Research On The Existence Of Solutions Of Nonlinear Differential Equations

The present thesis consists of two parts.In the first part, we introduce the developing history of nonlinear functional analysis, the discussed problems of the present thesis and some important results.In the second part, we discuss the existence of solutions of differential equations, the uniqueness of solutions and multiple solutions.To begin with, we make use of the theory of fixed-point to discuss second order differential equations with Dirichlet boundary problems. We obtain the sufficient condition of the existence of two positive solutions. We take an example to explain these conditions feasible. Then we establish the new Green function and get the sufficient condition of the existence of multiple solutions of fourth order differential equation group.What is more, we discuss the existence of positive solutions of singular initial problem. We give the conditions of the existence of function integral. Without using Green function, the present thesis makes use of diagonal sequence and the principle of approximate solution to get the existence of positive solutions. For the results of the existence of solutions, we commonly make use of the theory of fixed-point. But the present thesis will use the index theory of fixed-point and obtain the sufficient conditions of the existence of positive solutions. Moreover we give examples.Finally we use the maximum theory and the fixed-point theory to obtain the sufficient condition of the existence and uniqueness solution of Dirichlet boundary problems.