Solutions Of Several Classes Of Nonlinear Differential Equations And Applications

In this paper, by using the cone theory, the monotone iterative method, the fixed point index theory and the critical point theory, we study the existence and multiplicity of solutions for nonlinear differential equations. Some new conclusions are obtained. This paper includes four chapters.In the first chapter, we introduce the background of the discussed problems of this thesis briefly, and we state the main results of this thesis concretely.In the second chapter, by transforming the initial value problem for nonlinear fourth-order impulsive integro-differential equations into the initial value problem for second-order impulsive differential equations and using the cone theory, the monotone iterative method, the new comparison result and the principle of Banach contraction mapping, we investigate the existence of maximal and minimal solutions of the initial value problem for nonlinear fourth-order impulsive integro-differential equations in Banach space: Two examples are presented to illustrate the application of the obtained results.In the third chapter, by using fixed point index theory of strictly set contraction mapping, we study the existence of solutions for two-point boundary value problems of sixth-order ordinary differential equations in Banach space:The sufficient condition for the boundary value problems to have at least one solution is obtained. In order to illustrate the application of the obtained results, we give an example.In the fourth chapter, by using the critical point theory, especially by using the Morse theory, we study the existence of multiple solutions of two-point boundary value problems for a class of nonlinear 2mth-order ordinary differential equations:The sufficient conditions for the boundary value problems to have infinitely many solutions are obtained. Two examples are given to illustrate the application of the obtained results.