On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

一类高阶有理差分方程解的收敛性(英文)

本文考查下列高阶有理差分方程xn+1=(α+B1xn-1+B3xn-3+…+B2k+1xn-2k-1)/(A+B0xn+B2xn-2+…+B2kxn-2k,),n=0,1,…,其中k是非负整数,参数α,A,Bi,i=0,1,2,…,2k+1和初始值x-2k-1,x-2k,x-2k+1,…,x0是非负实数.给出充分条件,在此条件下当且仅当∑k+1i=1B2i-1=A时,方程的每个解收敛于方程的一个二周期解.     

On the period five trichotomy of all positive solutions of xn+1=(p+xn−2)/xn

We study the behavior of all positive solutions of the difference equation in the title, where p is a positive real parameter and the initial conditions x₋₂,x₋₁,x₀ are positive real numbers. For all the values of the positive parameter p there exists a unique positive equilibrium x? which satisfies the equation

     

On the dynamics of a higher order rational difference equations

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation

$y_{n+1}=a{y}_n+b{y}_{n?t}+c{y}_{n?l}+\frac{d{y}_{n?k}+e{y}_{n?s}}{\alpha y_{n?k}+\beta y_{n?s}},n=0,1,\dots ,$

with positive parameters and non-negative initial conditions. Numerical examples to the difference equation are given to explain our results.     

On the Global Character of the Difference Equation X n + 1 =

We investigate the global stability, the periodic character, and the boundedness nature of solutions of the difference equation x n +1 = f + n x n m (2 k +1) + i x n m 2 l A + x n m 2 l , n =0,1,… where k and l are non-negative integers, the parameters f , n , i , A are non-negative real numbers with f + n + i >0, and the initial conditions are non-negative real numbers. We show that the solutions exhibit a trichotomy character depending upon the parameters n , i and A .     

On the Dynamics of

We investigate the existence of unbounded solutions and the period-two convergence of solutions of the equation in the title with the parameter γ positive, the remaining parameters nonnegative, and with nonnegative initial conditions.     

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation

     

The boundedness character of two Stevi?-type fourth-order difference equations

The boundedness character of positive solutions of two nonlinear fourth-order difference equations, which are particular cases of two large classes of difference equations by Stevi?, are studied in this paper.     

Asymptotic Behavior of Solutions to the Difference Equation Using the Second Differences of a Liapunov Function

We introduce the concept of the second Liapunov differences in difference equations and show that, in the autonomous case, the second variations can guarantee the existence of solutions tending to zero while others starting arbitrarily near 0 go to ∞. These are based on Yorke's results for ordinary differential equations.     

On a class of third-order nonlinear difference equations

This paper studies the boundedness character of the positive solutions of the difference equation