Positive Decreasing Solutions of Higher-Order Nonlinear Difference Equations

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

(1.1)

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

(1.2)

(1.3)

Then (1.1) has a strictly decreasing solution such that

(1.4)

(1.5)

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

(1.6)

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

(1.7)

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,

(1.8)

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

(1.9)

with coefficients

(1.10)

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

(1.11)

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

(1.12)

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

(1.13)

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

(1.14)

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

(1.15)

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

(1.16)

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

(1.17)

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

(1.18)

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

(1.19)

where is the unique root of the equation

(1.20)

in the interval .

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

(1.21)

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

(2.1)

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

(2.2)

exist. We will assume that

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

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

(2.8)

where is the characteristic polynomial corresponding to (2.4).

Proof.

Equation (2.1) can be written in the form

(2.9)

where

(2.10)

Let be the -transform of defined by

(2.11)

where is the radius of convergence given by

(2.12)

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

(2.13)

satisfies the inequality

(2.14)

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

(2.15)

it follows by easy calculations that

(2.16)

where is the polynomial given by (1.11) and

(2.17)

with given by

(2.18)

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

(2.19)

Since is monotone decreasing, it follows for ,

(2.20)

and hence

(2.21)

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

(2.22)

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

(2.23)

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

(2.24)

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

(2.25)

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

(2.26)

From (2.1), we obtain for ,

(2.27)

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

(2.28)

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

(2.29)

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,

(2.30)

Then

(2.31)

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

(2.32)

of is given by

(2.33)

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

(2.34)

where is given by (1.11) and

(2.35)

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

(2.36)

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

(2.37)

Since is monotone decreasing, we have

(2.38)

Therefore,

(2.39)

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

(2.40)

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

(2.41)

Summation from to infinity and the definition of yield

(2.42)

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

(2.43)

satisfies

(2.44)

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

(2.45)

Moreover, since , the function defined by

(2.46)

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 ,

(2.47)

Hence

(2.48)

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

(2.49)

for some , , . Since , we have

(2.50)

Therefore,

(2.51)

Hence

(2.52)

where . This, together with (2.43), implies

(2.53)

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

(2.54)

From (2.43) and (2.52), we obtain

(2.55)

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

(2.56)

This, together with (2.54), yields

(2.57)

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 ,

(2.58)

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

(2.59)

we obtain

(2.60)

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

(2.61)

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

(2.62)

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

(2.63)

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

(2.64)

3. Proofs of the Main Theorems

Proof of Theorem 1.2.

Since , from (1.6), we find for ,

(3.1)

Thus, is a solution of (2.1) with coefficients

(3.2)

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

(3.3)

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),

(3.4)

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

(3.5)

Consequently, is a solution of the equation

(3.6)

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

(3.7)

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

(3.8)

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

(3.9)

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

(3.10)

Hence

(3.11)

The characteristic polynomial corresponding to (3.8) is

(3.12)

We have

(3.13)

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

(3.14)

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

(3.15)

exists and or . Condition (1.5) implies that

(3.16)

Therefore,

(3.17)

From this and (1.4), we find that

(3.18)

Hence

(3.19)

Since , we have that and (1.19) holds.

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

(3.20)

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

(3.21)

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).