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A Note on Asymptotic Equilibrium for Fuzzy Differential Equations
The asymptotic equilibrium results for fuzzy differential systems are investigated, where satisfies the compactness-type and satisfies the dissipative-type conditions. It is worth mentioning that the uniformly continuous conditions of are removed in Song et al. (2005). That is to say, the results of Song et al. (2005) are extended. In addition, the global existence and asymptotic equilibrium results of fuzzy differential systems are obtained.
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 the Cauchy problems which have a unique solution if is continuous and bounded. For a general reference to fuzzy differential equations, see a recent book by Lakshmikantham and Mohapatra  and references therein. In particular, Wu and Song [8–10] changed the initial value problem of fuzzy differential equations into abstract differential equations on a closed convex cone in a Banach space by the operator that is the isometric embedding from onto its range in the Banach space . They established the relationships between a solution and its approximate solutions to fuzzy differential equations. Furthermore, they obtained the local existence theorems under the compactness-type and the dissipative-type conditions. Park and Han  showed the global existence and uniqueness of fuzzy solutions of fuzzy differential equation using the properties of Hasegawa’s function and successive approximation. Song et al.  pointed out a variety of results which assure global existence of solutions to fuzzy differential equations. Song et al.  studied the asymptotic equilibrium for fuzzy differential equations: where and is a continuous. Since we are only interested in local solutions to , we assume and on for some , where and . The methods used by them are as follows: satisfied not only the compactness-type conditions but the uniformly continuity condition. Hence, it is of significance to lessen the growth conditions of .
Based on those preceding works, in this paper, we firstly give the existence theorems for without a uniform continuity assumption on . In addition, the more general systems than are considered: where , , , and and are the fuzzy number space, respectively. , satisfies the compactness-type, and satisfies the dissipative-type conditions. In particular, when and , we obtain that Corollary 24 is the promotion of the results of . When and , we obtain that Corollary 26 is the asymptotic equilibrium of fuzzy differential system under the dissipative-type conditions.
As preliminaries we recall some basic results on fuzzy number space and list several comparison theorems on classical ordinary differential equations. In Section 3, we will proof the existence theorems for without a uniform continuity assumption on . In Section 4, we will show that the asymptotic equilibrium for fuzzy differential system . Finally, in Section 5, we present some concluding remarks.
Let denote the family of all nonempty compact convex subsets 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:
Denote satisfies (1)–(4) 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 . Then from above (1)–(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 in [14, 15], we know that is a complete metric space. It is well known that application of fuzzy set theory very often involves the metric space of normal fuzzy convex set over , where is the supremum of the Hausdoff distance between corresponding level sets. This metric has been found very convenient in studying of fuzzy differential equations (see ).
The properties of the support functions can be referred to in  for details.
Theorem 2 (see [16, 17]). Let be the unit sphere of and the inner product in . Suppose , is the support function of ; then(1); that is, is bounded on , for each fixed ;(2) is nonincreasing and left continuous in and right continuous at , for each fixed ;(3) is Lipschitz continuous in and (4)if , , then .
Let be compact interval. The fuzzy mapping is called strong measurable if for all the set-valued mapping defined by is Lebesgue measurable, where is endowed with the topology generated by the Hausdorff metric . A mapping is called integrable bounded if there exists an integrable function such that for all .
Definition 4. Let . The integral of over , denoted by , is defined level-wise by the equation (see ):
A strongly measurable and integrable mapping is said to be integrable over if . From , we know that if is continuous, then it is integrable.
Let . If there exists a fuzzy number such that , is called the -difference of and that is denoted by . For brevity, we always assume that it satisfies the -difference when dealing with the operation of subtraction of fuzzy numbers throughout this paper.
Definition 5 (see ). A mapping is differentiable at if there exists a , such that the limits exist and are equal to .
Here the limits are taken in the metric space . At the endpoint of , we consider only one-side fuzzy derivatives. If is differentiable at , then we say that is the fuzzy derivative of at the point .
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 [19, 21]). 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 6 (see ). Let and . One says that is differentiable at if there exists an element , such that,(1)for all sufficiently small, there exist , and the limits (in the metric ) (2)for all sufficiently small, there exist , and the limits (3)for all sufficiently small, there exist , and the limits (4)for all sufficiently small, there exist , and the limits ( and at denominators mean and , resp.).
In addition, we define a continuous fuzzy-valued function by , where is an open set.
Theorem 7 (see ). Assume that . A function is a solution to the problem , if and only if it is continuous and satisfies the integral equation or for all .
Theorem 8 (see ). Let be an open set, , . Suppose that the maximum solution of initial value problem , is and its largest interval of existence of right solution is . If satisfies , for all and where is one of the four Dini derivatives and at most is a countable set on . Then one has , for all .
Theorem 9 (see , (Ascoli-Arzela)). A set is a relative compact set if and only if is equicontinuous and, for any , is relative compact set in .
3. On the Cauchy Problem for Fuzzy Differential Equations
Let be Kuratowski’s measure of noncompactness (see  for details).
Lemma 10. Let , are two bounded subsets of Banach space and a real number; then Kuratowski’s measure of noncompactness has following properties:(1) if and only if is a relatively compactness set,(2),(3), where ,(4), where ,(5), where is the bounded subsets of Banach space .
It is well known that continuity of is not sufficient for the existence of local solution to (see ). The extra conditions that have been imposed on are mainly estimates which guarantee that a certain sequence of approximate solutions has at least a uniformly convergent subsequence. In this section, we would like to dispense with the uniform continuity assumption of .
Next, we describe a class of uniqueness functions. Let be an interval. A function is said to satisfy Caratheodory’s condition on if is measurable in for , continuous in for and such that to each and compact interval there exists a function with in .
Definition 11 (see ). A function with is said to be class , denoted by , if a.e. on and for imply on for each absolutely continuous .
We will consider a sequence of fuzzy-number-valued functions behaving like the sequence of approximate solutions that we will use in next section.
Theorem 12. Let be a sequence of continuous fuzzy-number-valued functions from to such that there is some function with on . Let . Then is integrable on and
Proof. Fix ; we have for every partition and every with for . With the continuity of , we have An application of Fatou’s Lemma gives and with Lebesgue’s dominated convergence theorem; since for , , we have Therefore, we get
Theorem 13. Let be a sequence of continuously differentiable functions. Assume that there exists such that a.e. , . Let ; then is an absolutely continuous function and a.e. .
Proof. The absolute continuity of follows from We conclude by Theorem 12 that Dividing by and letting give
Proof. There is a sequence of approximate solutions to satisfying Let . We have by Theorem 13 that a.e. on . Moreover, by the properties of , we have We claim that is for . By the continuity of and the equicontinuity of the , we have with for uniformly in . Since , we get on . Arzala-Ascoli’s theorem gives a uniformly convergent subsequence of and a standard argument shows that the limit of this subsequence is a solution to on .
Next, consider the terminal value problem where is continuous, , and . This problem is equivalent to the following initial value problem: by means of the transformation on .
Definition 15 (see ). A function with is said to be of class , denoted by , if a.e. on and for imply on for each absolutely continuous .
Proof. Consider the following problem: By Theorem 14 and condition , the problem (29) has a solution . Let for . As a consequence of Theorem 13 and the choice of , we conclude similar to the proof of Theorem 14 that has a solution.
4. On the Asymptotic Equilibrium for Fuzzy Differential Equations
Theorem 17. For all , define and satisfy the following properties:(1),(2),(3)if is an open interval in and is a differential function on , then
Proof. and are easy consequences of the definition and follows from Lemma 3.6 in .
Definition 18. Assume that is a real continuous function, , , , is differentiable, and for any . If is a continuous and , the initial value problem has only solution and satisfies Then is called a function of kind.
Definition 19. Assume that is a continuous and ; the initial value problem has only solution and satisfies for any . Then is called a function of kind.
Definition 20 (see ). One says that fuzzy equation has asymptotic equilibrium if every solution of , such that , exists on and tends to a limit as , and, conversely, to every given vector there exists a solution for fuzzy system which tends to as .
Lemma 21 (see ). Assume that(1) is locally Lipschitzian in for ,(2), for all ,(3), is nondecreasing in for each , and maximal solution of the scalar initial value problem , exists throughout .
Then the largest interval of any solution of (1) with is . In addition, if is bounded on , then exists in .
Lemma 22 (see ). Under the assumptions of Lemma 21, given , there exist a and a sequence defined on , such that(1) is equicontinuous on ;(2) is uniformly bounded on ; that is, there exists , such that for all and for all ;(3)for each is a solution of , .
In the following, we give the main result of this paper.
Theorem 23. Under the assumptions of Lemma 21, is satisfied Lemma 22, and there exists such that is relatively compact in and satisfies the following.(1)Let is a bounded set; one has for all , where there exists such that and .(2)For all , , and there exists such that with respect to when ; we have when , where is a continuous function and .(3)For , , such thatwhere and is the limit point of .
Then the fuzzy differential system has asymptotic equilibrium.
Proof. Since is satisfied, and , according to Lemma 10 and Theorem 13, and the following fact: can produce a subspace of ; we let ; then we have
From the assumption and , we obtain on . Since is equicontinuous and uniformly bounded, from Ascoli-Arzelar’s theorem, then there exists a subsequence of which uniformly converges to on any finite closed subset of . Without any loss of generality, let uniformly converge to on any finite closed subset of and let converge too.
Now, we consider the initial problem
Next, we will prove that converges uniformly to on any finite closed subset of . Let ; then . And let , according to Theorem 17; we have Let converge uniformly to on and combine to condition ; we have where , . Then we have and when . Let ; then . According to the definition of , we easily know that satisfies the following: there exists such that and . From above, we get that is equicontinuous and uniformly bounded on , so there exists a subsequence converging uniformly to . Since and , from the hypotheses of the theorem, fixed , there exist such that when , , which means that implies Because , we have . Hence converge uniformly to ; that is, converge uniformly to . From above, we can get that converge uniformly to on . Again, from the arbitrariness of , it is easy to know and Since converge uniformly on a compact set of , we have which means that is a solution of fuzzy system , .
Next, we will show . Applying Lemma 22, let be the largest solution of , for . According to , we have and Since is nondecreasing in for every , we have So, there exists an integer which satisfies where . In addition, there exists ; when , we have Hence, Hence, we have .
When and , we have the following.
Corollary 24. Under the assumptions of Lemma 21, is satisfied Lemma 22, and there exists such that is relatively compact in and satisfies condition (1) in Theorem 23. Then the fuzzy differential system has asymptotic equilibrium.
Remark 25. Compared with Theorem 3.4 in , the uniformly continuous conditions are removed.
When and , we have the following.
Corollary 26. Under the assumptions of Lemma 21, is satisfied Lemma 22, and there exists such that is relatively compact in and satisfies where , , . Then the fuzzy differential system has asymptotic equilibrium.
In this paper, some global existence and asymptotic equilibrium results for fuzzy differential equations under Bede’s derivative for fuzzy-number-valued functions are proved. We apply our main result to the terminal value problem for ordinary differential equations in fuzzy number spaces, a particular situation where we would like to dispense with the uniform continuity assumption of . Our results improve the results given in [8–10, 12, 13] (where uniform continuity was required), as well as those referred to therein. For future research, we extend the asymptotic stability and global attractivity results for the so-called fuzzy systems in our paper.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the anonymous referees and Professor Junjie Wei for many valuable comments and suggestions which helped in improving the presentation of the paper. The authors would like to thank National Natural Science Foundation of China (no. 11161041) and Fundamental Research Fund for the Central Universities (no. 31920130004 and no. zyz2012079).
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