Abstract and Applied Analysis

Volume 2014 (2014), Article ID 932696, 8 pages

http://dx.doi.org/10.1155/2014/932696

Research Article

Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

Yabin Shao1,2 and Huanhuan Zhang¹

¹College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China

²Department of Mathematics, Sichuan University, Chengdu 610065, China

Received 29 April 2013; Revised 6 September 2013; Accepted 18 September 2013; Published 29 January 2014

Academic Editor: Márcia Federson

Copyright © 2014 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.

Abstract.

By using the strong fuzzy Henstock integral and its controlled convergence theorem, we generalized the existence theorems of solution for initial problems of fuzzy discontinuous integral equation.

1. Introduction

The fuzzy differential and integral equations are important part of the fuzzy analysis theory and they have the important value of theory and application in control theory.

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. Seikkala in [7] defined the fuzzy derivative and then some generalizations of that have been investigated in [8, 9]. Consequently, the fuzzy integral which is the same as that of Dubois and Prade in [10], by means of the extension principle of Zadeh, showed that the fuzzy initial value problem , , has a unique fuzzy solution when satisfies the generalized Lipschitz condition which guarantees a unique solution of the deterministic initial value problem. Kaleva [1] studied the Cauchy problem of fuzzy differential equation and characterized those subsets of fuzzy sets in which the Peano theorem is valid. Park et al. in [11–14] have considered the existence of solution of fuzzy integral equation in Banach space. In 2002, Xue and Fu [15] established solutions to fuzzy differential equations with right-hand side functions satisfying Caratheodory 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 [1] 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:

where

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 integral and it is a powerful tool. It is well known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue, and Newton integrals. 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 [16]. Recently, Wu and Gong [17, 18] have combined the fuzzy set theory and nonabsolute integral theory and discussed the fuzzy Henstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration. In order to complete the theory of fuzzy calculus and to transfer a fuzzy differential equation into a fuzzy integral equation, we [19, 20] have 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 [6, 21, 22] and using the concept of generalized differentiability [8], we will deal with the Cauchy problem of discontinuous fuzzy systems as follows:

where , , and , , are fuzzy-number-valued function and integrals which are taken in sense of strong fuzzy Henstock integration, and , are measurable functions such that , are continuous.

2. Preliminaries

2.1. Fuzzy Number Theory

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 [10]

Denote satisfies (1)–(4) , 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.

Then it is easy to see that is a fuzzy number space.

For , denote . Then from the above conditions (1)–(4), it follows that the -level set for all .

According to Zadeh's extension principle, we have addition and scalar multiplication in the fuzzy number space as follows [10]:

where and .

Define

where is the Hausdorff metric defined in . Then it is easy to see that is a metric in . Using the results in [23], we know that

(1) is a complete metric space;(2) for all , , ;(3) for all , , and .

The metric space has a linear structure; it can be embedded isomorphically as a cone in a Banach space of function , where is the unit sphere in , with an embedded function defined by

for all (see [23]).

Theorem 2 (see [24]). There exists a real Banach space such that can be embed as a convex cone with vertex 0 into . Furthermore the following conditions hold true:

(1)the embedding is isometric;(2)addition in induces addition in ;(3)multiplication by nonnegative real number in induces the corresponding operation in ;(4) is dense in ;(5) is closed.

It is well known that the -derivative for fuzzy-number-functions was initially introduced by Puri and Ralescu [5] 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 [8, 9]). 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 [8].

Definition 3 (see [8]). Let and . We say 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.).

2.2. The Strong Henstock Integrals of Fuzzy-Number-Valued Functions in

In this section we define the strong Henstock integrals of fuzzy-number-valued functions in the fuzzy number space and we give some properties of this integral.

Definition 4 (see [20]). A fuzzy-number-valued function will be termed piecewise additive on if there exists a division , such that is additive on each   .

Definition 5 (see [19, 20]). 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 exists a function and for any -fine division of we have

where such that is a fuzzy number and such that is a fuzzy number. We write .

Definition 6 (see [20]). A fuzzy-number-valued function defined on is said to be if for every there exists such that for every finite sequence of nonoverlapping intervals , satisfying where , for all , we have

where denotes the oscillation of over ; that is,

Definition 7 (see [20]). 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 .

For the strong fuzzy Henstock integrable we have the following theorems.

Theorem 8. Let . If a.e on , then is integrable on and .

Theorem 9. Let be integrable on and let for each . Then

(a)the function is continuous on ;(b)the function is differentiable a.e on and ;(c) is measurable.

Theorem 10 (controlled convergence theorem; see [20]). Suppose that is a sequence of SFH integrable functions on satisfying the following conditions:

(1) a.e. in as ;(2)the primitives of are uniformly in ;(3)the primitives converge uniformly on ;

then is also SFH integrable on and

3. Main Results

In this section we prove some existence theorems for the problem (4).

For any bounded subset of the Banach space , we denote by the Kuratowski measure of noncompactness of ; that is, the infimum of all such that there exists a finite covering of by sets of diameter less than . For the properties of we refer to [25], for example.

Lemma 11 (see [25]). Let be a family of strong equicontinuous functions; then

where denotes the Kuratowski measure of noncompactness in and the function is continuous.

Theorem 12 (see [25]). Let be a closed convex subset of and let be a continuous function from into itself. If, for ,

is relatively compact, then has a fixed point.

Theorem 13. If the fuzzy-number-valued function is integrable, then

where is the convex hull of , is an arbitrary subinterval of , and is the length of .

Proof. Because is abstract integrable in a Banach Space, by using the mean valued theorem of integrals, we have

On the other hand, there exists .

So, we have . And the set is a closed convex set; we have

Definition 14. A fuzzy valued function is a Caratheodory function if, for each , the fuzzy valued function is measurable in , and for almost all , the fuzzy valued function is continuous with respect to .

For , we define the norm of by

Let

Obviously, the set is closed and convex in .

We define the operator by

where integrals are taken in the sense of . Moreover, let .

Definition 15. A continuous function is said to be a solution of the problem (4), if satisfies

or

Theorem 16. Assume that, for each continuous function , is integrable, and is a Caratheodory function. Let , be measure functions such that , are continuous. Moreover, there exists and a Caratheodory function , with

such that the zero function is the unique continuous solution of the inequality

Suppose that is equicontinuous, equibounded, and uniformly on . Then there exists at least a solution of the problem (4) on for some with continuous initial function .

Proof. By equicontinuity and equiboundedness of , there exist some numbers such that

for and .

Next, we will prove that the operator is continuous. In fact, let . Because the function is a Caratheodory function, by the following equality

and Theorem 10, we have .

Observe that a fixed point of is the solution of the problem (4). Now we prove that has a fixed point using Theorem 12.

Suppose that satisfies condition for some ,  . Let , ; then is equicontinuous. By Lemma 11, is continuous on .

Let for any and let denote the mapping defined by , for each , . Obviously, , and

holds ture.

Using (27), Lemma 11, and the properties of measure of noncompactness , we have

Because , we have

By assumption, because the zero function is unique continuous solution of the last inequality, so we have . By Arzelá-Ascoli Theorem, is relatively compact. So, by Theorem 12, has a fixed point which is a solution of problem (4).

Next, we give another existence theorem for problem (4).

Let be the spectral radius of the integral operator defined by

Theorem 17. Assume that, for each continuous function , is integrable, and is a Caratheodory function and , are measure functions such that , are continuous. Moreover, there exists and such that

for each , . Suppose that is equicontinuous, equibounded, and uniformly on and . Then there exists at least a solution of the problem (4) on for some with continuous initial function .

Proof. By equicontinuity and equiboundedness of , there exist some numbers such that

for and . By assumption, the operator is well defined and maps into . Now, we show that the operator is continuous. In fact, let . Because the function is a Caratheodory function, by following equality

and Theorem 10, we have .

Observe that a fixed point of is the solution of the problem (4). Now we prove that has a fixed point using Theorem 12.

Suppose that satisfies condition for some , . Let , ; then is equicontinuous. By Lemma 11, is continuous on .

We divide the interval , where , . Let . By Lemma 11 and the continuity of there exists such that

In addition, by the definition of operator and Theorem 16 we have

for all , where and . So, we have

Using (35), (38) and the properties of measure of noncompactness , we have

where ; so we get

By continuity of we have and as . So, we have

Therefore, we have

for . Since , by the properties of measure of noncompactness , we have

and so in view of (44) it follows that

for . Because this inequality holds for all and , by applying Gronwall's inequality, we get that for . By Arzelá-Ascoli Theorem, is relatively compact. So, by Theorem 12, has a fixed point which is a solution of problem (4).

4. Conclusion

In this paper, we deal with the existence problems of discontinuous fuzzy integral equations involving the strong fuzzy Henstock integral in fuzzy number space. The functions of the equations are supposed to be discontinuous with respect to some variables and satisfy nonabsolute fuzzy integrability. Our result improves the result given in [15, 26] (where uniform continuity was required), as well as those referred to therein.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the anonymous referees and Professor Márcia Federson 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 (nos. 11161041 and 71061013) and the Fundamental Research Fund for the Central Universities (no. 31920130010).

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