Definitions and preliminaries
A fuzzy real number X is a fuzzy set on R, i.e., a mapping () associating each real number t with its grade of membership .
A fuzzy real number X is called convex if , where .
If there exists such that , then the fuzzy real number X is called normal.
A fuzzy real number X is said to be upper semicontinuous if for each , , for all , it is open in the usual topology of R.
The class of all upper semicontinuous, normal, convex fuzzy real numbers is denoted by .
Define by , for . Then it is well known that is a complete metric space.
A sequence of fuzzy real numbers is said to converge to the fuzzy number , if for every , there exists such that for all .
Let X be a nonempty set. Then a family of sets (power sets of X) is said to be an ideal if I is additive, i.e., and hereditary, i.e., , .
The fuzzy number is called the I-limit of the sequence of fuzzy numbers, and we write .
where for .
for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by de la Vallé-Poussin method.
We also note that Nuray and Savas  defined the sets of sequence spaces such as strongly σ-summable to zero, strongly σ-summable and strongly σ-bounded with respect to the modulus function, see .
In this article, we define some new sequence spaces of fuzzy real numbers by using Orlicz function with the 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 tools above.
3 Main results
Let I be an admissible ideal of N, let M be an Orlicz function. Let be a sequence of real numbers such that for all k, and . This assumption is made throughout the paper.
Further, when , for , the sets and are reduced to and , respectively.
If with as uniformly in m, then we write .
The following well-known inequality will be used later.
for all k and .
Lemma 3.1 (See )
Let , . Then if and only if , where .
Note that no other relation between and is needed in Lemma 3.1.
Theorem 3.2 Let . Then implies that . Let . If , then is unique.
Proof Let .
and, consequently, .
Let . Suppose that , and .
From (3) and (4), it follows that , and by the definition of an Orlicz function, we have .
Hence, , and this completes the proof. □
- (ii)Let . Then
- (ii)Let and . Let . Then for each k, , there exists a positive integer N such that
This completes the proof. □
Theorem 3.4 Let stand for , or and . Then the inclusion is strict. In general, for all and the inclusion is strict.
Proof Let us take .
Since the set on the right-hand side belongs to I, 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 k. □
The authors are very grateful to the referees for the very useful comments and for detailed remarks that improved the presentation and the contents of the manuscript. The work of the authors was carried under the Post Doctoral Fellow under National Board of Higher Mathematics, DAE, project No. NBHM/PDF.50/2011/64. The first author gratefully acknowledges that the present work was partially supported under the Post Doctoral Fellow under National Board of Higher Mathematics, DAE, project No. NBHM/PDF.50/2011/64. The second author also acknowledges that the part of the work was supported by the University Putra Malaysia, Grant ERGS 5527179.
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