Sheaf fuzzy problems for functional differential equations
© Tri et al.; licensee Springer 2014
Received: 26 August 2013
Accepted: 20 March 2014
Published: 27 May 2014
In this paper, we present the studies on two kinds of solutions to fuzzy functional differential equations (FFDEs) and sheaf fuzzy functional differential equations (SFFDEs). The different types of solutions to FFDEs and SFFDEs are generated by the usage of generalized Hukuhara derivative concepts of fuzzy derivative in the formulation of a differential problem. Some examples are given to illustrate these results.
Keywordsfuzzy sets fuzzy functional differential equations generalized Hukuhara derivative sheaf fuzzy problems
The study of fuzzy differential equations (FDEs) forms a suitable setting for mathematical modeling of real-world problems in which uncertainties or vagueness pervade. Most practical problems can be modeled as FDEs [1, 2]. There are several approaches to the study of FDEs. One popular approach is based on H-differentiability for fuzzy number value functions. Under this setting, mainly the existence and uniqueness of the solution of a FFDE are studied (see, e.g., [3–6]). However, this approach has the disadvantage that it leads to solutions which have an increasing length of their support. Recently, Bede and Gal  solved the above mentioned approach under strongly generalized differentiability of fuzzy-number-valued functions. In this case the derivative exists and the solution of a fuzzy differential equation may have decreasing length of the support, but the uniqueness is lost. Other researchers have proposed several approaches to the solutions of FDEs (e.g., [3, 8–15]). Therefore, our point is that the generalization of the concept of H-differentiability can be of great help in the dynamic study of fuzzy functional differential equations and sheaf fuzzy problems for fuzzy functional differential equations. Recently, several works, e.g., [4, 9, 16–20], studied fuzzy differential equations and fuzzy integro-differential equations, fractional fuzzy differential equations, and some methods for solving fuzzy differential equations [21, 22] were proposed.
where and the symbol ′ denotes the first type Hukuhara derivative (classical Hukuhara derivative). Author studied the local and global existence and uniqueness results for (1.1) by using the method of successive approximations and the contraction principle. Malinowski  studied the existence and uniqueness result of a solution to the delay set-valued differential equation under condition that the right-hand side of the equation is Lipschitzian in the functional variable. In this paper, under the generalized Lipschitz condition, we obtain the local existence and uniqueness of two solutions to (1.1). Besides, we establish the global existence and uniqueness of two solutions to (1.1) by using some dissipative conditions . We were inspired and motivated by the results of Stefanini and Bede [7, 24], Malinowski  and Lupulescu [10, 11]. The paper is organized as follows. In Section 2, we collect the fundamental notions and facts about a fuzzy set space, fuzzy differentiation. In Section 3, we discuss the FFDE with the generalized Hukuhara derivative. Under suitable conditions we prove the existence and uniqueness of the solution to FFDE by using two different methods. In Section 4, the existence and uniqueness of the sheaf solutions to sheaf fuzzy functional differential equations are studied. Finally, we give some examples to illustrate these results.
2 Preliminaries and notation
ω is normal, that is, there exists such that ;
- (ii)ω is fuzzy convex, that is, for ,
ω is upper semicontinuous;
is compact, where cl denotes the closure in .
Although elements of are often called the fuzzy numbers, we shall just call them the fuzzy sets. For , define . We will call this set an α-cut (α-level set) of the fuzzy set ω. For , one has that for every . In the case , the α-cut set of a fuzzy number is a closed bounded interval , where denotes the left-hand endpoint of and the right-hand endpoint of . It should be noted that for , , a triangular fuzzy number is given such that and are the endpoints of the α-cut for all .
for all and . Let . If there exists such that , then is called the H-difference of , and it is denoted by . Let us remark that .
If for fuzzy numbers there exist Hukuhara differences , , then and .
Definition 2.1 ()
The generalized Hukuhara differentiability was introduced in .
If satisfying (2.1) exists, we say that x is generalized Hukuhara differentiable (gH-differentiable for short) at t.
Theorem 2.1 ()
- (i)x is [(i)-gH]-differentiable at t if , are differentiable functions and we have(2.2)
- (ii)x is [(ii)-gH]-differentiable at t if , are differentiable functions and we have(2.3)
Definition 2.4 ()
We say that a point is a switching point for the differentiability of x, if in any neighborhood V of t there exist points such that
(type I) at (2.2) holds while (2.3) does not hold and at (2.3) holds and (2.2) does not hold, or
(type II) at (2.3) holds while (2.2) does not hold and at (2.2) holds and (2.3) does not hold.
Lemma 2.1 (Bede and Gal )
if x is [(i)-gH]-differentiable (i.e., Hukuhara differentiable), then ;
if x is [(ii)-gH]-differentiable, then .
Lemma 2.2 (see )
If x is [(i)-gH]-differentiable, then , are differentiable functions and .
If x is [(ii)-gH]-differentiable, then , are differentiable functions and we have .
Lemma 2.3 (see )
Definition 2.5 (see )
Corollary 2.1 (see )
Theorem 2.2 (see )
for every .
By help of these concepts we formulate some dissipative conditions for FFDEs (1.1) and, under these conditions, we establish the global existence and uniqueness of a solution of functional fuzzy differential equations.
3 Fuzzy functional differential equations
the metric on the space . Define , . Then, for each , we denote by the element of defined by , .
where , , and the symbol denotes the generalized Hukuhara derivative from Definition 2.2. By a solution to equation (3.1) we mean a fuzzy mapping that satisfies: for , x is differentiable on and for .
Lemma 3.1 (see )
Assume that and . Then the fuzzy mapping belongs to .
Remark 3.1 (see )
Under assumptions of the lemma above we have that the mapping is integrable over the interval I.
Remark 3.2 (see )
If is a jointly continuous function and , then the mapping is bounded on each compact interval I. Also, the function is bounded on I.
Definition 3.1 Let be a fuzzy function which is [(i)-gH]-differentiable. If x and its derivative satisfy problem (3.1), we say that x is a (i)-solution of problem (3.1).
Definition 3.2 Let be a fuzzy function which is [(ii)-gH]-differentiable. If x and its derivative satisfy problem (3.1), we say that x is a (ii)-solution of problem (3.1).
3.1 Local existence
The following comparison principle is fundamental in investigation of the global existence of solutions of functional fuzzy differential equations.
Theorem 3.1 ()
existing on . Then, if , we have , .
Under the generalized Lipschitz condition we obtain the existence and uniqueness of two solutions to FFDE. To prove this assertion, we use an idea of successive approximations.
and , ;
- (ii), and , , such that is nondecreasing on u, the IVP(3.7)
, , and .
for the case of [(ii)-gH]-differentiability (where such that the sequence (3.9) is well defined, i.e., the foregoing Hukuhara differences do exist), converge uniformly to two unique solutions and of (3.1), respectively, on where .
provided , where , proving that is continuous on .
Hence we infer that , for ().
The summands in the last expression converge to 0. Due to Lemma 3.2 the function is the (ii)-solution to (3.1).
Hence, by Theorem 3.1, we have for all , where is the maximal solution of IVP (3.7). Therefore , which completes the proof. □
Moreover, there exists such that .
for the case of [(ii)-gH]-differentiability, converge uniformly to two unique solutions and of (3.1), respectively, on where .
Proof The proof is obtained immediately by taking in Theorem 3.2, . □
3.2 Global existence
Next, we shall establish the global existence and uniqueness results for FFDE (3.1). For the global existence and uniqueness, we use the dissipative conditions which were introduced and studied in . We now prove a comparison theorem, which is a useful tool in proving the global existence theorem.
for . Then, if , are any (ii)-solutions of FFDE (3.1) such that exist for , we have , provided that .
The proof is complete. □
Remark 3.3 Under the assumptions of Theorem 3.3, satisfies , where . Then we obtain .
for every and with . Then the functional fuzzy differential equation (3.1) has a (ii)-solution on .
Then . Next, we define a partial order ≲ on as follows: if and only if and on . Then the standard application of Zorn’s lemma assures the existence of a maximal element z in . The proof is complete if we show that . Suppose that it is not true, so that .
for every , where . Therefore, there exists such that on . We infer that for every .
then it is clear that is a solution of (3.1) on . This contradicts the maximality of and hence . □
for . Then the functional fuzzy differential equation (3.1) has a unique (ii)-solution.
where is the solution of (3.19) on . By assumption , we obtain on . The proof is complete. □
for . By using Lemma 2.2, we have the following two cases.
where , , . In this example we shall solve (3.23) on .
4 Sheaf fuzzy functional differential equations
where , .
Definition 4.2 Let be a fuzzy sheaf which is [(i)-gH]-differentiable (i.e., [(i)-gH]-differentiable for each ). If x and its derivative satisfy problem (4.1) for each , we say that is a (i)-sheaf solution of problem (4.1).
Definition 4.3 Let be a fuzzy sheaf such that [(i)-gH]-differentiable (i.e., [(i)-gH]-differentiable for each ). If x and its derivative satisfy problem (4.1) for each , we say that is a (ii)-sheaf solution of problem (4.1).
Definition 4.4 A sheaf local solution is unique if for any , that is, a sheaf local solution to (4.1) on I.
In Section 3, under the generalized Lipschitz condition and dissipative condition, we proved the existence and uniqueness of the solution to both kinds of FFDE (3.1). In this section, we prove the existence and uniqueness of a sheaf solution to both kinds of sheaf fuzzy functional differential equation (SFFDE) (4.1) by using the results in Section 3.
Moreover, there exists such that . Then problem (4.1) has a unique sheaf solution for each case ([(i)-gH]-differentiable or [(ii)-gH]-differentiable type) on , where .
Therefore . This proof is complete. □
where , , , . In this example, we shall solve (4.4) on .
Suppose . We say that if and only if , and , . We can also define the fuzzy interval . Letting be two functions, we say that if for . Moreover, we define .
where , , , , , . In this example, we shall solve (4.4) on .
One can obtain the (i)-sheaf solution and (i)-sheaf solution by using the methods as in the above examples.
The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. This research is funded by the Foundation for Science and Technology Development of Ton Duc Thang University (FOSTECT), website: http://fostect.tdt.edu.vn.
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