The Strong Fuzzy Henstock Integrals and Discontinuous Fuzzy Differential Equations

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


Introduction
The Cauchy problems for fuzzy differential equations have been studied by several authors [1][2][3][4][5][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 [4], Nieto proved that the Cauchy problem has a uniqueness result if  is continuous and bounded.In [1,3,[7][8][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 [10] and references therein.In 2002, Xue and Fu [11] 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 [1] on certain intervals, and their solutions are not absolute continuous functions.To illustrate, we consider the following example.
Then, ℎ() =  |()| + Ã 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 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 [12] in 1963 and also independently by Kurzweil [13], 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 [14].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-numbervalued functions which extended Kaleva [1] 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 [19] and using the concept of generalized differentiability [20], 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 f :  →   is a strong fuzzy Henstock integrable function and  = {(, ) : | − | ≤ ,  ∈   , (, ) ≤ } .(5) 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.
According to Zadeh's extension principle, we have addition and scalar multiplication in fuzzy number space   as follows [21]: where , V ∈   and 0 ≤  ≤ 1.
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 f : [, ] →  F defined by f() =  ⋅ (), where  is a fuzzy number, ⋅ is the scalar multiplication (in the fuzzy context) and  : [, ] →  + , with   ( 0 ) < 0, is not differentiable in  0 (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 [20].

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 [18]).A fuzzy-number-valued function f is said to be termed additive on [, ] if, for any division Definition 4 (see [17,18]).A fuzzy-number-valued function f is said to be strong Henstock integrable on [, ] if there exists a piecewise additive fuzzy-number-valued function F on [, ] such that for every  > 0 there is a function () > 0 and for any -fine where
Hence, we have that { F} is uniformly  *  .We get the following theorem by Theorems 9 and 11.

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 [11]).Let  and  be fixed, and (iii)   () for almost everywhere  ∈ .Now we will state the existence theorem for the generalized solution of discontinuous fuzzy differential equation (4).
Theorem 20.Suppose that f satisfies the condition of Theorem 18; then there exists a generalized solution Φ of the discontinuous fuzzy differential equation (4) on some interval | − | ≤  which satisfies Φ() = .
We get the following existence theorem by Theorems 18 and 20.Finally, in this paper, we will show the continuous dependence of a solution on parameters by using Theorems 16,18,and 22.Let   be a connected set in .Let  > 0 and let  0 be fixed:

Conclusion
In this paper, we give the definition of the   for a fuzzynumber-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.
2, . . ., } such that F([ −1 ,   ]) is a fuzzy number and   = { ∈ {1, 2, . . ., } such that F([  ,  −1 ]) is a fuzzy number.One writes f ∈ SFH[, ].Definition 5. A fuzzy-number-valued function F defined on  ⊂ [, ] is said to be  *  () if for every  > 0 there exists  > 0 and () > 0 such that for any -fine partial division  = {([, V], )} with  ∈   satisfying ∑  =1 |V − | <  one has ∑ ( F[, V]) < .Definition 6.A fuzzy-number-valued function F is said to be  *  on  ⊂ [, ] if  is the union of a sequence of closed sets {  } such that on each   , F is *  (  ).Definition 7. The sequence of fuzzy-number-function { F } is   *  on  ⊂ [, ] if  is the sequence of subsets   such that { F } is   *  for each , independent of .Definition 8. Let { F } be a sequence of fuzzy-numberfunction defined on [, ], and let  ⊂ [, ] be measurable.(i) The sequence of fuzzy-number-function { F } is P-Cauchy on   if { F } converges pointwise on  and if for each  > 0 there exist () > 0 on  and a positive integer  such that (  (),   ()) <  for all ,  ≥  whenever  is -subordinate to ().(ii) The sequence of fuzzy-number-function { F } is generalized P-Cauchy on  if  can be written as a countable union of measurable sets on each of which { F } is P-Cauchy.Theorem 9. Let the following conditions be satisfied: (i) f, () → f a.e. on [, ] as  → ∞ where each f, is strong Fuzzy Henstock integrable on [, ]; (ii) the primitives F, of f, are   *  with closed set  in [, ].
b) A sequence { F } of fuzzy-number-valued function is uniformly  ∇ on [, ] if [, ] = ∪    ,where   are measurable sets and { F} is uniformly  ∇ on each   .