Existence And Multiplicity Of Solutions To Boundary Value Problems Of Some Third-order Differential Equations

The boundary value problem of nonlinear differential equation is one of the research branches in the field of differential equation.Many mathematical models in nature can be expressed by it,such as the deformation of beam,biological mathematical model,infectious disease system,economic growth model and so on.Difference equation is often regarded as the discrete form of differential equation,which has a strong practical application background.It is widely used in computer,information system,astronomy,physics and other fields,and has gradually become one of the hot issues.In this paper,we study the existence of solutions,the existence and multiplicity of positive solutions for some three-point boundary value problems of nonlinear differential equations,and the Hyers-Ulam stability of third-order linear nonhomogeneous h-difference equations.The main tools are upper and lower solution method,Schauder fixed point theorem,fixed point theorem and fixed point index theorem on cone,Z-transformation method,etc.The full paper is divided into fifth chapters.The first chapter introduces the background knowledge of nonlinear differential equations and difference equations,the current research status at home and abroad and the main content of this paper.The second chapter,we study the existence of solutions for two kinds of three-point boundary value problems of third-order nonlinear ordinary differential equations by using the method of upper and lower solutions.Different from the previous work,the boundary value condition is studied by adding parameters.By establishing a new comparison theorem and using the Schauder fixed point theorem,new results are obtained.By constructing the increasing operator,the multiplicity of solutions for a class of third-order differential equation three-point boundary value problems is obtained.The third chapter bases on the fixed point theorem and fixed point index theorem in cones,we study the existence and multiplicity of positive solutions for two kinds of nonlinear three-point boundary value problems of third-order differential equations.According to the characteristics of boundary conditions and the types of equations,by constructing appropriate cones in the C[0,1]andC~1[0,1]Banach space and combining the properties of Green functions and using the fixed point theorem of cones and fixed point index theorem,the existence and multiplicity of positive solutions for boundary value problems are obtained.The fourth chapter bases on the difference method and Z-transform method,the Hyers-Ulam stability of the third-order nonhomogeneous linear h-difference equation is studied.In the existing work,there are few papers on the Hyers-Ulam stability of the third-order h-difference equation.Under the given initial conditions,we obtain the results that the equation has the Hyers Ulam stability by Z-transformation,and give some relations of the solutions of the equation.The fifth chapter summarizes the main work of this paper and prospects the follow-up work.