scholarly journals Global Asymptotic Stability of the Second-Order Nonlinear Difference Equation xn+1=a/xn2+1/xn-1

2017 ◽  
Vol 07 (01) ◽  
pp. 16-19
Author(s):  
旭光 秦
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Second Order ◽  
Nonlinear Difference Equation

scholarly journals Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

scholarly journals Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

scholarly journals Global Stability of a Rational Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Guo-Mei Tang ◽  
Lin-Xia Hu ◽  
Gang Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Open Problem ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Rational Difference Equation

We consider the higher-order nonlinear difference equation with the parameters, and the initial conditions are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).

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 Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 825
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Arbitrary Order ◽  
Nonnegative Function ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotic Behaviors

Let k , l be two integers with k ≥ 0 and l ≥ 2 , c a real number greater than or equal to 1, and f a multivariable function satisfying f ( w 1 , w 2 , w 3 , ⋯ , w l ) ≥ 0 when w 1 , w 2 ≥ 0 . We consider an arbitrary order nonlinear difference equation with the indicated function f: z n + 1 = c ( z n + z n − k ) + ( c − 1 ) z n z n − k + c f ( z n , z n − k , w 3 , ⋯ , w l ) z n z n − k + f ( z n , z n − k , w 3 , ⋯ , w l ) + c , n ≥ 0 , where initial values z − k , z − k + 1 , ⋯ , z 0 are positive and w i , i ≥ 3 , are arbitrary functions of z j , n − k ≤ j ≤ n . We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution z ¯ = c .

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