Oscillation And Stability For Difference Equations

This dissertation is composed of four chapters and mainly deals with three problems about oscillation and stability of difference equations. The three problems are of the following:1. Oscillation criteria of second order nonlinear damped difference equations with continuous variable;2. The φ₀-boundedness and practical φ₀-stability of difference equations;3. Stability in terms of two measures for discrete hybrid systems.In chapter one, the methods of investigating about oscillation and stability of difference equations are introduced accompanied with some prints of present work as well as the main purpose of this paper.In chapter two, by using the iterated integral and generalized Riccati transformations, oscillatory criteria for second order nonlinear damped difference equations with continuous variable are discussed and an example was given to validate the results.In chapter three, the notions of φ₀-boundedness and practical φ₀-stability are extended to difference equations. Combining cone-valued Lyapunov functions method and perturbing Lyapunov functions method, various boundedness and stability results for a very general system of difference equations are obtained via the methods of perturbing cone-valued Lyapunov functions and comparison principle.In chapter four, variational comparison principle is obtained via variational Lyapunov method. By using this principle, stability in terms of two measures for discrete hybrid systems are given. These results also indicate that the solution of perturbed system has stronger properties under weaker conditions of the solution of unperturbed system.