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