Research On Several Kind Of Problems Of Differential Equation

This thesis investigates the existence of solutions for some class of differential equa-tions. The existence of harmonic and subharmonic solutions for two kinds of Duffing equations is obtained by using the Poincare-Birkhoff twist theorem. We investigate the existence of periodic solution for several class of high-order differential equations by ap-plying the coincidence degree theory and some new inequalities. We study the existence of positive solution for several classes of nonlinear differential equations by the fixed point theorems. The thesis consists of six chapters.Chapter 1 is preface, the historical background of the problems and the significance of this thesis are introduced. The recent developments and some knowledge for nonlinear differential equations in this thesis are given. And we introduce the Poincare-Birkhoff twist theorem, coincidence degree theory, fixed point theorems and some inequalities.Chapter 2 we obtain that the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity and quasilinear Duffing equation. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincare-Birkhoff twist.Chapter 3 we discuss the properties of the neutral operator (Ax)(t)=x(t) - cx(t-δ(t)) and by applying coincidence degree theory and fixed-point index theory, we obtain sufficient conditions for the existence, multiplicity and nonexistence of (positive) periodic solutions to two kinds of second-order differential equations with the prescribed neutral operator.Chapter 4 using Green's function for third-order differential equation and some fixed-point theorems, i.e., Leray-Schauder alternative principle and Schauder's fixed point theo-rem, we establish three new existence results of periodic solutions for nonlinear third-order singular differential equation.Chapter 5 by applying coincidence degree theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for a generalized high-order neutral differential equation in c=1 the critical case and c≠1 case, respectively.Chapter 6 is application of differential equation, in 6.1, an experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system x"+a(1+cos2t)x=1/x for 0< a< 1 is proved, using coincidence degree theory. In 6.2, based on the previously experimental results of the plastic dynamic analysis of metallic glasses upon compressively loading, a dynamical model is proposed. This model analysis quantitatively predicts that the loading rate can influence the transit of the plastic dynamics in metallic glasses from unstable (low loading rate range) to stable behavior (high loading rate range), which is consistent with the previous experimental results in the compression tests of a Cu50Zr45Ti5 metallic glass.