On Strongly Almost Convergent Sequence Spaces of Fuzzy Numbers

In this paper we define some new almost convergent sequence spaces of fuzzy numbers through a non-negative regular matrix and we also examine some topological properties and some inclusion relations for these new sequence spaces.


INTRODUCTION
In many branches of mathematics and engineering we often come across different types of sequences and certainly there are situations either the idea of ordinary convergence does not work or the underlying space does not serve our purpose.So to ideal with such situations we have to introduce some new type of measures which can provide a better tool and a suitable frame work.In particular, we are interested to put forward our studies in fuzzy like situations.The concepts of fuzzy sets and fuzzy set operation were first introduced by Zadeh [16] in 1965.Since then a large number of research papers have appeared by using the concept of fuzzy set numbers and fuzzification of many classical theories has also been made.It has also very useful application in various fields, e.g.population dynamics [4], chaos control [6], computer programming [7], nonlinear dynamical systems [8], fuzzy physics [9], etc.In this paper, we define some new almost convergent sequence spaces of fuzzy numbers through a non-negative regular matrix and we also examine some topological properties.

Let D be the set of all bounded intervals ,
define if and only if and , ( , ) max( , ).

A B A B A B d A B
Then it can be easily see that d defines a metric on D (see, [5]) and ( , ) D d is a complete metric space.
A fuzzy number is a fuzzy subset of the real line  which is bounded, convex and normal.Let   L  denote the set of all fuzzy numbers which are upper semicontinuous and have compact support, i.e. if
In [15] it was shown that   c F and ( ) l F  are complete metric spaces.
In this paper we define some new almost sequence spaces of fuzzy numbers through a non-negative regular matrices ( ), ( , 1, 2,...).

nk A a n k  
By the regularity of A we mean that the matrix which transform convergent sequence into a convergent sequence leaving the limit invariant ( Maddox,[10]).
The famous Silverman-Toeplitz conditions for the regularity of A are as follows: A is regular if and only if where E is a linear space which satisfies the following conditions: The space E is called the paranormed space with the paranorm g .
Recently E. Savas [17] have defined the following space of sequences of fuzzy numbers.

of fuzzy numbers is said to be almost convergent to a fuzzy number L if
This means that for every and for all n.
If the limit in (1) exists, then we write We are ready to define the following: be an infinite regular matrix of non-negative real numbers and let ( ) be a sequence of positive real numbers.We define and call them respectively the spaces of strongly almost A-convergent to zero, and strongly almost A-convergent to 0 X .We can specialize these spaces as follows.
, and which are defined as follows:  and further on taking 1 k p  for all k , these are reduced to following sequences spaces: Strongly almost sequence of fuzzy numbers is discussed in [13].
where   r k is a lacunary sequence, i.e. an increasing sequence of non-negative integers with 1 as .
Strongly almost lacunary sequence of fuzzy numbers is discussed in [3].
A metric d on L(  ) is said to be a translation invariant if since d is a translation invariant.
(ii) It follows easily by using (i) and induction.
If d is a translation invariant, we have the following straightforward result.
where d is a translation invariant.
Proof.Clearly ( ) 0, ( ) ( ). g g X g X     It can also be seen easily that Now for any  we have max (1, ), Taking  small enough we then have for all , has exactly the same proof.q p be bounded.Then ˆ, ( ) , ( ).
and it follows that , , , .
is convergent for all n,i and since , 1 uniquely.we have

Proposition 2 . 2 .
Let   k p be a bounded sequence of strictly positive real numbers.are linear spaces over the complex field C. are paranormed spaces with the paranorm g defined by and this completes the proof.