In this section, we will prove that the uniform asymptotic stability of the zero solution of positive Volterra difference equations (1.1) and (1.2) is, in fact, characterized by the invertibility of the operator
for
. To this end we need some observations on the spectral radius of the
-transform of the convolution kernel
.
First of all, we show the relation between the positivity of Volterra difference equations and that of the sequence of bounded linear operators
.
Proposition 3.1.
Equation (1.1) is positive if and only if all 
are positive.
Proof.
Suppose that all
are positive. Then from (2.11) each element
(
of the fundamental solution is also positive; so that by virtue of (2.12),
for
, where
and
are arbitrary. Hence, (1.1) is positive. Conversely, suppose (1.1) to be positive. Then we have in particular
for
, which implies that
. Let
and for
,
be any function such that
except
. Then it follows from (2.12) that
which implies that
for
. Thus,
for
since
is arbitrary.
By using (2.13) one can verify the following proposition quite similarly.
Proposition 3.2.
Equation (1.2) is positive if and only if all 
are positive.
In what follows, we assume that
and each
is compact. For any closed operator
on
we denote by
,
, and
the spectrum, the point spectrum, and the resolvent set of
, respectively. Also denote by
the interior of the unit disk
of the complex plane. Then for the uniform asymptotic stability of the zero solution of (1.1) we have the following criterion.
Theorem 3.3.
Suppose that (1.1) is positive. If
, the zero solution of (1.1) is uniformly asymptotically stable.
Proof.
In view of Theorem 2.4 it is sufficient to show that
is invertible for
with
. Suppose by contradiction that
is not invertible for some
with
. Then
and hence
since
and
is compact. Let
,
be an eigenvector of the operator
for the eigenvalue
. Then,
. We generally get
for any
. Notice that the spectral radius of
is less than
by the assumption. Therefore, it follows from the well known Gelfand's formula (see, e.g., [13, Theorem 10.13]) for the spectral radius of bounded linear operators that
, which implies
. On the other hand, since
,
are positive by Proposition 3.1, we get
by (2.7); and generally,
for any
. Therefore,
and it follows from the lattice norm property that
This is a contradiction, because we must get
by (3.5).
The converse of Theorem 3.3 also holds. To see this we need another proposition. Let
be the spectral radius of
for
.
Proposition 3.4.
Suppose that 
are all positive. Then,
is nonincreasing and continuous as a function on 
Proof.
Let
. Then,
Observe that the resolvent
of
(
is given by
whenever
. Then, we deduce from (3.6) that
Note also that under our assumption
is a positive operator for
since
is closed in
. In particular,
is also positive and hence for
,
by (2.7). Therefore
so that
Now, let us assume that
holds for some
and
with
. Since
is positive, it follows from [5, Chapter 5, Proposition 4.1] that
. Observe that if
and
, then
is invertible in
with
. Since
, the above observation leads to the fact that
as
; consequently, we get
as
. On the other hand it follows from
for
that
and the function
is continuous on
. Hence, we get
, which is a contradiction. Consequently,
for
.
We next show the left continuity of
on
. If
is not left continuous at some
, we have
Since
, we have
. Notice that
is an open set. Hence, it follows that if
is small enough,
that is,
for such an
. On the other hand, by virtue of [5, Chapter 5, Proposition 4.1] again, the positivity of
yields
; this is a contradiction.
is right continuous as well in
. Indeed, if it is not so, there exists a
such that
In view of the positivity of
we have
. Also, since
is compact,
is an isolated point of its spectrum
; in particular, there exists an
such that
. By the continuity of
there corresponds an
such that
is invertible in
for
. Moreover, by the fact that
for
, one can see
from which
readily follows because of the positivity of
. Therefore, passing to the limit
, we deduce that
. Let
be any number such that
. By the same reasoning, we see that
. Since
we get
Then, for
, it follows from (2.7) that
hence
for any
with
. This is a contradiction, because
as
. The proof is now completed.
Theorem 3.5.
Suppose that (1.1) is positive. If the zero solution of (1.1) is uniformly asymptotically stable, then
.
Proof.
Set
. To prove the theorem it is sufficient to show
. Assume that
. Since by Proposition 3.4
is continuous, there exists
such that
, that is,
. It follows from the positivity of
, together with its compactness, that
belongs to
; hence, there exists an
such that
, or equivalently
Setting
, we have
so that
is a solution of (1.2). By virtue of Theorem 2.4 and our assumption, (1.2) is uniformly asymptotically stable and therefore
as
, which is impossible because
for all
. Thus we must have
.
Combining the results above with Theorem 2.4, we have, for positive Volterra difference equations, the equivalence among the uniform asymptotic stability of the zero solution of (1.1) and (1.2), the summability of the fundamental solution and the invertibility of the operator
outside the unit disk.
Theorem 3.6.
Let the assumptions in Theorem 2.4 hold. If, in addition,
ac.
-
(i)
The zero solution of (1.1) is uniformly asymptotically stable.
-
(ii)
The zero solution of (1.2) is uniformly asymptotically stable.
-
(iii)
-
(iv)
The operator
is invertible in
for
.
Before concluding this section, we will give an example to which our Theorem 3.6 is applicable. In [6, 7], following the idea in [14], we have shown that Volterra difference equations on a Banach space
are naturally derived from abstract differential equations on
with piecewise continuous delays of type
where
denotes the Gaussian symbol and
is the inifinitesimal generator of a strongly continuous semigroup
, of bounded linear operators on
, and
,
are bounded linear operators on
such that
Recall that a function
with
is called a (mild) solution of (3.19) on
, if
is continuous on
, and satisfies the relation
In case of
for some
, the relation above yields that
Letting
, we get the Volterra difference equation
where
are bounded linear operators on
defined by
for
Conversely, if
satisfies (3.23) with
, then the function
extended to nonintegers
by the relation
is a (mild) solution of (3.19). Thus, abstract differential equations of type (3.19) lead to Volterra difference equations on
.
Now suppose that the semigroup
is compact. Then,
is continuous in
with respect to the operator norm ([15]) and also
, defined by the relation (3.24) are compact operators on
(see [6, Proposition 1]). Moreover, it follows from (3.20) that
. Moreover in the restricted case where
, are given by
for some
with
, we know by [7, Proposition 1] that the spectrum of the characteristic operator
of (3.23) is given by the formula:
Hence, in the restricted case, Theorem 2.4 implies that the zero solution of (3.23) is uniformly asymptotically stable if and only if
for all
and
.
We further assume that
is a complex Banach lattice, the compact semigroup
on
is positive, and that
are all nonnegative. Then the sequence
defined by (3.24) meets the assumptions in Theorem 3.6. Noticing that
, we know by Theorem 3.6 in the further restricted case that the zero solution of (3.23) is uniformly asymptotically stable if and only if
for all
.