Global asymptotic stability for a higher order nonlinear rational difference equations

Yan, Xing-Xue; Li, Wan–Tong; Zhu, Zhaopeng

doi:10.1016/j.amc.2006.06.019

Cited by 10 publications

(5 citation statements)

References 15 publications

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“…Other related results can be found in [10][11][12][13][14][15][16][17][18][19] . For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel.…”

Section: Introductionmentioning

confidence: 60%

Global stability of a rational difference equation

2007

Applied Mathematics and Computation

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We consider the higher-order nonlinear difference equation x n 1 p qx n−k / 1 x n rx n−k , n 0, 1, . . . with the parameters, and the initial conditions x −k , . . . , x 0 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 .

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“…Other related results can be found in [10][11][12][13][14][15][16][17][18][19] . For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel.…”

Section: Introductionmentioning

confidence: 60%

Global stability of a rational difference equation

2007

Applied Mathematics and Computation

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“…Even after a sustained effort by many researchers such as [24][25][26][27][28], there were some cases of the 49 different cases that have not been investigated till 2007. Amleh et al in [29,30] give an up-to-date account on recent developments related to (2) up to 2007.…”

Section: Introductionmentioning

confidence: 99%

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

Atawna

Abu-Saris

Ismail

et al. 2015

Journal of Difference Equations

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Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equationxn+1=α+βxn+γxn-1/A+Bxn+Cxn-1,n=0,1,2,…, where the parametersA≥0andB,C,α,β,γare positive real numbers and the initial conditionsx-1, x0are nonnegative real numbers such thatA+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).

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“…Even after a sustained effort by many researchers such as [5][6][7][8][9], there were some difference equations of form (1) that have not been investigated till 2007.…”

Section: Introductionmentioning

confidence: 99%

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

Atawna

Abu-Saris

Ismail

et al. 2015

Journal of Difference Equations

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This is a continuation part of our investigation in which the second order nonlinear rational difference equationxn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1),n=0,1,2,…, where the parametersA≥0andB,C,α,β,γare positive real numbers and the initial conditionsx-1,x0are nonnegative real numbers such thatA+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.

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Global asymptotic stability for a higher order nonlinear rational difference equations

Cited by 10 publications

References 15 publications

Global stability of a rational difference equation

Global stability of a rational difference equation

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

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

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