1. Introduction and Preliminaries

Fuzzy sets theory compared to other mathematical theories, is perhaps the most easily adaptable theory to practice. The main reason is that a fuzzy set has the property of relativity, variability and inexactness in the definition of its elements. Instead of defining an entity in calculus by assuming that its role is exactly known, we can use fuzzy sets to define the same entity by allowing possible deviations and inexactness in its role. This representation suits well the uncertainities encountered in practical life, which make fuzzy sets a valuable mathematical tool. The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [17] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming etc. Matloka [5] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. For details about sequence spaces (see [1], [2], [3], [9], [10], [11], [12], [13], [15], [16]) and references therein.

An Orlicz function is a function, which is continuous, non-decreasing and convex with for and .

Lindenstrauss and Tzafriri [4] used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences then

Which is called as an Orlicz sequence space. The space is a Banach space with the norm

It is shown in [4] that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for An Orlicz function can always be represented in the following integral form

Where is known as the kernel of , is right differentiable for is non-decreasing and

A fuzzy number is a fuzzy set on the real axis, i.e., a mapping which satisfies the following four conditions:

(1) is normal, i.e., there exist an such that

(2) is fuzzy convex, i.e., for and

(3) is upper semi-continuous;

(4) the closure of denoted by is compact.

Let is compact and convex }. The space has a linear structure induced by the operations

and

for and The Hausdorff distance between and of is defined as

where denotes the usual Euclidean norm in . It is well known that is a complete (non separable) metric space. For the -level set

is a non-empty compact convex, subset of , as is the support Let denote the set of all fuzzy numbers. The linear structure of induces addition and scalar multiplication in terms of -level sets, by

and

for each Define for each

and Clearly with Moreover is a complete, separable and locally compact metric space. We denote by the set of all sequences of fuzzy numbers.

Let be the set of all complex sequences normed by where the set of positive integers. Let denote the space whose elements are the sets of distinct positive integers. Given any elements we denote by the sequence which is such that otherwise. Further

the set of those whose support has cardinality at most and

Where where are real sequences see [5]. For Sargent [14] define the following sequence space

In [14], Sargent studied some of its properties and obtained its relationship with the space In [7], Nurray and Savas introduced the classes of sequences of fuzzy numbers,

where is a bounded sequence of positive real numbers. They proved that is a complete metric space with the metric defined by

where

Let be a one-to-one mapping of the set of positive integers into itself such that be an Orlicz function and be a bounded sequence of positive real numbers. In this paper we define the following classes of sequences of fuzzy numbers:

and

When we obtain the classes of sequences of fuzzy numbers as follows:

and

If we take we obtain the classes of sequences of fuzzy numbers as follows:

and

The following inequality will be used throughout the paper. Let be a bounded sequence of positive real numbers with and let

Then for the factorable sequences and in the complex plane, we have

(1)

The main aim of this paper is to study some topological properties and some inclusion relations between above defined sequence spaces.

2. Main Results

Theorem 2.1. Let be an Orlicz function and be a bounded sequence of positive real numbers, then the spaces are linear spaces over the field of complex numbers ℂ.

Proof. Let and then there exist positive numbers such that

and

Define Since is non-decreasing, convex and so by using inequality (1), we have

This proves that is a linear space. Similarly, we can prove that and are linear spaces.

Theorem 2.2. Let be an Orlicz function and be a bounded sequence of positive real numbers, then the space is a complete metric space, with the metric defined by

Proof. Let be a Cauchy sequence in . Then,

Hence

for all

Therefore is a Cauchy sequence in Since is complete, it is convergent so that for each Since is a Cauchy sequence for each there exists such that

So, we have

This implies that Since

then we obtain Therefore is a complete metric space. This completes the proof of the theorem.

Theorem 2.3. Let be an Orlicz function and be a bounded sequence of positive real numbers, then we have the following

(ⅰ) the space is a complete metric space, with the metric defined by

(ⅱ) the space is a complete metric space, with the metric defined by

Proof. It is easy to prove in view of Theorem 2.2, so we omit the details.

Theorem 2.4. Let be an Orlicz function and be a bounded sequence of positive real numbers, then if and only if

Proof. Let and Then

Therefore Hence

Conversely, let Suppose that then there exists a sequence of natural numbers such that Let Then,

Now, we have

Therefore which is a contradiction. Therefore This completes the proof of the theorem.

Theorem 2.5. Let be an Orlicz function and be a bounded sequence of positive real numbers, then if and only if and

Proof. The proof directly follows from Theorem 2.4.

Theorem 2.6. Let be an Orlicz function and be a bounded sequence of positive real numbers, then

Proof. Let then we have

Since is monotone increasing, so we have

Hence

Thus Therefore Next, let Then, we have

Thus

(on taking cardinality of to be 1). Thus Hence

This completes the proof of the theorem.

Theorem 2.7. Let be an orlicz function and be a bounded sequence of positive real numbers, then if and only if

Proof. It is clear that when for all By Theorem 2.4, if and only if Therefore by Theorem 2.6, if and only if

3. Conclusions

In this paper we have introduce some new sequence spaces defined by a Orlicz function. We have also studied some topological properties like linearity, completeness and interested inclusion relations between the spaces. According to our opinion these results are new and interesting and beneficial for young researchers those working in this area.