The Problem Of Positive Solution Of Several Kinds Of Nonlinear Differential Equation

In the field of science and technology such as physics, biology, cybernation, electronics and so on, there are plenty of problems described with differential equations, so is sphere of sociology. Generally, according to practical problems, we can fix mathematical model, differential equations, which may accurately reflected reality, then with the definite solutions, we may obtain constructive ways to solve the problems.The positive periodic solution of functional differential equation with infinite delay is deeply concentrated these years, some scholars study this subject by means of Lyapunov theorem, Schauder fixed point theorem, and cone extending and compression theorem (from thesis [l]-[5]), and they obtain the existence theorem of positive periodic solution. It is very advantageous to study the BVP by utilizing the fixed point index theorem. In the first chapter (1.1.1)-(1.1.4) is studied by means of fixed point index theorem, not only the existence theorem is obtained, but the multiplicity is also obtained.The development process of population in a certain country or zone is a dynamics process, the population system is a dynamics system, this result has been confirmed widely as well as scientifically. There are two models, which can depress the development process of population in a certain country or zone, continuous model: a partial differential function with certain boundary conditions, discrete model: a difference function with two linear controlling conditions. The second chapter studies the continuous model (2. 1. 1), under two cases: fixed case and random case, it obtains the local existence theorems of solutions for nonlinear evolutional equations with random migration perturbation, by utilizing the theorems of m -accretive operator as well as the nonlinear semigroup theory.The mathematical model can forecast the distribution of a certain disease or epidemics around some countries and zone. Furthermore, it can depress the generality of distribution of epidemics. The government may make constructive ways to control the epidemics and provide decent appraisal. The third chapter focuses on some neutral integral equations arising in infectious disease. By utilizing the fixed point theorem this chapter obtains theorems of multiple positive solutions for this kind of integral equations (3.1.1). Furthermore, it improves previous work on this subject (from thesis [30]-[32]).