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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 419701, 8 pages
The Strong Fuzzy Henstock Integrals and Discontinuous Fuzzy Differential Equations
1College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China
2Department of Mathematics, Sichuan University, Chengdu 610065, China
Received 24 April 2013; Accepted 24 September 2013
Academic Editor: Mehmet Sezer
Copyright © 2013 Yabin Shao and Huanhuan Zhang. 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.
We generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzy discontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem.
The Cauchy problems for fuzzy differential equations have been studied by several authors [1–6] on the metric space of normal fuzzy convex set with the distance given by the maximum of the Hausdorff distance between the corresponding level sets. In , Nieto proved that the Cauchy problem has a uniqueness result if is continuous and bounded. In [1, 3, 7–9], the authors presented a uniqueness result when satisfies a Lipschitz condition. For a general reference to fuzzy differential equations, see a recent book by Lakshmikantham and Mohapatra  and references therein. In 2002, Xue and Fu  established the solutions to fuzzy differential equations with right-hand side functions satisfying Carathéodory conditions on a class of Lipschitz fuzzy sets. However, there are discontinuous systems in which the right-hand side functions are not integrable in the sense of Kaleva  on certain intervals, and their solutions are not absolute continuous functions. To illustrate, we consider the following example.
Example 1. Consider the following discontinuous system:
Then, is not integrable in the sense of Kaleva. However, the above system has the following solution:
It is well known that the Henstock integral is designed to integrate highly oscillatory functions which the Lebesgue integral fails to do. It is known as nonabsolute integration and is a powerful tool. It is well known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue, and Newton integrals [12, 13]. Though such an integral was defined by Denjoy in 1912 and also by Perron in 1914, it was difficult to handle using their definitions. But with the Riemann-type definition introduced more recently by Henstock  in 1963 and also independently by Kurzweil , the definition is now simple, and furthermore the proof involving the integral also turns out to be easy. For more detailed results about the Henstock integral, we refer to . Recently, Wu and Gong [15, 16] have combined the fuzzy set theory and nonabsolute integration theory, and they discussed the fuzzy Henstock integrals of fuzzy-number-valued functions which extended Kaleva  integration. In order to complete the theory of fuzzy calculus and to meet the solving need of transferring a fuzzy differential equation into a fuzzy integral equation, Gong and Shao [17, 18] defined the strong fuzzy Henstock integrals and discussed some of their properties and the controlled convergence theorem.
In this paper, according to the idea of  and using the concept of generalized differentiability , we will prove other controlled convergence theorems for the strong fuzzy Henstock integrals, which will be of foundational significance for studying the existence and uniqueness of solutions to the fuzzy discontinuous systems. As we know, we inevitably use the controlled convergence theorems for solving the numerical solutions of differential equations. As the main outcomes, we will deal with the Cauchy problem of discontinuous fuzzy systems as follows: where is a strong fuzzy Henstock integrable function and
To make our analysis possible, we will first recall some basic results of fuzzy numbers and give some definitions of absolutely continuous fuzzy-number-valued function. In addition, we present the concept of generalized differentiability. In Section 3, we present the concept of strong fuzzy Henstock integrals, and we prove a controlled convergence theorem for the strong fuzzy Henstock integrals. In Section 4, we deal with the Cauchy problem of discontinuous fuzzy systems. And in Section 5, we present some concluding remarks.
Let denote the family of all nonempty compact convex subset of , and define the addition and scalar multiplication in as usual. Let and be two nonempty bounded subsets of . The distance between and is defined by the Hausdorff metric  as follows:
Denote that satisfies – below} is a fuzzy number space, where(1)is normal; that is, there exists an such that ;(2) is fuzzy convex; that is, for any and ;(3) is upper semicontinuous;(4) is compact.
For , denote that . Then from the above –(4), it follows that the -level set for all .
According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space as follows : where and .
Define where is the Hausdorff metric defined in . Then it is easy to see that is a metric in . Using the results , we know that(1) is a complete metric space,(2) for all ,(3) for all and .
Let . If there exist such that , then is called the -difference of and and is denoted by . As mentioned above which always is called the condition . It is well known that the -derivative for fuzzy-number-functions was initially introduced by Puri et al. [5, 23] and it is based on the condition of sets. We note that this definition is fairly strong, because the family of fuzzy-number-valued functions -differentiable is very restrictive. For example, the fuzzy-number-valued function defined by , where is a fuzzy number, is the scalar multiplication (in the fuzzy context) and , with , is not -differentiable in (see [20, 24]). To avoid the above difficulty, in this paper we consider a more general definition of a derivative for fuzzy-number-valued functions enlarging the class of differentiable fuzzy-number-valued functions, which has been introduced in .
Definition 2 (see ). Let and . One says that is differentiable at , if there exists an element , such that(1)for all sufficiently small, there exists and the limits (in the metric ) or(2)for all sufficiently small, there exists and the limits or(3)for all sufficiently small, there exists and the limits or(4)for all sufficiently small, there exists and the limits ( and at denominators mean and , resp.).
3. The Convergence Theorem of Strong Fuzzy Henstock Integral
In this section, we define the strong Henstock integrals of fuzzy-number-valued functions in fuzzy number space , and we give some properties and controlled convergence theorem of this integral by using new conditions.
Definition 3 (see ). A fuzzy-number-valued function is said to be termed additive on if, for any division , one has that exists and or that exists and . For convenience, denotes .
Definition 4 (see [17, 18]). A fuzzy-number-valued function is said to be strong Henstock integrable on if there exists a piecewise additive fuzzy-number-valued function on such that for every there is a function and for any -fine division of one has where such that is a fuzzy number and such that is a fuzzy number. One writes .
Definition 5. A fuzzy-number-valued function defined on is said to be if for every there exists and such that for any -fine partial division with satisfying one has .
Definition 6. A fuzzy-number-valued function is said to be on if is the union of a sequence of closed sets such that on each , is .
Definition 7. The sequence of fuzzy-number-function is on if is the sequence of subsets such that is for each , independent of .
Definition 8. Let be a sequence of fuzzy-number-function defined on , and let be measurable.(i)The sequence of fuzzy-number-function is -Cauchy on if converges pointwise on and if for each there exist on and a positive integer such that for all whenever is -subordinate to .(ii)The sequence of fuzzy-number-function is generalized -Cauchy on if can be written as a countable union of measurable sets on each of which is -Cauchy.
Theorem 9. Let the following conditions be satisfied:(i) a.e. on as where each is strong Fuzzy Henstock integrable on ;(ii)the primitives of are with closed set in .Then is strong fuzzy Henstock integrable on with the primitive .
Proof. By (ii), for every there exist a and , such that for any -fine partial division of satisfying we have . By Egoroff’s theorem [18, Theorem 3.4], there is an open set with such that for and . Consider the following, in which is a -fine division of and so that contains the intervals with the associated points and otherwise: Hence, for any -fine partial division of we have for . Therefore the fuzzy sequence is generalized -Cauchy on . Then, by (i), we have that is strong fuzzy Henstock integrable on with primitive .
Definition 10. (a) A sequence of fuzzy-number-valued function is uniformly on whenever to each there exist and such that(1), for each , (2)with .
(b) A sequence of fuzzy-number-valued function is uniformly on if , where are measurable sets and is uniformly on each .
Theorem 11. If is uniformly , then is uniformly .
Proof. Let be such that is uniformly on each . So, for each there exist and such that holds in Definition 10 for each satisfying condition . We take with and put and . So, we have . Then by condition in Definition 10, we have Hence, we have that is uniformly .
Theorem 12. Let the following conditions be satisfied:(i) a.e. in where each is strong fuzzy Henstock integrable on ;(ii)the primitives of are with closed set in .Then is strong fuzzy Henstock integrable on with primitive .
Next, we give the controlled convergence theorem for the strong fuzzy Henstock integrals by the definition of the for a fuzzy-number-valued function.
Definition 13. Let , and let . A fuzzy-number-valued function is on if for each there exist and on such that for . A fuzzy-number-valued function is on if is the union of a sequence of set such that the function is for each .
Definition 14 (see ). A fuzzy-number-valued function defined on is said to be if for every there exists such that for every finite sequence of non-overlapping intervals , satisfying where for all , one has , where denotes the oscillation of over ; that is, . A fuzzy-number-valued function is said to be on if is the union of a sequence of closed sets such that, on each , is .
Theorem 15. A fuzzy-number-valued function is if and only if it is on .
Theorem 16 (controlled convergence theorem). Let the following conditions be satisfied:(1) almost everywhere in as where each is strong fuzzy Henstock integrable on ;(2)the primitives of are uniformly in ;(3)the sequence converges uniformly to a continuous function on . Then is strong fuzzy Henstock integrable on and one has If conditions and are replaced by condition :(4) almost everywhere on , where and are strong fuzzy Henstock integrable.
Proof. By condition there exists a sequence such that in a bounded, closed set with bounds and , and put . We note that ; we have on , and hence on . By Theorem 15, on and hence on and also on and hence on .
Now we prove that a.e. on . In fact, let equal on , and extend linearly to the closed interval contiguous to . Likewise, we define from . We see that and are on . By condition (3), we have on . Let be the intervals contiguous to . Then we have . We define a fuzzy-number-valued function as follows: Consequently, converges on . Hence converges on a.e. Since on , then on . Therefore, we have a.e. on . Thus a.e. on by Theorem 15. Therefore, there exists an function on such that a.e. on . Hence is strong fuzzy Henstock integrable on , and we have
4. The Generalized Solutions of Discontinuous Fuzzy Differential Equations
In this section, a generalized fuzzy differential equation of form (4) is defined by using strong fuzzy Henstock integral. The main results of this section are existence theorems for the generalized solution to the discontinuous fuzzy differential equation.
Definition 17 (see ). Let and be fixed, and let a fuzzy-number-valued function be a Carathéodory function defined on a rectangle ; that is, is continuous in for almost all and measurable in for each fixed .
Theorem 18. Let a fuzzy-number-valued function be a function as given in Definition 17; then there exist two strong fuzzy Henstock integrable functions and defined on such that for all .
Proof. Note that is a Carathéodory function. Thus, there exist two measurable functions and defined on with values in such that for all . Next, we will show that and are fuzzy Henstock integrable by using controlled convergence Theorem 16. First, there exists a sequence of step functions defined on with values in such that almost everywhere as . Let . Then is uniformly in and equicontinuous. By controlled convergence Theorem 16, is strong fuzzy Henstock integrable. Similarly, is strong fuzzy Henstock integrable.
Definition 19. A fuzzy-number-valued function is said to be a solution of the discontinuous fuzzy differential equation (4) if satisfies the following conditions:(i) is on each compact subinterval of ;(ii) for ;(iii) for almost everywhere .
Now we will state the existence theorem for the generalized solution of discontinuous fuzzy differential equation (4).
Proof. Given for all and almost all with , we get . Let
Then is a Carathéodory function. Furthermore, for all , where , such that
By Carathéodory existence theorem (see Theorem 7 in ), there is a fuzzy-number-valued function on some interval such that almost everywhere in this interval and . Let
Then, for almost all , we have the following.
Case 1. Consider
Case 2. Consider The proof is complete.
Example 21. Consider fuzzy differential equation , where for all and is Kaleva integrable on and if and . Here is defined in Example 1. Note that is strong fuzzy Henstock integrable but not Kaleva integrable, and Thus, by Theorem 20, there exists a solution of with . For instance, if , then is a solution by using integrating factor.
Theorem 22. Let a fuzzy-number-valued function be a Carathéodory function defined on a rectangle . Let be strong fuzzy Henstock integrable on for any step function defined on with values in . Denote that . If is a step function} is uniformly in and equicontinuous on , then there exists a solution of on some interval with .
Let be a connected set in . Let and let be fixed: Let be a fuzzy-number-valued function defined on such that, for each fixed , the function is a Carathéodory function defined on for each fixed and continuous at for every . For , let have a solution on , where . Let be strong fuzzy Henstock integrable on for any step function with for .
Definition 23. Denote The family is said to be equicontinuous in and near if, for each , there exists an interval such that the family is equicontinuous at .
Theorem 24. Let be a fuzzy-number-valued function as given above. If the primitive of is uniformly in and and equicontinuous in and near , then there exists such that, for any fixed with , a solution of discontinuous fuzzy differential equation (29) exists over and as uniformly over .
Proof. Firstly, we will consider the case . By Theorem 22 and the equicontinuity of , all solutions of problem (29) with for some exist over some interval with . Then is equicontinuous on . Because , is uniformly bounded. Hence, for all sequence of with , there exists a subsequence which converges uniformly. That is to say, uniformly on as . Since we have
by Theorem 16 we get
Thus, is a solution of (29) for . Hence, on because the solution is unique. Consequently, uniformly on as . By Reductio ad absurdum, uniformly there as .
Secondly, we will extend the result over . We consider the case . Assume that there exists such that the result is valid over but not . Obviously, . Let be such that is contained in . Since is equicontinuous in and near , we may choose small enough such that for all step functions whenever and . Since uniformly on , there exists a such that for . Hence, we have Thus, for each with , a solution of (29) with exists on . Hence, can be continued to . So, in the case uniformly on . It leads to a contradiction, similarly, for the cases and . Therefore, the theorem holds over .
Example 25. Let where fuzzy number is defined in Example 1. Let . We define and for . Let defined on . Then, we have where or , and Note that is Kaleva integrable on every subinterval of and . Therefore, we have that is equicontinuous on in and near . Furthermore, is uniformly in and , where . On the other hand, by using integrating factor, for , we have with . Obviously, is unique. Thus, by Theorem 24, uniformly on .
In this paper, we give the definition of the for a fuzzy-number-valued function and the nonabsolute fuzzy integral and its controlled convergence theorem. In addition, we deal with the Cauchy problem and the continuous dependence of a solution on parameters of discontinuous fuzzy differential equations involving the strong fuzzy Henstock integral in fuzzy number space. The function governing the equations is supposed to be discontinuous with respect to some variables and satisfy nonabsolute fuzzy integrability. Our result improves the result given in [1, 11, 19, 20] (where uniform continuity was required), as well as those referred therein.
The authors are very grateful to the anonymous referees and Professor Mehmet Sezer for many valuable comments and suggestions which helped to improve the presentation of the paper. The authors would like to thank the National Natural Science Foundation of China (no. 11161041) and the Fundamental Research Fund for the Central Universities (no. 31920130010).
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