Qualitative Theoretical Research On Solutions Of Two Kinds Of Impulsive Differential Equations

Impulsive differential equations are a type of mathematical model that has a wide range of applications.In recent years,the theoretical research of impulsive differential equations has made great progress,especially the qualitative theoretical research of solutions(including periodic solutions,positive solutions,existence of multiple solutions,boundedness,etc.).However,some theoretical research is still in its infancy.This article mainly establishes two impulsive differential equation models.Inspired by the existing research results,we have obtained conclusions that are more practical to some extent.This paper focuses on the existence of multiple solutions of the fixed impulsive differential equations and stochastic impulsive differential equations.The corresponding solution frameworks are established for different types of impulsive differential equations,and then the semigroup theory,cone theory and topological degree theory are used to obtain the corresponding mild equations.Sufficient conditions for the existence of solutions and multiple solutions.The main content of this article includes the following four parts.Chapter 1 first introduces the development history of fixed impulsive and stochastic impulsive differential equations,the research progress of periodic solutions of impulsive differential equations and the theory of multiple solutions,then gives the main research content of this paper,and finally gives the preliminary knowledge of the paper.Chapter 2 studies the existence and uniqueness theory of asymptotically almost periodic mild solutions of a class of Riemannian fractional impulsive differential equations.First,use Riemannian fractional Laplace transformation to solve the general form of the differential equation solution,and define a new Then based on the semigroup theory,construct appropriate operators and use the Krasnoselskii fixed point theorem to prove the existence and uniqueness of the asymptotically almost periodic mild solutions of the equation.Finally,the correctness and validity of the conclusions obtained are verified by examples.Chapter 3 studies the existence theory of multiple solutions for a two-point boundary value problem of differential equations with random impulses.First,the system solutions of the equations are derived using stochastic analysis techniques and Green functions,and appropriate cones and totals are defined according to the characteristics of the solutions.Continuous operator.Then,under the p-norm,the fixed point index theorem and the Leggett-Williams theorem in the cone are used to give sufficient conditions for the existence of at least two and three solutions to the equation.Chapter 4 studies the existence theory of multiple solutions for a kind of stochastic impulsive differential equation boundary value problem with Nagumo's condition.First,the integral form of the equation solution is obtained with the aid of the Green function and the corresponding upper and lower solutions are defined.Then,when the nonlinear function satisfies Nagumo's Under the conditions,the Leray-Shauder degree theory combined with the upper and lower solution method is used in the defined space to obtain the existence results of the three solutions of the second-order stochastic impulsive differential equation.Finally,an example is used to prove the correctness and validity of the obtained results.