Positive Decreasing Solutions of Higher-Order Nonlinear Difference Equations
© B. Krasznai et al. 2010
Received: 23 December 2009
Accepted: 13 May 2010
Published: 13 June 2010
It is shown that the decay rates of the positive, monotone decreasing solutions approaching the zero equilibrium of higher-order nonlinear difference equations are related to the positive characteristic values of the corresponding linearized equation. If the nonlinearity is sufficiently smooth, this result yields an asymptotic formula for the positive, monotone decreasing solutions.
1. Introduction and the Main Results
where , so that and is a Lipschitz continuous function such that in and is identically in . As noted in , (1.1) is related to the discretization of traveling wave solutions of the Fisher-Kolmogorov partial differential equation. The main result of  is the following theorem about the existence of positive, decreasing solutions of (1.1) (see [1, Theorem ] and its proof).
Note that in , solutions of (1.1) satisfying conditions (1.4) and (1.5) of Theorem 1.1 are called fast solutions.
We will show that under appropriate assumptions the decay rates of these solutions are equal to the characteristic values of the corresponding linearized equation belonging to the interval . Our result, combined with asymptotic theorems from  or , yields asymptotic formulas for the positive, monotone decreasing solutions of (1.6).
where is a polynomial of degree less than , the multiplicity of as a root of . Such solutions are called characteristic solutions corresponding to . If is a nonempty set of characteristic values, then by a characteristic solution corresponding to the set , we mean a finite sum of characteristic solutions corresponding to values .
Now we can formulate our main results. The first theorem applies to the positive, monotone decreasing solutions of (1.6) provided that zero is a hyperbolic equilibrium of (1.6). Recall that the zero equilibrium of (1.6) is hyperbolic if the linearized equation (1.9) has no characteristic values on the unit circle .
In contrast to Theorem 1.2, the next result applies also in some cases when the zero equilibrium of (1.6) is not hyperbolic.
Note that conclusion (1.18) of Theorem 1.3 is stronger than (1.14).
The proofs of the above theorems are given in Section 3.
2. Preliminary Results
In this section, we establish some preliminary results on linear difference equations with asymptotically constant coefficients which will be useful in the proof of our main theorems.
Theorem 1.2 will be deduced from the following proposition.
The boundedness of implies that . By the application of a Perron-type theorem (see [4, Theorem ] or [5, Theorem ]), we conclude that for some characteristic value of (2.4). By virtue of (2.3), the characteristic values of (2.4) are nonzero. Therefore, .
Since the coefficients of the -transform are positive, according to Prinsheim's theorem (see [6, Theorem ] or [7, Theorem , page 262]) has a singularity at . Since and hence is holomorphic in the region , this implies that . Otherwise, (2.16) would imply that can be extended as a holomorphic function to a neighborhood of by . Thus, is a characteristic value of (2.4).
This proves the existence of the limit (2.6) with . Finally, conclusion (2.7) is an immediate consequence of [2, Theorem and Remark ].
Theorem 1.3 can be regarded as a corollary of the following result.
Before we give a proof of Proposition 2.2, we establish two lemmas.
The following lemma will play a key role in the proof of Proposition 2.2.
Relation (2.34), combined with Pringsheim's theorem [6, Theorem ], implies that . (Otherwise, can be extended as a holomorphic function to a neighborhood of by .) Since and the only root of in is , we have that .
Now we are in a position to give a proof of Proposition 2.2.
Proof of Proposition 2.2.
3. Proofs of the Main Theorems
Proof of Theorem 1.2.
From this, (1.10), (3.2) and the Lipschitz continuity of the partial derivatives , , it is easily shown that hypothesis (2.5) of Proposition 2.1 also holds. The conclusions of Theorem 1.2 follow from Proposition 2.1.
Proof of Theorem 1.3.
As noted in the proof of Theorem 1.2, is a solution of (2.1) with the asymptotically constant coefficients given by (3.2) and the limiting equation of (2.1) is the linearized equation (1.9). Further, by virtue of (1.7), condition (2.3) holds. Thus, all hypotheses of Proposition 2.2 are satisfied and the result follows from Proposition 2.2.
Proof of Theorem 1.4.
This shows that the convergence of the coefficients of (3.6) to their limits is exponentially fast. Thus, Proposition 2.1 applies and the limit relation (1.21) follows from the asymptotic formula (2.7).
This research was supported in part by the Hungarian National Foundation for Scientific Research (OTKA) Grant no. K 732724.
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