Oscillation And Related Problems Of The Second Order Linear Difference Equations (Systems)

This doctoral dissertation mainly deals with the oscillation and related problems of the second order linear difference equations (systems) mainly using Riccati transformations and inequality techniques. This thesis is made up by nine sections.Chapter 1 concentrates on the brief introduction of the historic background and significance for all the investigated problems. The qualitative theory of second order linear(semilinear) difference equations is an issue that has been paid close attention to at home and abroad academic circles all along, especially its oscillatory problem. In this section, the research status of second-order linear (semilinear) difference equations is reviewed and the classification and summary of several methods for studying the oscillation of difference equations are given. Finally, the main work in this dissertation are briefly introduced.In chapter 2, the reasch of the oscillation of the second order linear difference equation which does not satisfy some specific conditions is presented, and a Series of new results are obtained.In chapter 3, by using the fixed-point Schauder theorem, some necessary and sufficient conditions for existence of nonoscillatory solutions with some asymptotic behaviors for a class of secone order difference systems are derived.A class of secone order difference systems are considered in the fourth chapter, some necessary and sufficient conditions which have improved some known conclusions are obtained for the solutions of the systems to be oscillatory.The oscillation problem for the solutions of a class of secone order difference systems is discussed in the fifth chapter, and the conclusion is drawn from the results in the second chapter, which are de-veloped in this section.Several sufficient conditions for the oscillation of the solutions of a class of semilinear difference equations are established in the sixth chapter, which is a extension of the results of the second chapter on the other aspect.Some conjecture problems are resolved in the seventh and eighth chapter, which are the further reseach of the second chapter.Finally, in the ninth chapter, there are some examples to illustrate the application of the conclusions obtained in the oscillation of the first order difference equation. The results given in this doctoral dissertation play a certain function for the development and perfection of the qualitative theory of the difference equation.