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.
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).
Equation (2.1) can be written in the form
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 ,
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.
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.
Suppose (2.2) and (2.3) hold. Let be a positive, monotone decreasing solution of (2.1). Then there exists such that
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.
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,
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 .
where is the radius of convergence of the above power series given by
Since is monotone decreasing, we have
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
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 ,
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
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
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