Research On The Properties Of Solutions For Impulsive Dynamic Equations And Fractional Differential Equations

A time scale is a nonempty closed subset of a set of real numbers.The theory of dynamic equations on time scales unifies the theory of differential equations and the theory of difference equations,which provides a new tool for people to explore the relationship between the continuous domain and the discrete domain.Boundary value problems of differential equations are closely related to various fields.The study of boundary value problems can solve the problems of fluid mechanics and nonlinear optics,which has practical significance.Compared with integer-order calculus,fractional calculus can describe the memory and hereditary properties of various materials and processes,and provide a more accurate system model.In this paper,the oscillation of solutions of impulsive dynamic equations on time scales,the existence of solutions for Sturm-Liouville boundary value problems with impulses and the existence results for fractional impulsive functional differential equations with infinite delay are studied.The main contents are as follows:Firstly,we discuss the second order neutral impulsive dynamic equations with positive and negative coefficients on time scales.Under different pulse conditions,we use variable substitution and Riccati transformation.Through variable substitution,the neutral impulsive dynamic equations with positive and negative coefficients are transformed into the neutral impulsive dynamic equations with only positive coefficients,and the sufficient conditions for the oscillation of the equations are obtained.Secondly,we consider the existence of solutions for the Sturm-Liouville boundary value problems of second order impulsive differential equations.Through Green's function,we obtain the equivalent integral equation,then by the related properties of Green's function and the nonlinear alternative of Leray-Schauder type,Banach fixed-point theorem,Krasnoselskii's fixed-point theorem and Schaefer's fixed-point theorem,we obtain the existence and uniqueness of the solutions.Finally,we study the existence results for fractional impulsive neutral functional differential equations with infinite delay.According to the definitions of fractional integral and fractional derivative,we get the equivalent integral equation.By constructing the operator,the existence of the solutions of the equations is transformed into the existence of the fixed point problem of the operator,and then the relevant conclusions are obtained by using M¨nch fixed-point theorem,Sadovskii's fixed-point theorem and Banach fixedpoint theorem.