Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

and Applied Analysis 3 (1) for all h > 0 sufficiently small, there exists f(x 0 + h)− H f(x 0 ), f(x 0 )− H f(x 0 − h) and the limits (in the metricD)


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][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.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   () = (, ()), (0) =  0 , 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][12][13][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 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 integral and it is a powerful tool.It is well known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue, and 2 Abstract and Applied Analysis 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 fuzzynumber-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  ∈   = [0, ],  ∈  + , and , f, g :   →   are fuzzy-number-valued function and integrals which are taken in sense of strong fuzzy Henstock integration, and  1 ,  2 :   ×   →  + are measurable functions such that  1 (, ⋅),  2 (, ⋅) are continuous.
According to Zadeh's extension principle, we have addition and scalar multiplication in the fuzzy number space   as follows [10]: where , V ∈   and 0 ≤  ≤ 1.
The metric space (  , ) has a linear structure; it can be embedded isomorphically as a cone in a Banach space of function  * :  ×  −1 → , where  −1 is the unit sphere in   , with an embedded function  * = () defined by for all ⟨, ⟩ ∈  ×  −1 (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-numberfunctions 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 fuzzynumber-valued functions -differentiable is very restrictive.For example, the fuzzy-number-valued function 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 [8,9]).To avoid the above difficulty, in this paper we consider a more general definition of a derivative for fuzzynumber-valued functions enlarging the class of differentiable fuzzy-number-valued functions, which has been introduced in [8].

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 6 (see [20]).A fuzzy-number-valued function F defined on  ⊂ [, ] is said to be  * () if for every  > 0 there exists  > 0 such that for every finite sequence of nonoverlapping intervals {[  ,   ]}, satisfying ∑  =1 |  −   | <  where   ,   ∈  for all , we have where  denotes the oscillation of F over [  ,   ]; that is, Definition 7 (see [20]).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  * (  ).
For the strong fuzzy Henstock integrable we have the following theorems.(c) f is measurable.

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  > 0 such that there exists a finite covering of  by sets of diameter less than .For the properties of  we refer to [25], for example.
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
where conv f() is the convex hull of f(),  is an arbitrary subinterval of   , and || is the length of .
Proof.Because  ∘ f is abstract () integrable in a Banach Space, by using the mean valued theorem of () integrals, we have On the other hand, there exists Definition 14.A fuzzy valued function f :   ×   →   is a Caratheodory function if, for each  ∈   , the fuzzy valued function f(, ) is measurable in  ∈   , and for almost all  ∈   , the fuzzy valued function f(, ) 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  Observe that a fixed point of  is the solution of the problem (4).Now we prove that  has a fixed point using Theorem 12.