Advances in Difference Equations volume 2008, Article number: 598964 (2008)
For positive linear Volterra difference equations in Banach lattices, the uniform asymptotic stability of the zero solution is studied in connection with the summability of the fundamental solution and the invertibility of the characteristic operator associated with the equations. Moreover, the robust stability is discussed and some stability radii are given explicitly.
A dynamical system is called positive if any solution of the system starting from nonnegative states maintains nonnegative states forever. In many applications where variables represent nonnegative quantities we often encounter positive dynamical systems as mathematical models (see [1, 2]), and many researches for positive systems have been done actively; for recent developments see, for example, [3] and the references therein.
In this paper we treat the Volterra difference equations
together with
in a (complex) Banach lattice X, where 
is a sequence of compact linear operators on 
satisfying the summability condition 
, and we study stability properties of (1.1) and (1.2) under the restriction that the operators 
, are positive. In fact, the restriction on 
yields the positivity for the above equations (whose notion is introduced in Section 2). Also, without the restriction, in [4] the authors characterized the uniform asymptotic stability of the zero solution of (1.1), together with (1.2), in connection with the invertibility of the characteristic operator
of (1.1) for any complex numbers 
such that 
. In Section 3, we will prove that under the restriction that the operators 
, are positive, the invertibility of the characteristic operator reduces to that of the operator 
, and consequently the uniform asymptotic stability of the zero solution for positive equations is equivalent to the condition which is much easier than the one for the characteristic operator in checking (Theorem 3.6). Moreover, we will discuss in Section 4 the robust stability of (1.1) and give explicit formulae of some stability radii.
Let 
, 
, 
, 
, 
, 
and 
be the sets of natural numbers, nonnegative integers, nonpositive integers, integers, nonnegative real numbers, real numbers and complex numbers, respectively.
To make the presentation self-contained, we give some basic facts on Banach lattices which will be used in the sequel (see, e.g., [5]). Let 
be a real vector space endowed with an order relation 
Then 
is called an ordered vector space. Denote the positive elements of 
by 
. If furthermore the lattice property holds, that is, if 
for 
then 
is called a vector lattice. It is important to note that 
is generating, that is,
Then, the modulus of 
is defined by 
If 
is a norm on the vector lattice 
satisfying the lattice norm property, that is, if
then 
is called a normed vector lattice. If, in addition, 
is a Banach space, then 
is called a (real) Banach lattice.
We now extend the notion of Banach lattices to the complex case. For this extension all underlying vector lattices 
are assumed to be relatively uniformly complete, that is, if for every sequence 
in 
satisfying 
and for every 
and every sequence 
in 
, it holds that
Now, let 
be a relatively uniformly complete vector lattice. The complexification of 
is defined by 
The modulus of 
is defined by
A complex vector lattice is defined as the complexification of a relatively uniformly complete vector lattice equipped with the modulus (2.4). If is normed, then
defines a norm on 
satisfying the lattice norm property; in fact, the norm restricted to is equivalent to the original norm in , and we use the same symbol 
to denote the (new) norm. If is a Banach lattice, then 
equipped with the modulus (2.4) and the norm (2.5) is called a complex Banach lattice, and is called the real part of 
.
For Banach spaces 
and 
, we denote by 
the Banach space of all bounded linear operators from 
to 
equipped with the operator norm, and use the notation 
in place of 
. Let 
and 
be Banach lattices with real parts 
and 
, respectively. An operator 
is called real if 
. A linear operator 
from 
to 
is called positive, denoted by 
, if 
holds. Such an operator is necessarily bounded (see [5]) and hence real. Denote by 
and 
the sets of real operators and positive operators between 
and 
, respectively:
Then we observe that
Indeed, it is clear that the inequality holds true for any 
, the real part of 
. Let 
with 
and 
. Since 
by (2.4), we get
with 
and 
. Then, it follows from the modulus (2.4) that
as required. We also emphasize the simple fact that
(see, e.g., [5, page 230]). Moreover, by the symbol 
we mean 
for 
Throughout this paper, 
is assumed to be a complex Banach lattice with the real part 
and the positive convex cone 
.
For any interval 
we use the same notation 
meaning the discrete one 
, for example, 
for 
. Also, for an 
-valued function 
on a discrete interval 
, its norm is denoted by 
. Let 
and a function 
be given. We denote by 
the solution 
of (1.1) satisfying 
on 
. Similarly, for 
and a function 
, we denote by 
the solution 
of (1.2) satisfying 
on 
. We then recall the representation formulae of solutions for initial value problems of (1.1) and (1.2) (see [6–8]). Let 
be the fundamental solution of (1.1) (or (1.2)), that is, the sequence in 
satisfying
for 
Then, the solution 
is given by
for arbitrary initial function 
, and also for arbitrary initial function 
, the solution 
of (1.2) is given by
for 
, where we promise 
for 
.
Here, we give the definition of the positivity of Volterra difference equations.
Definition 2.1.
Equation (1.1) is said to be positive if for any 
and 
, the solution 
for 
. Similarly,(1.2) is said to be positive if for any 
and 
, the solution 
for 
.
Also, we follow the standard definitions for stabilities of the zero solution.
Definition 2.2.
The zero solution of (1.1) is said to be
uniformly stable if for any 
there exists a 
such that if 
and 
is an initial function on 
with 
then 
for all 
;
uniformly asymptotically stable if it is uniformly stable, and if there exists a 
such that, for any 
there exists an 
with the property that, if 
and 
is an initial function on 
with 
then 
for all 
.
Definition 2.3.
The zero solution of (1.2) is said to be
uniformly stable if for any 
there exists a 
such that if 
and 
is an initial function on 
with 
then 
for all 
;
uniformly asymptotically stable if it is uniformly stable, and if there exists a 
such that, for any 
there exists an 
with the property that, if 
and 
is an initial function on 
with 
then 
for all 
.
Here and subsequently, 
denotes the Z-transform of 
; that is, 
, which is defined for 
under our assumption 
. Then, 
is called the characteristic operator associated with (1.1) (or (1.2)). In [6, 7], under some restrictive conditions on 
, we discussed the uniform asymptotic stability of the zero solution of (1.2) in connection with the summability of the fundamental solution 
, as well as the invertibility of the characteristic operator 
; see also [9–12] for the case that 
is finite dimensional. Moreover, we have shown in [4] the equivalence among these three properties without such restrictive conditions; more precisely, we have established the following.
Theorem 2.4 (see [4, Theorem 1]).
Let 
, and assume that 
are all compact. Then the following statements are equivalent.
The zero solution of (1.1) is uniformly asymptotically stable.
The zero solution of (1.2) is uniformly asymptotically stable.
.
For any 
such that 
, the operator 
is invertible in 
.
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.
The zero solution of (1.1) is uniformly asymptotically stable.
The zero solution of (1.2) is uniformly asymptotically stable.
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 
.
Let (1.1) be uniformly asymptotically stable, that is, the zero solution of (1.1) is uniformly asymptotically stable, and consider a perturbed difference equation of the form
where 
are given operators corresponding to the structure of perturbations and 
is an unknown (disturbance) parameter. Here 
and 
are also assumed to be complex Banach lattices. Our objective in this section is to determine various stability radii of (1.1) provided that 
are all positive; for this topic in case that the space 
is finite dimensional, see, for example, [16] and the references therein. By the stability radius of (1.1) we mean the supremum of positive numbers 
such that the uniform asymptotic stability of the perturbed (4.1) persists whenever the size of the perturbation 
, measured by the 
-norm 
, is less than 
(for precise definitions see the paragraph preceding Theorem 4.3).
Here and hereafter we also assume that for any perturbation 
, 
, 
, are all compact, although this assumption is not necessary in the case that at least one of the operators 
and 
is compact. In what follows, we define 
by convention.
Theorem 4.1.
Let 
, 
, be positive. Suppose that (1.1) is uniformly asymptotically stable and 
, 
are both positive. Then the perturbed (4.1) is still uniformly asymptotically stable if
We need the following lemma to prove the theorem.
Lemma 4.2.
Under the same assumptions as in Theorem 4.1 we have
Proof.
Since (1.1) is uniformly asymptotically stable, 
, the spectral radius of 
, is less than 
by Theorem 3.5. In particular, it follows that 
is convergent and coincides with 
. Let 
, 
, be given. As in the proof of Theorem 3.3, one can see 
for 
and 
; hence 
, so that 
. Therefore,
The positivity of 
and 
then implies
for 
, and we thus obtain
which completes the proof.
Proof.
Assume that perturbed (4.1) is not uniformly asymptotically stable for some 
satisfying 
, that is,
Then, by Theorem 2.4 there exists a 
with 
such that 
is not invertible. So 
since 
is compact. Hence, there corresponds an 
with 
satisfying 
. By virtue of the uniform asymptotic stability of (1.1) we know that 
is invertible; and therefore we get
so that 
and 
. In view of Lemma 4.2,
which gives 
, a contradiction to (4.7). The proof is completed.
Let 
be the set of all compact operators mapping 
into 
. We introduce three classes of perturbations defined as 
, 
and 
. Then the complex, real and positive stability radius of(1.1) under perturbations is defined, respectively, by
where the convention 
is used. By definition, it is easy to see that 
. On the other hand Theorem 4.1 yields the estimate
provided that the assumptions of the theorem are satisfied. In fact, these three radii coincide, that is, we have the following.
Theorem 4.3.
Let 
, 
, be positive. Suppose that (1.1) is uniformly asymptotically stable and 
, 
are both positive. Then
Proof.
By the fact 
, combined with (4.11), it is sufficient to prove that
We may consider the case 
since otherwise the theorem is trivial. Suppose by contradiction that (4.13) does not hold. Then there is an 
, 
, such that
By the same reasoning as in the proof of Lemma 4.2, one can see that 
and hence 
. Also by (2.10) one may choose 
with 
such that
Now let 
. Since 
, there exists a positive 
, 
, satisfying 
(see [17, Proposition 1.5.7]). Consider a map 
defined by
Then, it is easy to see that 
and 
. Hence,
By setting 
, we have
Notice that 
because 
, and that 
, or equivalently 
. Define a perturbation 
by
Then 
. Moreover,
and in particular 
, which means, by Theorem 3.6 (or Theorem 2.4), that the perturbed (4.1) with 
is not uniformly asymptotically stable. Therefore,
which contradicts (4.14). Consequently we must have (4.13), and this completes the proof.
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The first author is partly supported by the Grant-in-Aid for Scientific Research (C), no. 19540203, Japan Society for the Promotion of Science.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Murakami, S., Nagabuchi, Y. Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices. Adv Differ Equ 2008, 598964 (2008). https://doi.org/10.1155/2008/598964
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DOI: https://doi.org/10.1155/2008/598964