| Compared with integer order calculus,fractional order calculus can describe better the materials and processes with heredity and memory.Impulsive differential equations can reflect actual changes more accurately than differential equations without impulses.Variational methods and critical point theory are important tools for studying fractional differential equations,but have seldom been used to investigate fractional differential equations with impulse effects,and it is worth further exploration.Therefore,the research on fractional Impulsive differential equations by using variational methods and critical point theory has important theoretical significance and application value.In this thesis,by using fractional calculus theory,impulsive differential equation theory,variational methods,critical point theory and some inequality techniques,we studied the existence and multiplicity of nontrivial weak solutions and classical solutions for two classes of fractional order impulsive systems.The details are as follows:(1)The Dirichlet boundary value problem for a class of fractional order impulsive systems is studied.Firstly,the variational framework is established and energy functional is defined.We transform the existence of the solution for the equation into the existence of functional critical point.Then,the Morse theory and local connection theorem are used to obtain the sufficient conditions for the existence of the classical solution for the equation;Secondly,by using Morse theory,Clark’s theorem,Brezis and Nirenberg’s connection theorem,the sufficient conditions for the existence of at least three classical solutions and the existence of at least k pairs classical solutions are obtained.Finally,an example is given as an application of the result.(2)The Dirichlet boundary value problem for a class of N-dimensional fractional order impulsive systems is studied.First,the definitions of N-dimensional functional and the corresponding norm are given.By the basic idea that dealing with N-dimensional problem,the functional space under N-dimensional is established,and the energy functional with the corresponding inner product are defined.Then,we transform the existence of the solution for the equation into the existence of the critical point of the functional.At last,the tools,such as Morse theory and local connection theorem are used to obtain the sufficient conditions for the existence of the solution of the system. |
