The Existence Of Solutions For Several Classes Of Ordinary Differential Equations And Integral Equations

Boundary value problems of ordinary differential equations are one of major component parts of the ordinary differential equation discipline,which have extremely important applications in classical mechanics,engineering and finance and other fields.Meanwhile,many practical problems can be transformed to boundary value problems of differential equations or integral equations to study.This thesis researches the existence of positive solutions of the boundary value problems of several classes of nonlinear ordinary differential equations and numerical solutions of integral equations.The main works of this dissertation are arranged as follows:The first chapter briefly introduces the development history and research status of the boundary value problems of nonlinear ordinary differential equations and integral equations,and gives common concepts and theorem used in this thesis.The second chapter mainly studies the existence of positive solutions of nonlinear Dirichlet type three point boundary value problem.By using the fixed point theorem of cone expansion-compression type in nonlinear functional analysis,some more specific existence results for positive solutions of nonlinear Dirichlet three-point boundary value problem are obtained.Then,an concrete example is given as an application.The third chapter mainly studies the existence of positive solutions of the generalized n-point boundary value problems.By the cone expansion and compression fixed point theorem,some more specific existence conditions for positive solutions of the generalized n-point boundary value problems are got.Finally,the application example is given.The fourth chapter mainly studies the existence of two positive solutions of the semi-positive value problem of impulsive differential equation boundary.By using basic properties of the fixed point index,the conditions for the existence of two positive solutions of the boundary value problem of more general the semi-positive impulsive singular differential equation are given.The fifth chapter mainly studies the numerical solution of nonlinear Volterra integral equation.It introduces two methods for solving nonlinear second kind Volterra integral equations,i.e.successive approximation method and the NewtonKantorovich method.Finally,two application examples are given respectively,solved by using MATLAB software,and their convergence rates are compared.In the sixth chapter,the summary and prospects of this thesis are given.