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Basic theorems for fuzzy differential equations in the quotient space of fuzzy numbers
Advances in Difference Equations volume 2014, Article number: 303 (2014)
Abstract
In this paper, we study the fuzzy differential equations in the quotient space of fuzzy numbers. We solve the initial value problem for fuzzy differential equations provided that the involved mappings are continuous, of uniformly bounded variation, and are bounded functions. Then we establish a variety of comparison results for the solutions of fuzzy differential equations.
MSC:34A07, 46S40.
1 Introduction
Fuzzy sets were introduced in 1965 by Zadeh [1] with a view to reconcile mathematical modeling and human knowledge in the engineering science. Fuzzy set theory and its applications have been extensively developed since the 1970s and the industrial interest in fuzzy control has dramatically increased since then [2–11].
The fuzzy differential equation was first introduced by Kandel and Byatt [12]. Since then, it has been extensively investigated on the metric space of a normal fuzzy convex set with the distance D given by the maximum of the Hausdorff distance between the corresponding level sets. The fuzzy differential equation was studied by Kaleva [9, 13, 14] and Wu and Song [15–17], for the fuzzy-valued function of a real variable whose values are normal, convex, upper semicontinuous, and are compactly supported fuzzy sets in . Many authors studied the existence and uniqueness of solutions of the initial value problems for fuzzy differential equations under various kinds of conditions and obtained many meaningful results in [13, 18–25]. One found the local existence and uniqueness theorems for the Cauchy problem when the fuzzy-valued function f satisfies the generalized Lipschitz condition [16]. The existence theorems under compactness-type conditions were studied in [17]. Based on these works, the global existence of solutions of the Cauchy problem was investigated in [15].
The above results of the fuzzy differential equation were based on the well-known and widely used Hukuhara difference, proposed by [26], and the H-differentiability of Puri and Ralescu [25], which generalized the Hukuhara differentiability of a set-valued mapping. In [27, 28], Mareš presented a natural equivalence relation between fuzzy quantities. This equivalence relation can be used to partition the set of fuzzy quantities into equivalence classes having the desired group properties under the addition operation [29–31]. Hong and Do [32] defined a more refined equivalence relation than Mareš [27] and improved Mareš’s results. In [33], Qiu et al. showed that the method of finding the inverse operation of fuzzy numbers in the sense of Mareš is very intuitive. The authors introduced a new concept of convergence under which the quotient space in complete. As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there would be more differentiable fuzzy functions than what is found in the literature. In [34] Qiu et al. further investigated the differentiability and integrability properties of such functions and gave an existence and uniqueness theorem for a solution to a fuzzy differential equation in the quotient space of fuzzy numbers.
In this paper, we will study the basic theory of fuzzy differential equations in the quotient space of fuzzy numbers. In Section 2, we recall some related concepts. In Section 3, we will solve the initial value problem for fuzzy differential equations. In Section 4, we will establish a variety of comparison results for the solutions of fuzzy differential equations, which form the essential tools for studying the fundamental theory of fuzzy differential equations. The comparison discussion shows how one can develop the theory of differential inequalities with the minimum linear structure.
2 Preliminaries
We start this section by recalling some pertinent concepts and key lemmas from the functions of bounded variation, fuzzy numbers, and fuzzy equivalence classes which we need for the discussion below.
Definition 2.1 [35]
Let . f is said be of bounded variation if there exists a such that
for every partition on . The set of all functions of bounded variation on is denoted by .
Definition 2.2 [35]
Let be a function of bounded variation. The total variation of f on is defined by
where p represents all partitions of .
Lemma 2.1 [35]
For any constants , if , then so is cf and
Lemma 2.2 [35]
For any constants , if , then so is and
Lemma 2.3 [35]
Let f be a function of bounded variation on , and let V be a function defined by . Then if f is continuous from the left (or right) at a point , so is V.
Lemma 2.4 [35]
If , then
Lemma 2.5 [35]
Every monotonic function f is of bounded variation, and .
Lemma 2.6 [35]
If , then so is fg and
where , .
A fuzzy set of ℝ is characterized by a membership function . An α-level set of is for each . We define the set by , where denotes the closure of a crisp set A. A fuzzy set is said to be a fuzzy number if it satisfies the following conditions [36]:
-
(1)
is normal, i.e., there exists an such that ;
-
(2)
is convex, i.e., , for all and ;
-
(3)
is upper semicontinuous;
-
(4)
is compact.
Equivalently, a fuzzy number is a fuzzy set with nonempty bounded level sets for all , where denotes a closed interval with the left end point and the right end point . We denote the class of fuzzy numbers by ℱ. Notice that the real numbers ℝ can be embedded in ℱ by defining a fuzzy number as
for each . Thus we will represent the singleton by for any real number and in particular is just the usual zero.
For any and , owing to Zadeh’s extension principle [37–39], addition and scalar multiplication are defined for any by
and
For any , we define the fuzzy number by , i.e., , for all . It is well known that
and
for all and . In particular, . By the level set representations of the fuzzy numbers , , , we can get
which implies ℱ is a commutative semigroup under addition. We say that a fuzzy number is symmetric [27], if
for all , i.e., . The set of all symmetric fuzzy numbers will be denoted by .
Definition 2.3 [32]
Let . We say that is equivalent to and write if and only if there exist symmetric fuzzy numbers such that
The equivalence relation defined above is reflexive, symmetric, and transitive [27]. Let denote the equivalence class containing the element and denote the set of equivalence classes by .
Definition 2.4 [29]
For a fuzzy number , we define a function by assigning the midpoint of each α-level set to for all , i.e.,
Then the function will be called the midpoint function of the fuzzy number .
Lemma 2.7 [33]
For any , the midpoint function is continuous from the right at 0 and continuous from the left on . Furthermore it is a function of bounded variation on .
It is interesting to note that the midpoint function is actually a gradual number and all of the gradual numbers form a group structure under addition [40, 41]. Among applications of the decomposition of a fuzzy interval using a symmetric fuzzy number and a midpoint function is the study of the fuzzy variance of a fuzzy interval [42, 43].
Definition 2.5 [28]
Let and let be a fuzzy number such that for some , if for some and , then . Then the fuzzy number will be called the Mareš core of the fuzzy number .
In [33], Qiu et al. not only pointed out that is equivalent to if and only if they have the same midpoint function, which implies that each equivalence class corresponding to each midpoint function, but they also pointed out that for each fuzzy number has only one Mareš core, and is equivalent to if and only if they have the same Mareš core, which implies that all elements of an equivalence class have the same Mareš core . Hence, the Mareš core of is also called the Mareš core of the equivalence class that contains the fuzzy number . It is natural to define the midpoint function for an equivalence class as follows.
Definition 2.6 [34]
For an equivalence class , we define a midpoint function by
for all , where is the Mareš core of .
Definition 2.7 [34]
For any , we define by
It is obvious that , for all . Moreover, it follows from Definitions 2.3 and 2.7 that
for any . In [34] Qiu et al. have given an example to show that the above inclusion can become strict.
Remark 2.1 The addition operation defined by Definition 2.7 is a group operation over the set of equivalence classes up to the equivalence relation in Definition 2.3. It means that
for any . For the details of the discussion, please see [44, 45].
is a group.
For any , the midpoint functions and are continuous from the right at 0 and continuous from the left on , and they are functions of bounded variation on [33]. Then the function is also continuous from the right at 0 and continuous from the left on . In addition, by Lemma 2.6, we know that is a function of bounded variation on . Hence, is the midpoint function for some fuzzy number equivalence class.
Definition 2.8 [34]
Let . If
for all , then what we called is the product of and , i.e., .
Definition 2.9 For any and , we define by
It is obvious that , for all . If is the Mareš core of , then is the Mareš core of . In fact, let be the Mareš core of . Then, by Theorems 3.10 and 3.13 in [33], and Lemma 2.1, we have
and
for all . Hence, we get for all , i.e., .
Definition 2.10 Define by
for all .
is a metric space [33]. From Definition 2.10, we list here some simple properties of the metric :
-
(1)
;
-
(2)
;
-
(3)
for all and , where the addition and scalar multiplication on are defined in Definitions 2.7 and 2.9, respectively.
3 Initial value problem
In this section, we will firstly give some definitions and basic results for the fuzzy differentials and fuzzy integrals. Then we will solve the initial value problem for fuzzy differential equations.
Definition 3.1 [34]
Let . A mapping is differentiable at if there exists an such that
If (or ), then we consider only (or ). If F, G are differentiable at t, then we have and , .
Lemma 3.1 [34]
If is differentiable, then it is continuous with respect to .
Definition 3.2 [34]
A mapping is measurable if F is measurable with respect to .
A mapping is called integrably bounded if there exists an integrable function such that for all and ; a mapping is said to be of uniformly bounded variation with respect to (for short, uniformly bounded variation) if there exists a constant such that
for all [34].
Definition 3.3 [34]
Let be measurable. The integral of F over T, denoted or , is a mapping , which is defined by the equation
for all . The mapping is said to be integrable over T if there exists an such that . In this case, we denote the integral by
If is a measurable, integrably bounded, and uniformly bounded variation of a mapping, then F is integrable on T and [34]. Also the following properties of the integral are valid. If are integrable on T and , then:
Lemma 3.2 [34]
Let be continuous with respect to and of uniformly bounded variation. Then the integral is differentiable and for all .
Lemma 3.3 [34]
Let be differentiable and the derivative be integrable over T. Then for all , we have , .
Lemma 3.4 [34]
Let be continuous with respect to and of uniformly bounded variation and . Then for we have
Assume that is continuous and of uniformly bounded variation. Consider the initial value problem
From Lemmas 3.1, 3.2, and 3.3, a lemma immediately follows.
Lemma 3.5 A mapping is a solution to the problem (1) if and only if it is continuous, uniformly bounded variation and satisfies the integral equation
for all .
Theorem 3.1 Let be continuous with respect to and of uniformly bounded variation on T. Suppose there exists an such that
for all and . Then the problem (1) has a solution on T.
Proof Since f is of uniformly bounded variation, there exists a constant such that
for all and . Let . Denote by the set of all continuous and of uniformly bounded variation mappings φ from T to and for all . We metricize by setting
for . Since is a metric space, it is easy to show that also is metric space. Next, we shall show that is complete. By Definition 2.10, we have
Let is a Cauchy sequence in , i.e., for any , there exists a positive integer N such that
for all . Denote by the set of all continuous function with respect to the first variable . Let
for any . Then is a Cauchy sequence in , which implies that for any given and , is a Cauchy sequence. Thus, there exists an such that
Now we shall show that . Firstly, we have
for all and . Thus, letting we have
whenever , which implies that converges uniformly to with respect to on . Thus, we find that is continuous with respect to , i.e., . For any given , let be a nondecreasing sequence converging to . For any given , we have
Since is continuous from the left on with respect to α and converges uniformly to with respect to t, we see that the limit of the right-hand side of the above inequality is equal to 0 as . Hence is continuous from the left on with respect to α. Similarly, we can find that is continuous from the right at 0. Since are of bounded variation on and converge uniformly to with respect to α, by Theorem 4.8 in [33] we obtain
Hence, by Lemma 2.7, we know that is a midpoint function of some fuzzy number equivalence class, we still use to denote this fuzzy equivalence class for any . It is obvious that is a mapping of uniformly bounded variation from T to and for arbitrary there exists a positive integer N such that
for all . Since is continuous, for any given and the same , there exists a such that if , then
Thus, we have
which implies that
Thus, we get . Hence, is complete.
For define Gξ on T by the relation
Then for any , we have
By Lemma 2.2, we get
Thus, we see that Gξ is of uniformly bounded variation. Hence, by Corollaries 4.1 and 4.3 in [34], we have . Let
For any and :
(1) For any given , if , then
(2) By Lemma 2.2 and Definitions 2.7 and 2.9, we have
(3) By definition of D, we have
Thus, , i.e., B is convex subset of .
Let the set and the sequence converge to ξ in B. Since f is continuous with respect to , for any , there exists such that
for all and . Thus, we get
for all . By Lemma 3.4, we have
for all . Thus we get
Hence, G is a continuous mapping of B into itself, i.e., . For any , , and , by Lemma 3.4, we have
which implies that GB is equicontinuous. Since GB is totally bounded if and only if it is equicontinuous, we see that GB is totally bounded. By Ascoli’s theorem, we conclude that GB is a relatively compact subset of . By Schauder’s fixed point theorem, G has a fixed point, and by Lemma 3.5, we know that this fixed point is a solution of (1). □
Example 3.1 Define by the level sets of the fuzzy mapping
where is the Mareš core of , for all . Thus, we have
for all and . It is obvious that is continuous from the right at 0 and continuous from the left on with respect to α. Since is increasing with respect to α and by Lemma 2.5, we get
Thus, we find that is of uniformly bounded variation. Since is uniformly continuous with respect to , we see that is continuous with respect to . Define by
It is obvious that f is continuous with respect to . Since
for all and by Lemma 2.6, we get
Thus, f is of uniformly bounded variation. Since
for all and . Then satisfies the assumptions of Theorem 3.1 and hence the initial value problem
has a unique solution on . By the proof of Theorem 3.1, we know that the unique solution of the initial value problem is
Then we get
Thus, we have
for all . By Theorems 3.10 and 3.13 in [33], we see that the left end points and the right end points of level sets of the Mareš core for , respectively, obey
4 Comparison theorems
Using the properties of , integrals, and differential inequalities, we will establish the comparison principles in this section.
Definition 4.1 Define by
for all .
Theorem 4.1 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where and is nondecreasing with respect to φ for all . Suppose the maximal solution of the scalar differential equation
exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and we have
for all .
Proof Let . Then . Thus, we get
for all . By Theorem 1.9.2 in [46], we obtain for all . □
Example 4.1 Define by the level sets of the fuzzy mapping
where is the Mareš core of , for all . Thus, we have
It is obvious that F is continuous with respect to . Since
for all , we see that F is of uniformly bounded variation. Define by
where the multiplication in is defined by Definition 2.6. It is obvious that f is continuous with respect to and of uniformly bounded variation. By Definition 2.10, we have
for all and . Define the scalar differential equation
where the function and . It is obvious that for all , and is nondecreasing with respect to φ for all . Then
for all and . Hence, we obtain
for all and . Let and , such that . By Theorem 4.1, we conclude that
for all , where , are two solutions of fuzzy differential equation
through , on .
Corollary 4.1 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where g satisfies the assumptions of Theorem 4.1. Then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.1. □
Theorem 4.2 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where . Suppose the maximal solution of the scalar differential equation (2) exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and we have
for all .
Proof Let . Then and for any fixed and with , we have
Since
and
we get
Thus, by Definition 4.1, we get
By Theorem 1.4.1 in [46], we obtain , for all . □
Corollary 4.2 Let be continuous with respect to and of uniformly bounded variation on T. If
for all and , where g satisfies the assumptions of Theorem 4.2, then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.2. □
Theorem 4.3 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where . Suppose the maximal solution of the scalar differential equation (2) exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and , we have
for all .
Proof Let . Then and for any fixed and with , from the proof of Theorem 4.2, we have
which implies that
By Theorem 1.4.1 in [46], we obtain , for all . □
Corollary 4.3 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where g satisfies the assumptions of Theorem 4.3. Then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.3. □
Example 4.2 Define two fuzzy mappings by the level sets
and
for all , where and are the Mareš core of and , respectively, for all . Thus, we have
for all and . It is obvious that and are continuous from the right at 0 and continuous from the left on with respect to α. Since and are decreasing with respect to α and by Lemma 2.4, we get
and
Thus, we see that and are of uniformly bounded variation. Since and are uniformly continuous with respect to , we see that and are continuous with respect to . Define by
where the addition and multiplication in are defined by Definitions 2.7 and 2.8, respectively. It is obvious that f is continuous with respect to and of uniformly bounded variation. By Definition 2.10, we get
for all sufficiently small. Hence
for all and . Define the scalar differential equation
where the function and . It is obvious that for all , . Then
for all and . Hence, we obtain
for all and . Let and , such that . By Theorem 4.3, we conclude that
for all , where , are two solutions of the fuzzy differential equation
through , on .
5 Conclusions
In this paper, we have researched the basic theory of fuzzy differential equations in the quotient space of fuzzy numbers. We have solved the initial value problem for the fuzzy differential equations provided that f is a continuous with respect to , of uniformly bounded variation on T, and is a bounded function, and then we have established a variety of comparison results for the solutions of fuzzy differential equations which form the essential tools for studying the fundamental theory of fuzzy differential equations. The comparison discussion shows how, with the minimum linear structure, one can develop the theory of differential inequalities that are important in comparison principles. We also hope that our results in this paper may lead to significant, new, and innovative results in related fields.
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Acknowledgements
This work was supported by The National Natural Science Foundations of China (Grant Nos. 11201512 and 61472056) and The Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).
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Qiu, D., Lu, C., Zhang, W. et al. Basic theorems for fuzzy differential equations in the quotient space of fuzzy numbers. Adv Differ Equ 2014, 303 (2014). https://doi.org/10.1186/1687-1847-2014-303
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DOI: https://doi.org/10.1186/1687-1847-2014-303
Keywords
- fuzzy number
- equivalence classes
- fuzzy differential equation
- comparison theorems