Research On Several Classes Of Nonlinear Fractional Systems

Fixed point theory and contraction mapping principle are the main tools to research the existence and uniqueness of solutions for fractional systems.In the first two parts of this doctoral dissertation,we mainly study the existence,uniqueness and stability of solutions for fractional delay difference equations,impulsive fractional difference equations,variable fractional difference equations,fractional delay differential equations,fractional delay differential equations with non-instantaneous impulses and fractional differential inclusions with non-instantaneous impulses by using the addressed two methods.In the last part,we use the theory of correlation stochastic differential equations to obtain the averaging principle results of delay fractional stochastic system and impulsive fractional stochastic system.This dissertation consists of the following four chapters and the arrangement is as follows:In Chapter one,an overview of the historical background,significance,research status for the discussed problems,the main works of this paper,as well as preliminary facts on fractional calculus used in this dissertation are provided.In Chapter two,according to the proof by contradiction and Gronwall inequality,we first research the uniqueness and finite-time stability of solution for the fractional delay difference equation with disturbance.Then we prove the existence and uniqueness of solutions for impulsive fractional delay difference equation by contraction mapping principle and obtain the finite-time stability.Finally,we discuss the existence of solutions by using the Krasnoselskii’s fixed point theorem,and get the Ulam-Hyers stability results of the variable-order fractional discrete equation.In Chapter three,by using the contraction mapping principle and proof by contradiction,we discus the existence and uniqueness,boundness,and Ulam-Hyers for a kind of Ψ-Hilfer fractional differential equation with time-varying delays in the first section.In the second section,with the help of the Krasnoselskii’s fixed point theorem,we discuss the existence of solutions for fractional delay differential equations with impulses,and obtain the Ulam-Hyers stability.In the third section,by virtue of the Schauder’s fixed point theorem and contraction mapping principle,we investigate the existence,uniqueness and finite-time stability of solutions for theΨ-Hilfer fractional delay differential equations with non-instantaneous impulses.In the next,with the help of the Lyapunov functions along with the generalized Gronwall inequality,we study the existence and finite-time stability of solutions for the fractional delay differential equations with uncertain term.Finally,we consider the initial value problem for a class of impulsive Ψ-Hilfer fractional differential inclusions and get the existence and stability results of solutions.In Chapter four,with the aid of the Burkholder-Davis-Gundy inequality,Cauchy-Schwarz inequality,Jensen’s inequality and some new conditions,we first study the averaging principle of solutions for the fractional delay stochastic system.Subsequently,in order to explore the effects of impulses on the averaging principle of solution for stochastic system,we continue with the above methods and some nonlinear analysis techniques,and then obtain the results of the averaging principle problem of the solutions for impulsive fractional stochastic systems.