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# Positive Decreasing Solutions of Higher-Order Nonlinear Difference Equations

*Advances in Difference Equations*
**volume 2010**, Article number: 973432 (2010)

## Abstract

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

Let , , and be the set of real and complex numbers and the set of integers, respectively. The symbol denotes the set of nonnegative integers.

Recently, Aprahamian et al. [1] have studied the second-order nonlinear difference equation

where , so that and is a Lipschitz continuous function such that in and is identically in . As noted in [1], (1.1) is related to the discretization of traveling wave solutions of the Fisher-Kolmogorov partial differential equation. The main result of [1] is the following theorem about the existence of positive, decreasing solutions of (1.1) (see [1, Theorem ] and its proof).

Theorem 1.1.

In addition to the above hypotheses on , suppose that there exist and such that

Then (1.1) has a strictly decreasing solution such that

Note that in [1], solutions of (1.1) satisfying conditions (1.4) and (1.5) of Theorem 1.1 are called *fast solutions*.

In this paper, we give an asymptotic description of the decreasing fast solutions of (1.1) described in Theorem 1.1. We will also consider a similar problem for the more general higher-order equation

where and , with being a convex open neighborhood of in . Throughout the paper, we will assume that the partial derivatives , , are continuous on and so that (1.6) has the zero equilibrium; moreover, the mild technical assumption

holds. We are interested in the asymptotic properties of those positive, monotone decreasing solutions of (1.6) which tend to the zero equilibrium, that is,

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 [2] or [3], yields asymptotic formulas for the positive, monotone decreasing solutions of (1.6).

Before we formulate our main theorems, we introduce some notations and definitions. Associated with (1.6) is the *linearization* about the zero equilibrium, namely, the linear homogeneous equation

with coefficients

By a *characteristic value* of (1.9), we mean a complex root of the *characteristic polynomial*

Thus, the set of all characteristic values of (1.9) is given by

To each , there corresponds solutions of (1.9) of the form

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 .

Theorem 1.2.

In addition to the above hypotheses on , suppose that the partial derivatives , , are Lipschitz continuous on compact subsets of . Assume also that zero is a hyperbolic equilibrium of (1.6). Let be a positive, monotone decreasing solution of (1.6) satisfying (1.8). Then the limit

exists and is a characteristic value of the linearized equation (1.9) belonging to the interval . Moreover, there exists such that the asymptotic representation

holds, where is a positive characteristic solution of the linearized equation (1.9) corresponding to the set

In contrast to Theorem 1.2, the next result applies also in some cases when the zero equilibrium of (1.6) is not hyperbolic.

Theorem 1.3.

In addition to the hypotheses on , suppose that the linearized equation (1.9) has exactly one characteristic value in the interval . Assume also that is the only characteristic value of (1.9) on the circle and

where is the multiplicity of as a root of the characteristic polynomial . Let be a positive, monotone decreasing solution of (1.6) satisfying (1.8). Then

Note that conclusion (1.18) of Theorem 1.3 is stronger than (1.14).

For the second-order equation (1.1), we have the following theorem which provides new information about the decreasing fast solutions obtained by Aprahamian et al. [1].

Theorem 1.4.

Adopt the hypotheses of Theorem 1.1. Let be a monotone decreasing solution of (1.1) satisfying conditions (1.4) and (1.5). If the (finite) right-hand derivative exists, then

where is the unique root of the equation

in the interval .

If is Lipschitz continuous on , then (1.19) can be replaced with the stronger conclusion

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.

Consider the linear homogeneous difference equation

where the coefficients , , are *asymptotically constant*, that is, the (finite) limits

exist. We will assume that

If we replace the coefficients in (2.1) with their limits, we obtain the *limiting equation*

Theorem 1.2 will be deduced from the following proposition.

Proposition 2.1.

Suppose (2.2) and (2.3) hold. Assume that the convergence in (2.2) is exponential, that is, there exists a constant such that

Let be a positive, monotone decreasing solution of (2.1). Then the limit

exists and is a characteristic value of the limiting equation (2.4) belonging to the interval . Moreover, there exists such that the asymptotic representation

holds, where is a positive characteristic solution of the limiting equation (2.4) corresponding to the set of characteristic values

where is the characteristic polynomial corresponding to (2.4).

Proof.

Equation (2.1) can be written in the form

where

Let be the -transform of defined by

where is the radius of convergence given by

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, .

From (2.5) and (2.12), we see that the radius of convergence of the -transform of given by

satisfies the inequality

Therefore, is holomorphic in the region . Taking the -transform of (2.9) and using the shifting properties

it follows by easy calculations that

where is the polynomial given by (1.11) and

with given by

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).

According to [4, Theorems and ], we have

Since is monotone decreasing, it follows for ,

and hence

Letting in the last inequality, and using (2.19), we find that

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.

Proposition 2.2.

Suppose (2.2) and (2.3) hold. Assume that the limiting equation (2.4) has exactly one characteristic value in the interval . Assume also that is the only characteristic value of (2.4) on the circle and

where is the multiplicity of as a root of the characteristic polynomial corresponding to (2.4). Let be a positive, monotone decreasing solution of (2.1). Then

Before we give a proof of Proposition 2.2, we establish two lemmas.

Lemma 2.3.

Suppose (2.2) and (2.3) hold. Let be a positive, monotone decreasing solution of (2.1). Then there exists such that

Proof.

The second inequality in (2.25) follows from the monotonicity of . In order to prove the first inequality, suppose by the way of contradiction that there exists a strictly increasing sequence in such that

From (2.1), we obtain for ,

From this and from the fact that is monotone decreasing, we find for ,

Writing in the last inequality, letting , and using (2.2), (2.3), and (2.26), we obtain

a contradiction. Thus, (2.25) holds for some .

The following lemma will play a key role in the proof of Proposition 2.2.

Lemma 2.4.

Suppose that (2.4) has exactly one characteristic value in the interval . Assume also that is the only characteristic value of (2.4) on the circle and (2.23) holds. Let be a positive, monotone decreasing biinfinite sequence satisfying (2.4) on , that is,

Then

Proof.

We will prove the lemma by using a similar method as in the proof of [8, Lemma ]. The radius of convergence of the -transform

of is given by

Since is bounded on , , and [4, Theorem ] or Lemma 2.3 implies that . Taking the -transform of (2.4), we obtain

where is given by (1.11) and

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 .

Define

where is the radius of convergence of the above power series given by

Since is monotone decreasing, we have

Therefore,

and hence . As a solution of a constant coefficient equation, is a sum of characteristic solutions. Therefore,

and hence . From (2.30), we find for and ,

Summation from to infinity and the definition of yield

with as in (2.35). This, combined with Prinsheim's theorem, implies that . Since and the only root of in is , we have that . Thus, . This, together with (2.34) and (2.42), implies that the holomorphic function defined on the open set by

satisfies

By hypotheses, is the only root of on the circle . Therefore, (2.44) implies that can be extended as a holomorphic function to by

Moreover, since , the function defined by

has a removable singularity at . Thus, it can be regarded as an entire function. Using (1.11), (2.35), and (2.44), we obtain for ,

Hence

By the application of the Extended Liouville Theorem [6, Theorem ], we conclude that is a polynomial of degree at most 2, that is,

for some , , . Since , we have

Therefore,

Hence

where . This, together with (2.43), implies

From this, in view of the uniqueness of the -transform, we obtain

From (2.43) and (2.52), we obtain

From this and (2.36), in view of the uniqueness of the coefficients of the power series, we obtain

This, together with (2.54), yields

From this, taking into account that for all , we see that . Therefore, (2.57) reduces to (2.31).

Now we are in a position to give a proof of Proposition 2.2.

Proof of Proposition 2.2.

Let be an arbitrary accumulation point of so that for some strictly increasing sequence in ,

In order to prove (2.24), it is enough to show that . From conclusion (2.25) of Lemma 2.3 and the relation

we obtain

From (2.60), it follows by the standard diagonal choice (see the proof of [8, Theorem ]) that there exists a subsequence of such that the limits

exist and are finite. Let be fixed. Writing in (2.1) and dividing the resulting equation by , we obtain

where is so large that for . From this, letting and using (2.2) and (2.61), we see that the biinfinite sequence satisfies (2.30). Further, it is easily seen that inherits the monotone decreasing property of . Finally, from estimates (2.60), we see that on . By the application of Lemma 2.4, we conclude that has the form (2.31). Since (see (2.61)), (2.31) yields

From this and (2.61), taking into account that is a subsequence of , we obtain

## 3. Proofs of the Main Theorems

Proof of Theorem 1.2.

Since , from (1.6), we find for ,

Thus, is a solution of (2.1) with coefficients

By virtue of (1.8), the limits in (2.2) exist and are given by (1.10), that is, the limiting equation (2.4) coincides with the linearized equation (1.9). By virtue of (1.7), hypothesis (2.3) of Proposition 2.1 is satisfied. Let

By virtue of the boundedness of , . As noted in the proof of Proposition 2.1, according to [4, Theorem ], for some . Since zero is a hyperbolic equilibrium of (1.6), for all . Therefore, . Choose . By virtue of (3.3),

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.

Equation (1.1) can be written in the form

Consequently, is a solution of the equation

Suppose that is finite. Condition (1.4), together with the positivity of and , implies

Therefore, the coefficients in (3.6) are asymptotically constant and the corresponding limiting equation is

Since and , we have . From (1.2), it follows

with the last but one equality being a consequence of L'Hospital's rule. This, together with (1.3), yields

Hence

The characteristic polynomial corresponding to (3.8) is

We have

Further, it is easily shown that (3.11) implies that

Therefore, has a root in and the second root of belongs to the interval . By the application of Poincaré's theorem (see [5, Section ] or [9, Section ]), we conclude that the limit

exists and or . Condition (1.5) implies that

Therefore,

From this and (1.4), we find that

Hence

Since , we have that and (1.19) holds.

Now suppose that is Lipschitz continuous on . For all large , we have

where is the Lipschitz constant of on . From this and (3.18), we see that

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).

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## Acknowledgment

This research was supported in part by the Hungarian National Foundation for Scientific Research (OTKA) Grant no. K 732724.

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Krasznai, B., Győri, I. & Pituk, M. Positive Decreasing Solutions of Higher-Order Nonlinear Difference Equations.
*Adv Differ Equ* **2010**, 973432 (2010). https://doi.org/10.1155/2010/973432

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DOI: https://doi.org/10.1155/2010/973432