The Convergence Of The Solutions Of Two Classes Of Difference Equations

Difference equations (or recurrence sequence) appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in economics, biology, computing science and engineering. We generally investigate its final characters , including periodicity,convergence, boundedncss, and oscillatory properties, etc. This article investigate the convergence of the solutions of two classes of nonlinear difference equations.In Chapter One, we introduce briefly the historic background and the current situation of nonlinear difference equation and some known results about our theorems.In Chapter Two, we consider the nonlinear difference equationx_n+1 = f(x₍n-ls+1,x_n-2ks+1, n = 0, 1, … ,under appropriate assumptions, where s,k,l ∈ {1,2, …} , gcd(2k,l) = 1 and the initial values x_-α,x_α+1, … , x₀∈ (0, +∞) with α = max{ls — 1,2ks — 1}. We give sufficient conditions under which every positive solution of this equation converges to a ( not necessarily prime ) 2s-periodic solution.In Chapter Three, we consider the following nonlinear difference equationx_n+1 = f(P_n, X₍n-m,x_n-t(k+1)+1, n = 0, 1, 2, … ,where m ∈ {0,1,2, …} and k, t ∈ {1,2, …} with 0 ≤ m < t(k + 1) - 1, the initial values x_-t(k+1)+1,x_-t(k+1)+2, …,x₀∈(0, +∞) and {p_n}_n=0^∞ is a positive sequence of period k + 1. We give sufficient conditions under which every positive solution of this equation tends to the period k + 1 solution.