Definite Solution Problems Of Fuzzy Differential Equations And A Fuzzy Optimization Problem

Fuzzy numbers, calculus theory of fuzzy number valued functions, fuzzy differential equations and fuzzy optimization problems are important parts of fuzzy mathematics. We study the following problems in this dissertation: two kinds of special fuzzy numbers, i.e., the continuous fuzzy numbers u∈F bst with the unique center and the continuous fuzzy numbers u∈Ec; the differential and integral of fuzzy number valued functions that map to F bst or Ec ; the initial value problems of fuzzy differential systems and n ? order fuzzy differential equations; the two-point boundary value problems of fuzzy differential equations under H-differential; the two-point boundary value problems of fuzzy differential equations under differential inclusion, that is, the two-point boundary value problems of differential inclusion and so called fuzzy transportation problems. The main work of this dissertation can be summarized as follows:1. We introduce a simple and direct representation for the continuous fuzzy numbers u∈F bst with the unique center and the continuous fuzzy number u∈Ec, i.e., the parametric representation of the fuzzy numbers, then the fuzzy number u can be considered as a continuous curve in 2 , or a point in Banach space C [0,1]×C[0,1]. Based on these, we establish the calculus theory of fuzzy number valued functions from to F bst or Ec . Especially we introduce the concept of relative derivative, and obtain the similar methods of differentiation and integration by parts. Then we can study fuzzy differential equations (systems) under this new frame.2. We first prove the existence and uniqueness of solutions to the initial value problems of fuzzy differential equations. Then, we introduce the definition of maximal extended solutions and their esxistence and uniqueness. Under this new frame, we obtain the structure of domains on which maximal extended solutions can be considered as fuzzy number valued functions with several variables, continuity on domains and continuous dependence on the initial values for maximal extended solutions.3. V. Lakshmikantham and D. O'Regan et al proved that the two point boundary value problem of fuzzy differential equations is equivalent to some fuzzy integral equation under H-differential. But B. Bede used a counterexample to point out this result is wrong, and gave the method to rectify it in 2006, that is, under what conditions, the two-point problems of fuzzy differential equations have solutions under H-differential. Under the new frame, this problem can be solved completely by using relative derivative and the differentiation and integration by parts which is similar to usual, moreover we point out that the main results can not be improved any more essentially.4. In order to overcome the limitations in studying the fuzzy differential equations by H-differential, we use the methods that deal the initial value problems of fuzzy differential equations by differential inclusions under the new frame established. Using relative derivative, differetiation and integration by parts methods, we discuss the two-point boundary value problems of fuzzy differential equations under differential inclusions, that is, the two-point boundary value problems of fuzzy differential inclusions, and prove that there are unique large solutions under some conditions, especially periodic solutions. We obtain some important results about the relationship between large solutions and small solutions to the two-point boundary value problems for indefinite dynamic systems independent of velocity, and point out that large solutions fully descript the range of solution orbits of the two-point boundary volue problems for indefinite dynamic systems.5. We study the so called fuzzy transportation problems, and establish the mathematical model. Moreover, we give the method to find the optimal solutions. Our fuzzy transportation pattern and solving method extend results of other people. At last, we give a dynamical optimal model for the fuzzy transportation problems of expense concerning equipment maintenance. Fuzzy transportation problems has important practical value, because it comes from real world, and it is a subproblem of a project funded by Japan Culture Department.