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.  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 ).
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 ).
For any with , there exists a unique such that ( and ).
Conversely, for any with and , there exists a unique such that .
Note (see ).
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 ).
The complete metric on is defined by
for any , which satisfies .
Let , then
Definition 2.5 (see ).
The derivative of a fuzzy process is defined by
provided that the equation defines a fuzzy .
Definition 2.6 (see ).
The fuzzy integral , is defined by
provided that the Lebesgue integrals on the right-hand side exist.