scholarly journals GLOBAL ASYMPTOTIC STABILITY OF A HIGHER ORDER DIFFERENCE EQUATION

2007 ◽  
Vol 44 (3) ◽  
pp. 439-445 ◽  
Author(s):  
Alaa E. Hamza ◽  
R. Khalaf-Allah
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Higher Order ◽  
Order Difference ◽  
Order Difference Equation

scholarly journals Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-4 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Nonnegative Integer ◽  
Equilibrium Solution ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Order Difference ◽  
Order Difference Equation

Let k be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c),  n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.

scholarly journals The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Stevo Stević ◽  
Kenneth S. Berenhaut
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Positive Solutions ◽  
Global Asymptotic Stability ◽  
Second Order ◽  
Order Difference ◽  
All Solutions ◽  
Second Order Difference Equation ◽  
Order Difference Equation

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equationxn=f(xn−2)/g(xn−1),n∈ℕ0, wheref,g∈C[(0,∞),(0,∞)]. It is shown that iffandgare nondecreasing, then for every solution of the equation the subsequences{x2n}and{x2n−1}are eventually monotone. For the case whenf(x)=α+βxandgsatisfies the conditionsg(0)=1,gis nondecreasing, andx/g(x)is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, thenf(x)=c1/xandg(x)=c2x, for some positivec1andc2.

scholarly journals Multiplicity of Periodic Solutions for a Higher Order Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Ronghui Hu
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Higher Order ◽  
Critical Groups ◽  
Order Difference ◽  
Reduction Methods ◽  
Order Difference Equation

We study a higher order difference equation. By Lyapunov-Schmidt reduction methods and computations of critical groups, we prove that the equation has fourM-periodic solutions.

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