On Double Statistical -Convergence of Fuzzy Numbers

For sequences of fuzzy numbers, Nanda [1] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Kwon [2] introduced the definition of strongly p-Cesàro summability of sequences of fuzzy numbers. Savas [3] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. Savas [4] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy and he continue to study in [5, 6]. Recently Mursaleen and Basarir [7] introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. Quite recently, Savas and Mursaleen [8] defined statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of double statistical convergence and introduce the definition of double strongly -Cesàro summabilty of sequences of fuzzy numbers.

Since the theory of fuzzy numbers has been widely studied, it is impossible to find either a commonly accepted definition or a fixed notation. We therefore being by introducing some notations and definitions which will be used throughout.

Let . The spaces has a linear structure induced by the operations

(2.1)

for and . The Hausdorff distance between and of is defined as

(2.2)

It is well known that is a complete (not separable) metric space.

A fuzzy number is a function from to satisfying

(1) is normal, that is, there exists an such that ;

(2) is fuzzy convex, that is, for any and ,

(2.3)

(3) is upper semicontinuous;

(4)the closure of ; denoted by , is compact.

These properties imply that for each , the -level set

(2.4)

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

(2.5)

for each .

Define for each ,

(2.6)

and clearly with if . Moreover is a complete, separable, and locally compact metric space [9].

Throughout this paper, will denote with . We will need the following definitions (see [8]).

Definition 2.1.

A double sequence of fuzzy numbers is said to be convergent in Pringsheim's sense or -convergent to a fuzzy number , if for every there exists such that

(2.7)

and we denote by . The number is called the Pringsheim limit of .

More exactly we say that a double sequence converges to a finite number if tend to as both and tend to independently of one another.

Let denote the set of all double convergent sequences of fuzzy numbers.

Definition 2.2.

A double sequence of fuzzy numbers is said to be -Cauchy sequence if for every there exists such that

(2.8)

Let denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.3.

A double sequence of fuzzy numbers is bounded if there exists a positive number such that for all and ,

(2.9)

We will denote the set of all bounded double sequences by .

Let be a two-dimensional set of positive integers and let be the numbers of in such that and . Then the lower asymptotic density of is defined as

(2.10)

In the case when the sequence has a limit, then we say that has a natural density and is defined as

(2.11)

For example, let , where is the set of natural numbers. Then

(2.12)

(i.e., the set has double natural density zero).

Definition 2.4.

A double sequence of fuzzy numbers is said to be statistically convergent to provided that for each ,

(2.13)

In this case we write and we denote the set of all double statistically convergent sequences of fuzzy numbers by .

Definition 2.5.

A double sequence of fuzzy numbers is said to be statistically -Cauchy if for every there exist and such that

(2.14)

That is, , a.a.( ).

Let denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.6.

A double sequence of fuzzy and let be a positive real numbers. The sequence is said to be strongly double -Cesaro summable if there is a fuzzy number such that

(2.15)

In which case we say that is strongly -Cesaro summable to .

It is quite clear that if a sequence is statistically -convergent, then it is a statistically -Cauchy sequence [8]. It is also easy to see that if there is a convergent sequence such that a.a.( ), then is statistically convergent.

Theorem 3.1.

A double sequence of fuzzy numbers is statistically -Cauchy then there is a -convergent double sequence such that a.a.( ).

Proof.

Let us begin with the assumption that is statistically -Cauchy this grant us a closed ball that contains a.a.( ) for some positive numbers and . Clearly we can choose and such that contains a.a.( ). It is also clear that a.a.( ); for

(3.1)

we have

(3.2)

Thus is a closed ball of diameter less than or equal to 1 that contains a.a.( ). Now we let us consider the second stage to this end we choose and such that In a manner similar to the first stage we have , a.a.( ). Note the diameter of is less than or equal . If we now consider the ( )th general stage we obtain the following. First a sequence of closed balls such that for each ( ), , the diameter of is not greater than with , a.a.( ). By the nested closed set theorem of a complete metric space we have . So there exists a fuzzy number . Using the fact that , a.a.( ), we can choose an increasing sequence of positive integers such that

(3.3)

Now define a double subsequence of consisting of all terms such that and if

(3.4)

Next we define the sequence ( ) by

(3.5)

Then indeed if and then either is a term of . Which means or and diameter of . We will now show that a.a.( ). Note that if then

(3.6)

and by (3.3)

(3.7)

Hence the limit as is 0 and a.a.( ). This completes the proof.

Theorem 3.2.

If is strongly -Cesaro summable or statistically -convergent to then there is a -convergent double sequences and a statistically -null sequence such that and .

Proof.

Note that if is strongly -Cesaro summable to then is statistically -convergent to . Let and and select two increasing index sequences of positive integers and such that and , we have

(3.8)

Define and as follows: if and , set and . Suppose that and , then

(3.9)

We now show that . Let be given, pick be given, and pick and such that , thus for , since if and if we have .

Next we show that is a statistically -null double sequence, that is, we need to show that . Let if such that then for all . From the construction of , if and then only if . It follows that if and then

(3.10)

Thus for and and then

(3.11)

this completes the proof.

Corollary 3.3.

If is a strongly -Cesaro summable to or statistically -convergent to then has a double subsequence which is -converges to .

In recent years the statistical convergence has been adapted to the sequences of fuzzy numbers. Double statistical convergence of sequences of fuzzy numbers was first deduced in similar form by Savas and Mursaleen as we explain now: a double sequences is said to be -statistically convergent to provided that for each ,

(4.1)

Since the set of real numbers can be embedded in the set of fuzzy numbers, statistical convergence in reals can be considered as a special case of those fuzzy numbers. However, since the set of fuzzy numbers is partially ordered and does not carry a group structure, most of the results known for the sequences of real numbers may not be valid in fuzzy setting. Therefore, this theory should not be considered as a trivial extension of what has been known in real case. In this paper, we continue the study of double statistical convergence and also some important theorems are proved.