Dynamical Behaviors Of Some Difference Equations

As a powerful tool to model various evolution systems from natural sciences and social sciences, difference equations have a broad spectrum of applications in biology, ecology, physiology, physics, engineering and economics. Besides, difference equations are indispensable in some other areas such as algorithm analysis and iterative solutions of (algebraic or differential) equations.This thesis focuses itself on the study of dynamical behaviors of three classes of difference equations. First of all, we briefly review the history and the state-of-the-art of difference equations. Second, the fundamental knowledge in concern with this thesis is introduced. On this basis, this thesis makes the following contributions.(1) A new part-metric-related inequality chain is established, which is then applied to successfully prove the global asymptotic stability of a generic difference equation.(2) A class of higher order difference equation with real powers is studied. Specifically, a sufficient condition for their global asymptotic stability is presented, and their oscillation character is investigated. Our results extend some known results.(3) The asymptotic behavior of positive solutions to a difference equation with maximum is examined.All of the above researches are accompanied with computer simulations. The experimental results not only illustrate the evolutionary tendency of the solutions, but also validate the presented results.