Dynamics for Nonlinear Difference Equation
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Research Article Dynamics for Nonlinear Difference Equation p xn1 αxn−k /β γxn−l Dongmei Chen,1 Xianyi Li,1 and Yanqin Wang2 1
College of Mathematics and Computational Science, Shenzhen University, Shenzhen, Guangdong 518060, China 2 School of Physics & Mathematics, Jiangsu Polytechnic University, Changzhou, 213164 Jiangsu, China Correspondence should be addressed to Xianyi Li, [email protected] Received 19 April 2009; Revised 19 August 2009; Accepted 9 October 2009 Recommended by Mariella Cecchi We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones. Copyright q 2009 Dongmei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Consider the following higher order difference equation: xn1
αxn−k
p
β γxn−l
,
n 0, 1, . . . ,
1.1
where k, l, ∈ {0, 1, 2, . . .}, the parameters α, β, γ and p, are nonnegative real numbers and the initial conditions x− max{k, l} , . . . , x−1 and x0 are nonnegative real numbers such that p
β γxn−l > 0,
∀n ≥ 0.
1.2
It is easy to see that if one of the parameters α, γ, p is zero, then the equation is linear. If β 0, then 1.1 can be reduced to a linear one by the change of variables xn eyn . So in the sequel we always assume that the parameters α, β, γ, and p are positive real numbers.
2
Advances in Difference Equations The change of variables xn β/γ1/p yn reduces 1.1 into the following equation: yn1
ryn−k p
1 yn−l
,
n 0, 1, . . . ,
1.3
where r α/β > 0. Note that y1 0 is always an equilibrium point of 1.3. When r > 1, 1.3 also possesses the unique positive equilibrium y 2 r − 11/p . The linearized equation of 1.3 about the equilibrium point y 1 0 is zn1 rzn−k ,
n 0, 1, . . . ,
1.4
so, the characteristic equation of 1.3 about the equilibrium point y1 0 is either, for k ≥ l, λk1 − r 0,
1.5
λl−k λk1 − r 0.
1.6
or, for k < l,
The linearized equation of 1.3 about the positive equilibrium point y 2 r − 11/p has the form zn1 −
pr − 1 zn−l zn−k , r
n 0, 1, . . . ,
1.7
with the characteristic equation either, for k ≥ l, λk1
pr − 1 k−l λ −10 r
1.8
pr − 1 0. r
1.9
or, for k < l, λl1 − λl−k
When p 1, k, l ∈ {0, 1}, 1.1 has been investigated in 1–4. When k 1, l 2, 1.1 reduces to the following form: xn1
αxn−1 p
β γxn−2
,
n 0, 1, . . . .
1.10
El-Owaidy et al. 3 investigated the global asymptotical stability of zero equilibrium, the periodic character and the existence of unbounded solutions of 1.10.
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