Oscillation And The Distribution Of Zeros Of Solutions Of Functional Differential Equations

This PH.D.Thesis is mainly concerned with the oscillatory and asymptotic behavior for second-order nonlinear ordinary differential equations,higher order nonlinear functional differential equations and dynamic equations on time scales,and the distribution of zeros of solutions of first-order functional differential equation and the neutral functional differential equation.The results of this report improve,extend,unify and complement a number of existing results,and they also handle a number of cases not covered by known criteria.Some interesting examples are included to show the versatility of our results.This PH.D.Thesis is composed of five chapters.In Chapter I,we introduce the historical background and the recent development of ordinary differential equations,functional differential equations,and dynamic equations on time scales.The main results of this thesis are also briefly introduced.In Chapterâ…¡,by using the generalized Riccati technique,the averaging technique and differential inequalities,new oscillation criteria for second-order nonlinear differential equations are established and asymptotic behavior for forced equations is also discussed.Then by using the generalized Gronwall inequalities,the oscillation for second-order self-adjoint differential equations is discussed and new oscillation criteria are obtained.In Chapterâ…¢,by considering functions H(t,s)k(s)that may not have a nonpositive partial derivative on s and employing integral averaging technique and Hardy inequality,the oscillation for a class of high order nonlinear differential equations and high order nonlinear damped differential equations is discussed,new oscillation criteria are established. In Chapterâ…£,the distribution of zeros of solutions of the first-order functional differential equation with variable delays is studied.The estimate fbr the distance between adjacent zeros of oscillatory solutions of the above equation is improved.The results are generalized to first-order neutral functional differential equation.Chapterâ…¤is concerned with oscillation properties of second order nonlinear dynamic delay equations and neutral dynamic delay equations on time scales.The results extend some known results for difference equations and differential equations.