Multiple Solutions For Nonlinear Operators And Boundary Value Problems Of Differential Equations

Nonlinear functional analysis has been one of the most important branches of learning in modern mathematics at present. It provides an effect theoretical tool for studying many nonlinear problems. It plays an important role in dealing with nonlinear equations and differential equations arising in applied mathematics.The study of boundary value problems for nonlinear ordinary differential equations has interested people for a long time and the research in this field is still very active. Recently, the existence of solutions of boundary value problems for nonlinear ordinary differential equations attracts close attention, in which the nonlinear function is endowed with the conditions of some different kinds in much literature. The eigenvalue and the spectral radius of linear completely continuous operators are very important and significant indexes. In the first two Chapter of this thesis, we devote ourselves to give the eigenvalue criteria for the existence of (positive) solutions to several different kinds of boundary value problems of nonlinear ordinary differential equations (in abstract spaces), that is, the nonlinear function is given the conditions concerning the eigenvalues corresponding to the relevant linear operators. The results extend and improve the conclusions in many recent work. Chapter 3 deals with Sturm-Liouville problems, we give some existence of multiple solution for Sturm-Liouville problems by use of the bifurcation theorem of Rabinowitz. In the final Chapter, we introduce some new concepts so that we get some new multiple solutions theorems. The tools used in this thesis are global structure, the cone theory and the fixed point index.