Existence And Uniqueness Of Solutions Of Hilfer Fractional Equations With Integral Boundary Value Conditions And Stability Of Solutions For Linear Systems

Fractional differential equations are extension and extension of integer order.Fractional differential equations with boundary value conditions are widely used in engineering and science.The existence of solutions to fractional differential equations have attracted a high degree of attention at home and abroad.The use of equation boundary value problems to describe mathematical and physical models has been proposed many times in practice.Therefore,the research of this type of problem has great value in promoting the solution of real problems.This paper mainly discussed the unique sufficient conditions for the existence of solutions to boundary value problems of a class of Hilfer fractional differential equations and the stability of solutions for a class of Hilfer fractional differential autonomous systems.There are four chapters in the thesis,and the main contents are arranged as follows:In the first chapter,the research context was introduced,as well as growth status of Hilfer-type differential equations.The second chapter was the preliminaries,which mainly introduced some definitions of fractional calculus and the necessary basic lemmas.Chapter 3 discussed the existence and uniqueness of solution of a class of Hilfer fractional differential problems with integral boundary value conditions:(?) Where d≥0 represents a parameter,D0+α,β stands for the Hilfer fractional derivative,1<α<2,0≤β≤1.And D0+γ-1 is the γ-1 order Riemann-Liouville fractional derivative with zero as the lower limit,f(t,y):[0,1]×R+→R+ is a continuous function.This chapter discussed the properties of Green function in Banach space,and proved the existence and uniqueness of the solution to this problem by using the single-upper solution or the single-lower solution and the monotonic iterative sequence method.In chapter 4,we studied a class of n-dimensional fractional differential systems with Hilfer type and a class of fractional differential systems with integral boundary value problems:(?) Where D0+α,β notes as the Hilfer fractional derivatives of the order of α with type β,0<α<1,0<β≤1,I0+1-γ represents the Riemann-Liouville fractional integrals the order of 1-γ with the lower limit of zero,u=(u1,u2,…un)T∈Rn,A=(aij)n×n is the non-degenerate matric.u0=(u10,u20,…,un0)T,uio≥0,1≤i≤n.The stability and asymptotic stability of the solution in different situations were analyzed by the established stability concept,Laplace transform and the Mittag-Leffler function,and the sufficient conditions for judging the stability of the system were given at last.