Global behaviors of some nonlinear difference equations and some systems of rational difference equations

In this thesis, we give a systematic study of the local and global asymptotic stability of equilibrium points, the periodic nature, the boundedness, the oscillation, the semi-cycle analysis of positive solutions of some specific rational non-linear difference equations and systems of difference equations. The organization of this thesis is as follows: Chapter 1 is a brief overview of what this thesis is about and also is an introduction of difference equation theory. Chapter 2 contains some basic definitions and results which are used throughout the thesis. In this regard, this is a self-contained monograph. In Chapter 3 some results will be given about the periodic character of positive solutions of the difference equation , for where is an odd integer, and the initial conditions are arbitrary positive numbers. In chapter 4 it will be studied the dynamical behaviour of positive solutions for a system of rational difference equations of the following form , , for where the parameters and the initial values are positive real numbers. In Chapter 5 it will be investigated the local asymptotic stability of equilibria, the periodic nature of solutions, the existence of unbounded solutions and the global behaviour of solutions of the rational difference equation , for where the parameters are non-negative numbers and the initial values are positive numbers.