Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 832831, 10 pages

http://dx.doi.org/10.1155/2013/832831

Research Article

Some Fuzzy-Wavelet-Like Operators and Their Convergence

R. Ezzati, F. Mokhtarnejad, and N. Hassasi

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Received 25 January 2013; Accepted 20 May 2013

Academic Editor: Tofigh Allahviranloo

Copyright © 2013 R. Ezzati et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Firstly, we define some new fuzzy-wavelet-like operators via a real-valued scaling function to approximate the continuous fuzzy functions of one and two variables. Then, by using the modulus of continuity, we prove their pointwise and uniform convergence with rates to the fuzzy unit operator . Using these fuzzy-wavelet-like operators, we present some numerical examples to illustrate the applicability of the proposed method. Also, we give a new method to approximate the integration of continuous fuzzy real-number-valued function of two variables by using the fuzzy-wavelet-like operator.

1. Introduction

Approximating functions in a given space is an old problem. For this purpose, many authors have studied approximation of fuzzy functions on a finite set of distinct points; see [1–6]. For instance, the authors of [2] proposed a new method to approximate fuzzy functions by distance method. Indeed, they considered the problem for fuzzy data and fuzzy functions using the defuzzification function introduced by Fortemps and Roubens. Also, they introduced a fuzzy polynomial approximation as -approximation of a fuzzy function on a discrete set of points.

Wavelet theory is a relatively new and an emerging area in mathematical research. Also, wavelets are the suitable and powerful tool for approximating functions based on wavelet basis functions. In [1], the author defined some fuzzy-wavelet-like operators and presented their pointwise and uniform convergence with rates to the fuzzy unit operator . Recently, the authors of [7] constructed fuzzy Haar wavelet and applied it on solving linear fuzzy Fredholm integral equation of second kind.

Here, we propose some new fuzzy-wavelet-like operators via a real-valued scaling function to approximate the continuous fuzzy functions of one and two variables. Also, we prove their pointwise and uniform convergence with rates to the fuzzy unit operator by using modulus of continuity. It is noticed that we do not suppose any kind of orthogonality condition on the scaling function, and the operators act on fuzzy-valued continuous functions over and .

The rest of the paper is organized as follows. In Section 2, we review some elementary concepts of the fuzzy set theory and modulus of continuity. In Section 3, we prepare some theorems for pointwise and uniform convergence of defined fuzzy-wavelet-like operators with rates to the fuzzy unit operator . In Section 4, we give two numerical examples for applicability of the proposed methods. In Section 5, we present an application to approximate the integration of continuous fuzzy real-number-valued function of two variables by using the fuzzy-wavelet-like operator. Finally, this paper is concluded in Section 6.

2. Preliminaries

Definition 1 (see [8]). A fuzzy number is a function with the following properties:

(1)is normal; that is, there exists such that ,(2) is fuzzy convex set

(3) is upper semicontinuous on , (4)the is compact set.

The set of all fuzzy numbers is denoted by .

Definition 2 (see [9]). Suppose that . The -level set of is denoted by and defined by , where . Also, is called the support of , denoted by , and it is given as . It follows that the level sets of are closed and bounded intervals in .

It is well known that the addition and multiplication operations of real numbers can be extended to . In other words, for , and , one defines uniquely the sum and the product by

where means the usual addition of two intervals (as subsets of ) and means the usual product between a scalar and a subset of . One uses the same symbol both for the sum of real numbers and for the sum (when the terms are fuzzy numbers).

Definition 3 (see [9]). An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions , , which satisfy the following requirements:

(1) is a bounded left continuous nondecreasing function over [], (2) is a bounded left continuous nonincreasing function over [], (3), .

The addition and scaler multiplication of fuzzy numbers in are defined as follows:

(1) ,(2)

Definition 4 (see [10]). For arbitrary fuzzy numbers , , the quantity

is the distance between and . It is proved that is a complete metric space with the properties [11, 12]

(1) for all ,(2) for all for all ,(3) for all .

Definition 5 (see [10]). Let be fuzzy real number valued functions. The uniform distance between , is defined by

Definition 6 (see [10]). Let . is fuzzy-Riemann integrable to if for any , there exists such that for any division of with the norms , one has

where denotes the fuzzy summation. In this case it is denoted by .

Theorem 7 (see [13]). If are fuzzy continuous functions, then the function defined by is continuous on , and

Definition 8 (see [14]). A fuzzy real number valued function is said to be continuous in , if for each there is such that , whenever and . One says that is fuzzy continuous on if is continuous at each and denotes the space of all such functions by .

Definition 9 (see [14]). Let . One calls a uniformly continuous fuzzy real number valued function, if and only if for any there exists whenever ; , implies that . One denotes it as . Also, one denotes the space of all fuzzy continuous functions by .

Similarly we have the following.

Definition 10. Let . One calls a uniformly continuous fuzzy real number valued function, if and only if for any there exists whenever: ; , implies that

One denotes it as .

Definition 11 (see [14]). Let , then is called a nondecreasing function if and only if whenever , , one has that ; that is, and , for all .

Definition 12 (see [10, 15]). Let be a bounded function, then function ,

where is the set of positive real numbers, is called the modulus of continuity of on .

For , the modulus of continuity is defined as follows:

where and . Observe that, for all and , one has

where is defined to be the greatest integer less than or equal to .

Some properties of the modulus of continuity are presented below.

Theorem 13 (see [15]). The following properties hold:

(1)  for any  ,(2)  is increasing  function of  ,(3), (4)for any  ,  and  ,(5)  for any , ,  and ,(6)  for any ,  where    is the ceiling of the number, any .(7)If    then .

In [1], the following theorem is proved.

Theorem 14. Let and the scaling function a real-valued bounded function with , , , such that on . For , , put

which is a fuzzy-wavelet-like operator. Then

for all and . If , then as one gets and , pointwise and uniformly with rates.

3. Main Results

Theorem 15. All assumption here are as in Theorem 14. Define for , , the fuzzy-wavelet-like operator

Then

for all , . When then as one gets and .

Proof. We want to estimate

Clearly, we have

Therefore, we conclude that

and hence

Theorem 16. All assumptions here are as in Theorem 14. Define for , , the fuzzy-wavelet-like operator

Then

when then as one gets and .

Proof. We need to estimate

Here, we have

Now, by using the above inequality in , we can write

and hence

Theorem 17. All assumptions here are as in Theorem 14. Define for , the fuzzy-wavelet-like operator

where

Then

When then as one gets and .

Proof. We have the following:

Notice that

Therefore,

This completes the proof.

Theorem 18. Let and the scaling function

a real-valued bounded function with ,  , ,  , such that on . For , , put

which is a fuzzy-wavelet-like operator. Then

for all and . If , then as one gets and , pointwise and uniformly with rates.

Proof. We have the following:

According to this fact that

we conclude that

and hence

This completes the proof.

Theorem 19. All assumptions here are as in Theorem 18. For , , put

which is a fuzzy-wavelet-like operator. Then

for all and . If , then as one gets

and , pointwise and uniformly with rates.

Proof. We have the following:

According to this fact that

we conclude that

and hence

This completes the proof.

Theorem 20. Let and the scaling function a real-valued bounded function with , , such that

(1) on , (2)there exists such that is nondecreasing for and is nonincreasing for

(the above imply ). Let be nondecreasing fuzzy function, then is nondecreasing fuzzy valued functions for any .

Proof. The proof is similar to the proof of [15, Theorem 12.14]. Let such that , as , then , by fuzzy continuity of . But we have

That is, , for all as , respectively. Therefore , for all , that is, real-valued continuous functions on . Since is fuzzy nondecreasing by Definition 11 we get that are non-decreasing, for all , respectively.

Next we observe that

That is,

So whenever we get

Therefore, is nondecreasing.

Theorem 21. Let and be as in Theorem 20. Then is a nondecreasing fuzzy valued function for any .

Proof. The proof is similar to the proof of Theorem 20.

4. Numerical Examples

In this section, we present two examples. Also, we apply the following scaling function [1]:

Example 1. In this example, we want to apply the fuzzy-wavelet-like operator defined in Theorem 15 to approximate a given function , where , . To do this, we consider . Comparing the numerical result with is done in Figure 1.

Figure 1: Comparing and fuzzy-wavelet-like operator with   in .

Example 2. Consider , where , . Now, we apply the fuzzy- wavelet-like operator defined in Theorem 18 to approximate . We also compare the result of using with in , . For more details, see Figures 2 and 3.

Figure 2: The fuzzy number in  , .

Figure 3: Approximation of by using fuzzy wavelet like operator with   in  , .

5. An Application

In this section, we approximate the integration of continuous fuzzy real number valued function of two variables by using the fuzzy-wavelet-like operator defined in Theorem 18. Consider the following fuzzy integral:

By using the fuzzy-wavelet-like operator , we get

So, we have

Theorem 22. Let . Then

Proof. We have the following:

Example 3. Consider the following fuzzy integral:

where . The exact solution of this example is , , where

Let , . By using proposed method in (53), we present approximate solution to this example for different values of . To compare the exact and the approximate solutions, see Tables 1, 2, and 3.

Table 1: Numerical results for Example 3 with .

-level

Table 2: Numerical results for Example 3 with .

-level

Table 3: Numerical results for Example 3 with .

-level

6. Conclusions

To approximate the continuous fuzzy functions of one and two variables, some new fuzzy-wavelet-like operators via a real-valued scaling function are defined. Convergence of defined fuzzy like operators is investigated by using the modulus of continuity. To approximate the integration of continuous fuzzy real number valued function of two variables by using the fuzzy-wavelet-like operator defined in Theorem 18, an application is presented.

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