Existence Of Solutions For Impulsive Differential Equations And Permanence For Impulsive Model

This thesis is mainly concerned with the existence of solutions for impulsive differential equations and permanence for impulsive model. It consists of five chapters.As an introduction, in Chapter 1, the background and history of periodicity and boundary value problems for impulsive differential equations and mathemat-ical models with impulsive effect are briefly addressed, and the main results of this work are stated.In Chapter 2, we focus on multiplicity of boundary value problems for im-pulsive differential equations. We study the existence of solutions to the Duffing equation with impulses. Existence of infinitely many solutions is proved by means of the Poincarc-Birkhoff fixed point theorem under given conditions. We discuss the multiplicity of Dirichlct problem for second-order impulsive equations. The existence of solutions is established. The proof is based upon critical point theo-rems.Chapter 3 concerns multi-point boundary value problems for a second-order impulsive functional differential equation. We deal with the existence of extremal solutions of a multi-point boundary value problem for a class of second order impulsive functional differential equations. We introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extremal solutions.By using a fixed-point theorem due to Avery and Peterson, we consider a multi-point boundary value problem on the half-line with impulses. We obtain existence of at least three positive solutions.In Chapter 4, we consider periodic boundary value problems for a second-order impulsive differential equation. By using Schauder's fixed point theorem, we discuss an anti-periodic boundary value problem for second order impulsive differential equations. Some sufficient conditions for existence of solution are obtained. Three examples are presented to illustrate our main results. Using the method of upper and lower solutions, coupled with monotone iterative technique, we study solutions of nonlinear boundary value problems for a class of first order functional differential equations. Again we introduce a new concept of lower and upper solutions. In the view of the existence of upper and lower solutions, we obtain existence result about nonlinear boundary value problem. and extend previous results. We discuss existence of extremal solutions of boundary value problems for a class of first-order impulsive functional equations with nonlinear boundary conditions. In presence of a lower solution a and an upper solutionβwithβ≤α, we establish existence results of extremal solutions by using the method of upper and lower solutions and the monotone iterative technique.In Chapter 5, we consider the permanence for impulsive model. We base on the classical stage-structured model and Lotka-Volterra predator-prey model. We proposed and investigated an impulsive nonautonomous two dimensional delayed differential equations with HollingⅡtype functional response to model the process of periodically releasing natural enemies at fixed times for pest control. Conditions arc obtained for global attractivity of the "pest-cxtinction" periodic solution and permanence of the population of the model depending on time delay and impulses. We discuss a non-autonomous Logistic type impulsive equation with infinite delay. For the general non-autonomous case, we obtain several sufficient conditions which guarantee the permanence.