The Research On The Existence Of Positive Solutions For Two Point Boundary Value Problems Of Impulsive Differential Equations

With the development of the differential equations,the impulsive differential equations gradually evolves into an important branch of differential equations.It not only reflects the pulse phenomenon of the development,but also fully reflects the influence of the phenomenon on the entire process state.In addition,it can be applied in physics,population dynamics,chemical science,biological science,economics and so on.This thesis studies the existence of positive solutions for two point boundary value problems of impulsive differential equations,which is divided into three chapters.In the first chapter,firstly,we introduce the background and existed results.Secondly,it gives some definitions and basic lemmas.Lastly,lower and upper solution is established for the second order impulsive differential equations,which plays an important role as a key tool in the whole thesis.The second chapter focuses on the nonlinear second order impulsive differential equations and sufficient conditions for the existence of positive solutions to this nonlinear problem.In the third chapter,we consider two types of nonlinear impulsive differential problem where △u|t=1/2=u(1/2+)-u(1/2),△u’|t=1/2=u’(1/2+)-u’(1/2-),0<β2<β1,f:[0,1]×R→R is a continuous function.On the one hand,we study the boundary value problem of a class of nonsingular impulsive differential equations.Based on the lemma 1.2.1 in the first chapter,we obtain the existence of the positive solutions by constructing the upper and lower solution of the nonsingular impulse equations.This paper discusses the existence of positive solutions of f(u)for the following three equations:On the other hand,we discuss the boundary value problem of a class of singular impulsive differential equations.We study the existence of positive solutions for f(u)to satisfy the following three equations:(1)f(u)=u-p,where 0<p<+∞；(2)f(u)=uβ+u-α,where α>0,0<β<1 is a constant;(3)f(u)=λuβ-u-α,where 0<α<1,0<β<1,λ>0 is a constant.First,we obtain the existence of the positive solutions by constructing the upper and lower solution of the nonsingular impulse equations.Secondly,when,we obtain the existence of positive solutions for boundary value problems with singular impulsive differential equations by using the consistent bounded equality theorem.