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