Multiple sequences of fuzzy numbers and their statistical convergence

Purpose

The purpose of this paper is to extend a generalized convergence method, namely, statistical convergence to sequences of fuzzy numbers of multiplicity greater than two.

Methods

We use analytic method to obtain our results.

Results

Certain theorems on statistical convergence of real double sequences obtained by Savaş et al. and Móricz are also extended to multiple sequences of fuzzy numbers. Finally, we define Cesàro summable and strongly p-Cesàro summable multiple sequences of fuzzy numbers and obtained their relations with statistical convergence.

Conclusions

Although, we prove our results only for triple sequences, but all these results remain true for d-multiple sequences as well.

Statistical convergence for real number sequences was introduced by Fast [1] and Schonenberg [2] independently. Later, the idea was further investigated from sequence space point of view and linked with summability theory by Fridy [3], Šalát [4] and many others. The idea is based on the notion of natural density of subsets of , the set of positive integers. For any subset A of N, the natural density of A is denoted by and is defined by

where vertical bars denote the cardinality of the enclosed set. Using this definition, the notions of statistical convergence and statistically Cauchy for a number sequence are defined (in [5]) as follows.

A sequence of numbers is said to be statistically convergent to some number L, in symbol: , if for each ,

i.e., : .

A sequence of numbers is said to be a statistical Cauchy if, for each , there is a positive integer m such that

Agnew [6] studied the summability theory of multiple sequences and obtained certain theorems which have already been proved for double sequences by the author himself. Móricz [7] continued with the study of multiple sequences and gave some remarks on the notion of regular convergence of multiple series. In 2003, the author extended statistical convergence from single to multiple real sequences and obtained some results for real double sequences. Savaş et al. [8] studied a similar method of convergence with the help of lacunary sequences for multiple sequences of numbers and called it lacunary statistical convergence. However, Şahiner et al. [9] and Sharma et al. [10], respectively, developed statistical convergence for triple sequences of real numbers and for sequences on probabilistic normed spaces.

On the other side, fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has a wide range of applications in various fields: population dynamics, chaos control, computer programming, nonlinear dynamical systems, etc. Fuzzy topology is one of the most important and useful tools to deal with such situations where the use of classical theories breaks down. While studying fuzzy topological spaces, we face many situations where we need to deal with convergence of sequences of fuzzy numbers. The concept of usual convergence of fuzzy numbers sequences was introduced by Matloka [11], where he proved some basic theorems. Nanda [12] continued with this study and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space. In recent years, statistical convergence has also been adapted to the sequences of fuzzy numbers. The credit goes to Nuray and Savaş [13], who first defined the concepts of statistical convergence and statistically Cauchy for sequences of fuzzy numbers. They proved that a sequence of fuzzy numbers is statistically convergent, if and only if, it is a statistically Cauchy. Nuray [14] introduced lacunary statistical convergence of fuzzy numbers sequences, whereas Kwon [15] obtained relationship between statistical convergence and strong p-Cesàro summability of fuzzy numbers sequences. For further development on statistical convergence of fuzzy number sequences, we refer Savaş [16], and Savaş et al. [17]. Let = {: A is compact and convex}. The space has a linear structure induced by the operations

for ; . The Hausdroff distance between A and B is defined by

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

Definition 2.1

A fuzzy number is a function X from to [0,1], which satisfies the following conditions:

(i) X is normal, i.e., there exists such that .

(ii) X is a fuzzy convex, i.e., for any and,

(iii) X is upper semi-continuous.

(iv) The closure of the set : , denoted by , is compact.

The properties (i)-(iv) imply for each , the -level set,

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

for each . Define, for each ,

and . Clearly, with if . Moreover, is a complete, separable and locally compact metric space.

Throughout the paper, d will denote with , and will denote the usual product set . We now quote the following definitions which will be needed in the sequel.

Definition 2.2

A triple sequence of fuzzy numbers is said to be convergent to a fuzzy number if for each , there exist a positive integer m such that

The fuzzy number is called the limit of the sequence and we write .

Definition 2.3

A triple sequence of fuzzy numbers is said to be a Cauchy sequence if, for each , there exists a positive integer such that

for every .

Definition 2.4

A triple sequence of fuzzy numbers is said to be bounded if there exists a positive number M such that

Let denote the set of all bounded triple sequences of fuzzy numbers.

In present paper, we introduce statistical convergence of sequences of fuzzy numbers having multiplicity greater than two. Certain Theorems regarding uniqueness of limit, algebraic characterization and closedness of the subspace are obtained. We also give the following important characterization of statistical convergence for sequences of fuzzy numbers having multiplicity greater than two. "A triple sequence of fuzzy numbers is statistical convergent to a fuzzy number , if and only if, there exists a subset such that and ". Finally, we define the notions of statistically Cauchy, Cesàro summable, Cesàro summable for these kinds of sequences and establish the Cauchy convergence criterion.

In this section, we shall, for brevity, state and prove our results only for triple sequences. The reader will see that our methods can readily be applied also to double sequences of fuzzy numbers and to sequences of fuzzy numbers of any multiplicity greater than three. For , the natural density of K is defined by

provided that the limit exists. Here, vertical bars denote the cardinality of the enclosed set.

Definition 3.1

A triple sequence of fuzzy numbers is said to be statistically convergent to some fuzzy number , in symbol: , if for each ,

Here and in the sequel, l, m and n tend to infinity independently of one another. We shall denote the set of all statistically convergent triple sequences of fuzzy numbers by . Since the asymptotic density of finite subsets of is zero, it follows that every convergent triple sequence of fuzzy number is statistically convergent, although the converse is not necessarily true, as seen from the following example.

Example 3.1

For every , define a sequence of fuzzy numbers as follows.

If i, j and k are all squares, define

Otherwise, where is given by

Now, for , we have

Since, the later set has triple density zero, it follows that , and consequently is statistically convergent to . But, the sequence is not ordinarily convergent to .

In the following theorems, we give the uniqueness and algebraic characterization of statistical limit for triple sequences of fuzzy numbers. However, the proofs are straightforward and therefore omitted.

Theorem 3.1

If a triple sequence of fuzzy numbers is statistically convergent to some limit, then it must be unique.

Theorem 3.2

Let be two triple sequences of fuzzy numbers.

(i) If is statistically convergent to and , then is statistically convergent to .

(ii) If and are statistically convergent to fuzzy numbers and , respectively, then is statistically convergent to.

Theorem 3.3

A triple sequence of fuzzy numbers is statistically convergent to a fuzzy number , if and only if, there exists a subset such that and .

Proof

Let be statistically convergent to . For each , if we denote

then , and therefore, . Furthermore, . Since , it follows that K is an infinite set as otherwise . Let . Now, to prove the result, it is sufficient to prove that is convergent to . Suppose that is not convergent to . By definition, there exists such that for infinitely many terms. Let

Clearly, . Also, for all and , we have

Thus, i.e. . Furthermore, for , which is impossible as . Hence, is convergent to .

Conversely, suppose that there exists a subset such that and . By definition, there exists a positive integer p such that for all . Since

it follows that

Hence, X is statistically convergent to .

Theorem 3.4

The set is a closed linear subspace of the normed linear space .

Proof

Let and . Since , therefore, there exists fuzzy number such that

Furthermore, implies that there exists a positive integer M such that for every and ,

Also, by Theorem 3.3, there exists subsets such that and

Now, the set is infinite as . Choose , then we have, from Equations (2) and (3),

Hence, for every and , we have, from Equations (1) to (4),

This shows that is a Cauchy sequence and, hence, convergent. Let

Next, we show that X is statistically convergent to Y . Since , so for each , there exists and such that

Also from Equation 5, we have, for every , such that

Furthermore, by virtue of the fact that is statistically convergent to , there is a set such that , and for each , there exists such that, for , we have

Let . Now, for and ,

This shows that X is statistically convergent to Y , i.e., . This shows that is a closed linear subspace of , and therefore, the proof of the theorem is complete.

Definition 3.2

A triple sequence of fuzzy numbers is said to be a statistically Cauchy if, for each , there exist integers and such that

Theorem 3.5

A triple sequence of fuzzy numbers is statistically convergent, if and only if, it is a statistical Cauchy.

Proof

Let be statistically convergent to . By definition, for each we have

We can choose numbers L, M and N such that . If we denote

then it is clear that and consequently . Hence is statistically Cauchy.

Conversely, suppose that is a statistically Cauchy. We shall prove that is statistically convergent. To this effect, let be a strictly decreasing sequence of numbers converging to zero. Since is a statistically Cauchy, therefore, there exists three strictly increasing sequences ( and ) of positive integers such that

Clearly, for each p and q pair of positive integers, we can select such that

It follows that

Thus, is a Cauchy sequence and satisfies the Cauchy convergence criterion. Let converge to . Since , so for , there exists such that

Now, consider arbitrary. By Equation (7),

where, by Equation (6),

This shows that is statistically convergent to , and therefore, the proof of the theorem is complete.

Definition 3.3

A triple sequence of fuzzy numbers is said to be -summable or Cesàro summable to provided that

Definition 3.4

Let p be a positive real number. A triple sequence of fuzzy numbers is said to be strongly p-Cesàro summable to a fuzzy number if

We denote the space of all strongly p-Cesàro summable triple sequences of fuzzy numbers by .

Remark 3.1

(i) If , (Holder inequality) and .

(ii) If is convergent but unbounded, then is statistically convergent; however, need not to be Cesàro nor strongly Cesàro.

(iii) If is a bounded convergent triple sequence of fuzzy numbers, then it is also , and statistically convergent.

Theorem 3.6

(a) Let . If a triple sequence of fuzzy numbers is strongly p-Cesàro summable to a fuzzy number , then it is also statistically convergent to .

(b) Let . If a triple bounded sequence of fuzzy numbers is statistically convergent to a fuzzy number , then it is strongly p-Cesàro summable to .

Proof

(a) Let . Now, we have

Since is strongly p-Cesàro summable to , therefore, we have

Hence,

as it cannot be negative. This shows that is statistically convergent to .

(b) Let and , where is the sup-norm for bounded triple sequences . Since is bounded and statistically convergent, we can choose a positive integer such that, for all , we have

Now, for all ,

This shows that is strongly p-Cesàro summable to .

The concepts and results presented in the previous section can be extended to d-multiple sequences of fuzzy numbers where d is a fixed positive integer. Let = . The d-tuple kn, where k = and n = , if and only if, for at least one j. Moreover, the partial order on is defined as follows.

For , we say that kn if, and only if, for each j. The natural density of a set S can be defined as

provided that this limit exists. With the help of -density, the notions of statistical convergence and statistical Cauchy for multiple sequences of fuzzy numbers can be define as follows.

Definition 4.1

A d-tuple sequence of fuzzy numbers is said to be statistically convergent to some fuzzy number if, for each ,

where

Definition 4.2

A d-tuple sequence of fuzzy numbers is said to be statistically Cauchy if for each and there exist = such that and

All the results presented in previous sections remain true for d-multiple sequences as well.

The authors declare that they have no competing interests.

The authors are thankful to the reviewers of the paper for careful reading and suggestions.

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