The Eventual Periodicity of { x-I I I FORF AND GIN x } x, , ,-,l-x,-- x 1-x x x-

The central objective of this thesis is to analyze the eventual periodic properties of solutions of the second order difference equation Xn+l = max{F{xn), G{xn_1 )}, n = 0,1,2, ... , (1.2) where F and G are any two, not necessarily distinct, functions in the system S = Ifx, x -1 ,_1__ ,!,1- X,} and max is the maximum function, max {x,y} = the 1 x I-x x x-I larger of x and y. Chapter 2 highlights some preliminary concepts and the proof that the system itself is closed under function composition. Chapter 3 introduces the Analysis of Interval Sequences, and its application to proving that every solution of Equation (1.2) is eventually periodic. The last part of the thesis follows with summary and recommendations. It is hoped that this paper will serve as an important contribution to promoting further research and analysis of nonlinear difference equations