Dynamics Of Two Classes Of Max Difference Equations

In this thesis, the dynamics, i.e., the period character and the boundedness nature of the solutions of two classes of max difference equations, are studied, which generalize some known results. We organize the thesis as follows:In chapter one, we introduce briefly the historic background, the recent development tendence, the basic notions of the difference equations and the main results of this paper.In chapter two, we investigate the dynamics of the difference equationwhere {A_n}_n=0⁸ is a sequence of positive real numbers with 0≤inf A_n≤supA_n=a<1,the initial conditions x_-m,x_-m+1,…,x₀∈(0,∞).We show that every positive solution ofthis equation is bounded and obtain the following results:(1) When m =1,let {x_n}_n=-1^∞be a positive solution of this equation and S=lim sup_{n→∞}x_n.(i) If there exist infinitely many n such that x_n≥S and x_n+1≥S, then {x_n}_n=-1^∞iseventually equal to 1.(ii) If there exists N such that x_N+2kN+2k-1≥S for all k≥0 ,thenx_N+2k→1／S and x_N+2k-1→S.(iii) If {A_n}_n=0^∞be an eventually periodic sequence, then {x_n}_n=-1^∞is eventually periodicwith period 2.(2) When m = 2k, every positive solutions of this equation is eventually periodic with period 2.In chapter three, we investigate the dynamics of the difference equationwhere {c_n}_n=0^∞is a periodic sequence with period k and c_n∈(0,∞)for every n≥0,gcd(3, k) = 1, and the initial conditions x_-2,x_-1,x₀∈(0,∞).We show that every positive solution of this equation is bounded.