Research On Existence Of Solutions For Several Classes To Non-instantaneous Impulsive Fractional Differential Equations

In this paper,we first consider the existence and uniqueness of solutions of integral-boundary value problems for non-instantaneous impulse fractional differential equations.Second,the existence and Hyers-Ulam stability of solutions of stochastic fractional differential equations with time-varying delay are studied.Thirdly,we consider the existence and Hyers-Ulam stability of solutions of stochastic fractional differential equations with time-varying delays for non-instant aneous impulses.Finally,the approximate properties of solutions of fractional stochastic differential equations for Poisson processes with timedelay are studied.This paper is divided into four chapters,and the specific arrangement is as follows:In the first chapter,we mainly introduce the background,current situation,main work and preliminary knowledge of the problem studied in this paper.In the second chapter,we study the integral boundary value problem of fractional differential equations with non-instantaneous impulses.By using some fixed point theorems,we obtain sufficient conditions for existence of a unique solution and at least one solution respectively.The new conclusion is a generalization of the results of the relevant literature.In the third chapter,we firstly consider the existence and Hyers-Ulam stability of solutions to time-varying delays stochastic fractional differential equations with non-Lipschitz coefficients.By using fractional calculus and stochastic analysis,we can obtain the existence and Hyers-Ulam stability result of solutions for stochastic fractional differential equations.Secondly,the existence and Hyers-Ulam stability of solutions for a class of fractional order stochastic differential equations involving time-varying delays and non-instantaneous impulses is studied.By using fractional calculus,stochastic analysis and Krasnoselskii's fixed point theorem,we establish the existence of solutions for the stochastic differential equation.Further,we derive the Hyers-Ulam stability of solutions for the proposed stochastic differential equations.The results are entirely new.In the fourth chapter,we consider approximation properties for solutions to time-delays stochastic differential equations with non-Lipschitz coefficients driven by fractional Brownian motion and Poisson point processes.By using fractional calculus and stochastic analysis,we can obtain the averaging principle and existence result of solutions for time-delayed stochastic differential equations.The new conclusion is to generalize the research results of relevant literature,and the improved theorem provides a new idea for the research of related fields.