Research On Problems Concerning Some Kind Of Uncertain Dynamical Systems

It is well known that fuzzy differential equations are mainly studied in thefollowing three ways: the first one is based on the H-derivatives and theirpromotion-Bede generalized derivatives, the second one is based on Zadehexpansion principle, the last one is based on the theory of differential inclusions.Fuzzy differential equations based on the three thoeries are different from eachother significantly.In2001, world-famous mathematician Lakshmikantham et al proved aequivalence theorem between a two-point boundary value problem of fuzzydifferential equation (i.e. a fuzzy two-point boundary value problem) based on H-derivative and a certain fuzzy integral equation, and here after all the researchpapers on fuzzy two-point boundary value problems in the sense of H-derivativesare based on this equivalence theorem. But in2006, Bede gave a counterexample toprove that this equivalence theorem is wrong and raised some open questions, i. e.“How does a fuzzy two-point boundary value problem based on the H-derivativeadmit solutions”, and “What about the two-point boundary value problems of fuzzydifferential questions in other senses?” etc. In2008, Chen Minghao et al establisheda new parameterized method of fuzzy numbers and a new framework of calculus forfuzzy valued functions. By using the relative derivative concept and the methodsimilar to the usual differential method and integration by parts, they gave acomplete answer to the first open problem above and pointed out that two-pointboundary value problems of fuzzy differential equations based on H-derivativesseldom have solutions while in many cases there is no solutions. On the other hand,any non-real (i.e., take at least one pure fuzzy value) fuzzy differential equationbased on H-derivatives has no periodic solutions. So fuzzy differential equationsbased on H-derivatives cannot reflect the rich variety nature of the usual ordinarydifferential equations, such as the periodicity, absorption and stability of solutions,and in many cases it cannot objectively reflect physical background and practicalproblems. This is the significant deficiency of fuzzy differential equations based onthe H-derivatives.In order to overcome the defects above, we learn from ideas of Hüllermeier etal using the theory of differential inclusions to deal with initial problems of fuzzydifferential equations. Consider some fuzzy differential equations as fuzzydifferential equations, i.e. uncertain dynamical systems, based on the theory ofdifferential inclusions. By using the theory of differential inclusion, functional analysis, set-valued analysis and the Sobolev space, we study the two-pointboundary value problems of uncertain dynamical systems, periodic problems ofsemi-linear uncertain dynamical systems, periodic problems of first order uncertaindynamical systems, etc. In this paper, the main work is as follows:1. Given the definition of solutions and big solutions to two-point boundaryvalue problems of second-order undamped fuzzy differential equations, i.e. twopoint boundary value problems of second-order undamped uncertain dynamicalsystems, based on the theory of differential inclusions. By means of the newparameterized method of fuzzy numbers and the new framework of calculus forfuzzy valued functions, using the theory of differential inclusions, functionalanalysis, set-valued analysis etc., completely solve the second problem mentionedabove raised by Bede in2006, which means that we prove an existence anduniqueness theorem of solutions for two-point boundary value problems of second-order undamped uncertain dynamical systems, and give the relationship of inclusionbetween solutions and big solutions, and point out that big solutions fully describedthe scope of trajectories of solutions for two-point boundary value problems ofsecond-order undamped uncertain dynamical systems; and prove an existence anduniqueness theorem of solutions for general two-point boundary value problems ofsecond-order uncertain dynamical systems.2. Given the definition of the solutions based on the theory of fuzzy differentialinclusion for periodic problems of quasi-linear first-order fuzzy differentialequations, i.e. periodic problems of semi-linear first-order uncertain dynamicsystems. By means of the new parameterized method of fuzzy numbers and the newframework of calculus for fuzzy valued functions, using the theory of differentialinclusion, functional analysis, set-valued analysis and Kakutani fixed point theorem,we prove an existence theorem of periodic solutions for semi-linear first-orderuncertain dynamic systems.3. Given the definition of solutions for periodic problems of first-ordernonlinear fuzzy differential equations, i.e. periodic problems of first-order nonlinearuncertain dynamic systems, based on the theory of differential inclusions. By meansof the new parameterized method of fuzzy numbers and the new framework ofcalculus for fuzzy valued functions, integratedly using the theory of differentialinclusions and functional analysis, set-valued analysis and Dugundji-Granas fixedpoint theorem, we prove an existence theorem of periodic solutions for first-ordernonlinear uncertain dynamic systems.It is necessary to point out that because of the absence of appropriate Green’sfunction, many domestic and foreign mathematicians think that the periodic problems of first-order uncertain dynamical systems are more difficult than thecorresponding problems of second-order uncertain dynamical systems.