Oscillation Of Solutions For Delay Differential And Difference Equations

This dissertation, consisting of five chapters, intends to describe the oscillatcry behavior of several classes of delay differential equations.Chapter 1 is a brief introduction to the historical background and the significance of this study and outlines the recent development for the oscillation for delay differential and difference equations.By introducing some new methods and techniques, in Chapter 2 I get some new oscillation criteria formulated directively in terms of the delay and coefficients for the first order delay difference equation with several delays, which intends to improve the existing results of the previous studies.In Chapter 3 I try to establish some oscillation criteria for the odd order delay difference equation with several delays, which extend the corresponding oscillatory results for the first order delay difference equation with several delays. For the even order delay difference equation with unstable type, I prove that the equation always has an unbounded and eventual solution by constructing a solution of the inequality corresponding the equation and by using Banach contraction principle. Some bounded oscillation criteria which guarantee all bounded solutions of the equation to oscillate are established. The obtained results are also the best for the second order delay difference equation with unstable type.Chapter 4 is a study on the even order superlinear delay differential equation with unstable type. By constructing a solution of the inequality corresponding the equation and by using Banach contraction principle, I prove that the equation always has an unbounded and eventually solution. Then I establish an almost sharp bounded oscillation and nonoscillation criterion.In the final Chapter I first obtain an almost "sharp" oscillation and nonoscillation criterion for the odd order superlinear delay differential equation with several delays. These extend the corresponding oscillatory results for the first order superlinear delay differential equation with several delays. Then by applying apply the obtained results to the odd order nonlinear delay differential equation and the the odd order superlinear neutral delay differential equation, I establish some oscillation criteria which guarantee all solutions of these equations to oscillate.