| Differential difference equation is a kind of powerful means to investigate the rule of natural phenomena, whose qualitative analysis has always been the hot spot subject to be studied in the recent years. In this dissertation we mainly consider some existing problems in the hot spot subject. Roughly speaking, we have dealt with work as follows:(i) Try to use a kind of new method to investigate the global asymptotic stability of difference equation;(ii) Solve several " Open Problems and Conjectures " in an international journal;(iii) Point out and correct some wrong work existing in two famous international journals; (iv) Improve many known results.Our work mainly concentrates on two aspects: the one is the stability theory of difference equation and the other is the oscillation of advanced differential equation. The whole doctoral dissertation is divided into five chapters.Some known work, including some known results and methods, are summed up in Section 1 of Chapter 1 whereas our main work are summed up in Section 2. For the sake of precise statements in the sequel, some basic theory for differential difference equations is stated in Chapter 2.The stability for difference equation is maily considered in Chapter 3, where, in Section 1, a new method, called " Semicycle Analysis Method ", is introduced to investigate global asymptotic stability of rational difference equations, and which is different from the existing ways, originates from the authors' trying to solve a conjecture presented in "Journal of Difference Equations and Applications" with respect to the global asymptotic stability of a detailed rational difference equation, and which can be used to solve the global asymptotic stability of a kind of rational difference equations which are verydifficult to be solved in the methods in the kno ,n literature. The global asymptotic stability and boundedness and persistence for the other t , o difference equations are investigated in Section 2, and the derived results solve an open problem and put the investigation for ,ard for a conjecture. The global attract!vity of a more general difference equation is considered in Section 3, ,here a ne, result is obtaied for an open problem. In Section 4, ,e point out and correct some existing errors for the asymptotic behavior of nonlinear second order difference equation in " Applied Mathematics Letters", and make some ne , conclusions. By citing a series of counterexamples, ,e demonstrate in Section 5 that famous Indian scholar E. Thandapani's and American scholar K. Ravy's classification methods are essentially ,rong for second order quasi-linear difference equation , ith damped term in "Computers and Mathematics ,ith Applications ", state ne , results and solve completely the problem for the classification of this kind of equation.The oscillatory theory for difference equation is mainly considered in Chapter 4. We investigate in Section 1 the oscillation and nonoscillation of nonlinear neutral difference equation ,ith continuous arguments, and get a sharp condition of the oscillation. Some comparison theorems are studied for the oscillation and nonoscillation of linear neutral difference equation ,ith continuous arguments in Section 2. In Section 3 ,e investigate the existence and asymptotic behavior of nonoscillatory solutions of second order neutral delay difference equation. Some open problems for Bob , hite Quail population model presented by G. Ladas in the first international conference for difference equation are still interesting to us. We obtain ne , results in Section 4 for the oscillation and boundedness and persistence of this model, including and improving many kno ,n ones. In section 5, ,e are in a position to consider some properties, mainly oscillation, cycle length and periodicity, etc, for a rational difference equation involving maximum function, partly solving an open problem.Several natures for Lyness equation are studied in Section 6. Firstly.for Lyness equation with constant coefficients,... |
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