A fuzzy set of is a function . For each fuzzy set , we denote by for any , its -level set.

Let be fuzzy sets of . It is well known that for each implies .

Let denote the collection of all fuzzy sets of that satisfies the following conditions:

(1) is normal, that is, there exists an such that ;

(2) is fuzzy convex, that is, for any , ;

(3) is upper semicontinuous, that is, for any , ;

(4) is compact.

We call an -dimension fuzzy number.

Wang et al. [9] defined -dimensional fuzzy vector space and investigated its properties.

For any , , we call the ordered one-dimension fuzzy number class (i.e., the Cartesian product of one-dimension fuzzy number ) an -dimension fuzzy vector, denote it as , and call the collection of all -dimension fuzzy vectors (i.e., the Cartesian product ) -dimensional fuzzy vector space, and denote it as .

Definition 2.1 (see [9]).

If , and is a hyperrectangle, that is, can be represented by , that is, for every , where with when , , then we call a fuzzy -cell number. We denote the collection of all fuzzy -cell numbers by .

Theorem 2.2 (see [9]).

For any with , there exists a unique such that ( and ).

Conversely, for any with and , there exists a unique such that .

Note (see [9]).

Theorem 2.2 indicates that fuzzy -cell numbers and -dimension fuzzy vectors can represent each other, so and may be regarded as identity. If is the unique -dimension fuzzy vector determined by , then we denote .

Let , be fuzzy subset of . Then .

Definition 2.3 (see [9]).

The complete metric on is defined by

for any , which satisfies .

Definition 2.4.

Let , then

Definition 2.5 (see [9]).

The derivative of a fuzzy process is defined by

provided that the equation defines a fuzzy .

Definition 2.6 (see [9]).

The fuzzy integral , is defined by

provided that the Lebesgue integrals on the right-hand side exist.