ON THE EXPONENTIAL CONVERGENCE TO A LIMIT OF SOLUTIONS OF PERTURBED LINEAR VOLTERRA EQUATIONS

We consider a system of perturbed Volterra integro- dieren tial equations for which the solution approaches a nontrivial limit and the dierence between the solution and its limit is inte- grable. Under the condition that the second moment of the kernel is integrable we show that the solution decays exponentially to its limit if and only if the kernel is exponentially integrable and the tail of the perturbation decays exponentially.


Introduction
In this paper we study the exponential decay of the solution of x (t) = Ax(t) + t 0 K(t − s)x(s) ds + f (t), t > 0, (1.1a) x(0) = x 0 , (1.1b) to a constant vector.Here the solution x is a vector-valued function on [0, ∞), A is a real matrix, K is a continuous and integrable matrixvalued function on [0, ∞) and f is a continuous and integrable vectorvalued function on [0, ∞).
The solution of (1.1) can be written in terms of the solution of an unperturbed version of the equation.This unperturbed equation is given by where the matrix-valued function R is known as the resolvent or fundamental solution of (1.1).The representation of solutions of (1.1) in 1991 Mathematics Subject Classification.Primary 45J05, 45D05, 34K20; Secondary 60K05.
Key words and phrases.Volterra integro-differential equations, Volterra integral equations, exponential stability, exponential asymptotic stability, John Appleby was partially funded by an Albert College Fellowship, awarded by Dublin City University's Research Advisory Panel.Siobhán Devin was funded by The Embark Initiative operated by the Irish Research Council for Science, Engineering and Technology (IRCSET).EJQTDE, 2005 No. 9, p. 1 terms of R is given by the variation of constants formula For this and other reasons, the asymptotic behaviour of R has long been a topic of study, and it is well known that uniform asymptotic stability for (1.1a) is associated with the solution R of (1.2) being integrable.In this case it is interesting to understand the relationship between the rate of decay of the kernel, and the rate of decay of solutions.Authors who have shown that some sort of exponential decay in the kernel can be identified with exponential decay of the resolvent include Murakami [8,9] and Appleby and Reynolds [2].Murakami shows that the exponential decay of the solution of (1.2) is equivalent to an exponential decay property on the kernel K under the restriction that none of the elements of K change sign on [0, ∞).A condition of this type will be employed in this paper to identify exponential convergence.In a similar spirit, various authors have identified decay conditions on K which give rise to particular decay properties in the resolvent.For example Burton, Huang and Mahfoud [3] have shown that the existence of the "moments" of the kernel can be identified with the existence of the moments of the solution.Appleby and Reynolds [1] have studied a type of non-exponential decay of solutions (called subexponential decay) which can in certain circumstances be identified with the subexponential decay of the kernel.Jordan and Wheeler [5] and Shea and Wainger [10] have studied the relationship between the existence of the kernel in a certain weighted L p -space and the existence of the solution in such spaces.
The case where the solutions of (1.2) are neither integrable, nor unstable, has also been considered.Krisztin and Terjéki [6] studied this case and determined conditions under which R(t) converges to a limit R ∞ , which need not be trivial, as t → ∞.In addition to determining a formula for R ∞ , they showed that the condition ∞ 0 t 2 K(t) dt < ∞ is crucial.MacCamy and Wong [7] dealt with a nonlinear version of (1.1).They showed that if the kernel and the perturbation satisfy an exponential decay constraint, then x converges to a nontrivial limit x ∞ exponentially fast.
In this paper we consider the case where the resolvent of (1.1) is not integrable.In the first instance, we find an equivalence between the exponential decay property of t → R(t) − R ∞ and an exponential decay property of the kernel; we also show for solutions of (1.1) that the exponential decay of t → x(t) − x ∞ can be identified with exponential decay in the kernel and the perturbation.EJQTDE, 2005 No. 9, p. 2

Mathematical Preliminaries
We introduce some standard notation.We denote by R the set of real numbers.Let M n (R) be the space of n × n matrices with real entries, and I be the identity matrix.We denote by diag(a 1 , a 2 , ..., a n ) the n×n matrix with the scalar entries a 1 , a 2 , ..., a n on the diagonal and 0 elsewhere.
All other norms on M n (R) are equivalent to • .If J is an interval in R and V a finite dimensional normed space, we denote by C(J, V ) the family of continuous functions φ : J → V .The space of Lebesgue integrable functions φ : (0, ∞) → V will be denoted by L 1 ((0, ∞), V ).The convolution of F and G is denoted by F * G and defined by We denote by N the set of natural numbers.We denote by C the set of complex numbers; the real part of z in C being denoted by Re z and the imaginary part by Im z.We now make our problem precise.Throughout the paper we assume that the function It is convenient to define the tail of the kernel K as and the tail of the perturbation f as The existence of K 1 and f 1 is assured by the integrability of K and f respectively.We define the function t → x(t; x 0 , f ) to be the unique solution of the initial value problem (1.1).Under the hypothesis (2.1), it is well-known that (1.2) has a unique continuous solution R, which is continuously differentiable.Moreover the solution of (1.1) for any initial condition x 0 is given by Where x 0 and f are clear from the context we omit them from the notation.
A fundamental result on the asymptotic behaviour of the solution of (1.1) is the following theorem due to Grossman and Miller [4]; under In this paper we consider the case where the solution of (1.1) approaches a constant vector x ∞ which need not be trivial, and so (2.6) does not necessarily hold.

Discussion of Results
In this section we explain the connection between the results on exponential decay presented by Murakami in [8,9] and those here.Murakami obtained the following result in the case where the solutions of (1.2) are integrable.
then the following are equivalent; (i) There exists a constant α > 0 such that In this paper we begin by considering the case where the solution of (1.2) approaches a constant matrix.EJQTDE, 2005 No. 9, p. 4 Theorem 3.2.Let K satisfy (2.1) and Suppose there exists a constant matrix R ∞ such that the solution R of then the following are equivalent; (i) There exists a constant α > 0 such that We can readily see the similarities between Theorem 3.1 and Theorem 3.2: the hypotheses (3.1) and (3.2) in Theorem 3.1 are identical to (3.6) and (3.7) in Theorem 3.2; moreover, the equivalence between (3.3) and (3.4) in Theorem 3.1 is mirrored by the equivalence between (3.8) and (3.9).The hypothesis in Theorem 3.2 which has no counterpart in Theorem 3.1 is (3.5); however, as we mention later, this hypothesis is natural and sometimes indispensible in the case R ∞ = 0.
It is possible to obtain results comparable to Theorem 3.2 for the solution of the perturbed equation (1.1).More precisely, it is possible to show that the exponential decay of x − x ∞ is equivalent to the exponential decay of the tail of the perturbation and the exponential integrability of the kernel.The following theorem makes this precise.Theorem 3.3.Let K satisfy (2.1) and (3.5), f satisfy (2.2), and f 1 be defined by (2.4).Suppose that for all x 0 there is a constant vector x ∞ (x 0 , f ) such that the solution t → x(t; x 0 , f ) of (1.1) satisfies
Murakami considered the case where the resolvent of (1.2) is integrable, which forces R(t) → 0 as t → ∞.In this paper, we consider the case where the solutions of (1.1) approach a constant vector, which may not necessarily be trivial, in which case the solution is not integrable.As a result it is not possible to apply Murakami's method of proof directly to our equation.Instead, we find it is necessary to appeal to a result of Krisztin and Terjéki [6] to obtain appropriate hypotheses for Theorem 3.2 and Theorem 3.3.
Before citing the relevant results from [6], we introduce some notation used there and adopted hereinafter.We let M = A + ∞ 0 K(s)ds and T be an invertible matrix such that T −1 M T has Jordan canonical form.Let e i = 1 if all the elements of the i th row of T −1 M T are zero, and e i = 0 otherwise.Put P = T diag(e 1 , e 2 , ..., e n )T −1 and Q = I − P .We now state the relevant theorem.Krisztin and Terjéki's result not only suggests the appropriate hypotheses for our theorems, but guarantees the existence of the constant matrix R ∞ as well as giving a formula for it.We note that under assumptions (3.5) and (3.6) that (2.6) fails at z = 0 if R ∞ = 0.

Preparatory Work
In order to prove Theorem 3.2 and Theorem 3.3 we must reformulate (1.2) as was done in Theorem 2 of [6].In order to make this reformulation precise we state the following lemma.where F (z) is defined for Re z ≥ 0 and z = 0 by and and Ĝ(z) is defined for Re z ≥ 0 and z = 0 by Proof.As conditions (3.5) and (3.6) hold we know from Proposition 3.4 that (3.14) holds.We now employ an idea used in [6, Theorem 2].Define the function Φ by Φ(t) = P + e −t Q for t ≥ 0. Taking the convolution of each side of (1.2) with Φ, we get Φ * R = Φ * (AR) + (Φ * K) * R, which after integration by parts becomes where and the function e is defined by e(t) = e −t , t ≥ 0. A further calculation yields where Since (2.1) holds we can take the Laplace transform of (4.7) to obtain (4.1)where F and Ĝ are given by (4.2) and (4.4) respectively for Re z > 0 and z = 0 and are given by (4.3) and (4.5) when z = 0. EJQTDE, 2005 No. 9, p. 7 Remark 4.2.If we assume that there exists a constant α > 0 such that (3.8) of Theorem 3.2 holds then the functions F and Ĝ defined by (4.2) and (4.4) respectively when can be extended into the negative half plane.
The following lemma may be extracted from [6, Theorem 2] and is necessary in the proof of Theorem 5.1.The following proposition may extracted from [8,9] and used is later in the proof of Theorem 5.2.Proposition 4.4.Let K be a continuous integrable function such that no entry of K changes sign on [0, ∞).Suppose that there is a continuous function z → B(z) defined for | Re z| ≤ α 1 and analytic for The proof is identical in all important details to that of Theorem 2 in [8]. where We begin by showing that (5.1) for some β 2 > 0.
Observe that since det[I + F (0)] = 0, H 1 (0) exists.Using (3.8) and the Riemann-Lebesgue Lemma we see that K(z) → 0 as |z| → ∞ for Re z ≥ −α, thus we can see from (4.Since z → (I + F (z)) is analytic on the domain Re z > −α, and its determinant is a continuous function of its entries then z → det[I + F (z)] is analytic on the domain Re z > −α.Thus it has at most a finite number of zeros in the set D, and so c 0 < 0. Take a constant β 2 > 0 so that β 2 < −c 0 .Consider the integration of the function H 1 (z)e −zt around the boundary of the box: Since H 1 (z) exists and is analytic in this box it follows that the integral over the boundary is zero, that is: Our claim will be verified if EJQTDE, 2005 No. 9, p. 9 Consider H 1 (z)e zt for z = λ + iT and −β 2 ≤ λ ≤ β 2 : then Using the continuity of H 1 (z)e zt and the above we can find constant m < ∞ such that for |Re z| ≤ β 2 , z = λ + iT , t > 0. Also Thus finishing the demonstration of (5.1).
It is necessary to choose an integrable function H 2 in order to obtain (3.9).We define the function H 2 (z) as follows: where EJQTDE, 2005 No. 9, p. 10 Thus Clearly z z−c 0 and z z+1 → 1 as |z| → ∞.As (3.8) holds we know from the Riemann-Lebesgue lemma that K(z) → 0 as |z| → ∞ with Re z ≥ −α thus (I + F (z)) → I also z(z + 1) Ĝ(z) − zL , c 0 (z + 1) Ĝ(z) and (z + 1) F (z)L are bounded for Re z ≥ −α.Now we have that Therefore we obtain  Suppose that for all x 0 there is a constant vector x ∞ (x 0 , f ) such that the solution t → x(t; x 0 , f ) of (1.1) satisfies (3.10).If there exists a constant α > 0 such that statement (i) of Theorem 3.2 holds, and there exist constants γ > 0 and c 3 > 0 such that statement (i) of Theorem 3.3 holds, then there exist constants β 3 > 0, independent of x 0 , and c 4 = c 4 (x 0 ) > 0, such that statement (ii) of Theorem 3.3 holds.
Remark 6.3.If we impose a weaker condition, that is if (3.12) of Theorem 3.3(ii) only holds for a basis of initial values, then the same result holds.
Proof of Theorem 6.1.Using (2.2) and (2.5) we have that Due to the fact that K obeys (3.8), by Theorem 3.2, it follows that R − R ∞ decays exponentially.We prove in the sequel that R decays EJQTDE, 2005 No. 9, p. 12 exponentially, f 1 also decays exponentially therefore the convolution of R and f 1 decays exponentially.By use of the above facts and the hypothesis (3.11) on f 1 , we have that each term on the right hand side of (6.1) decays exponentially, which yields (3.12).We now show that R decays exponentially.We can rewrite the resolvent equation (1.2) as (6.2) The first term on the right-hand side of (6.2) decays exponentially since (3.8) holds.We now provide an argument to show that the second term decays exponentially; since R(t) − R ∞ decays exponentially and (3.8) holds we can choose µ such that e µt K(t) and e µt (R(t) − R ∞ ) ∈ L 1 ((0, ∞), M n (R)).Because the convolution of two integrable functions is itself integrable, showing that the last term on the right-hand side of (6.2) also decays exponentially.
Finally we show that (A + ∞ 0 K(s) ds))R ∞ = 0. Integrating (6.2) and rearranging the terms yields Each term on the right-hand side of the equation is integrable thus (A + ∞ 0 K(s) ds)R ∞ = 0. From the above we see that R decays exponentially.

0 B
then the Laplace transform of B is formally defined to be B(z) = ∞ (t)e −zt dt.If ∈ R and ∞ 0 B(s) e − s ds < ∞ then B(z) exists for Re z ≥ and is analytic for Re z > .If B is a continuous function which satisfies B(t) ≤ ce βt for t > 0 then the inversion formula B(t) = lim )e zt dz holds for all > β.

e µt t 0 K
(t − s)(R(s) − R ∞ ) ds = t 0 e µ(t−s) K(t − s)e µs (R(s) − R ∞ ) ds ≤ c 6 ,so that the second term on the right-hand side of (6.2) decays exponentially.Note that(6.3)c 7 := ∞ 0 K(s) e αs ds ≥ ∞ t K(s) e αs ds ≥ e αt ∞ t K(s) ds ≥ e αt K 1 (t) , Proof of Theorem 5.2.Note that from our hypothesis Ŷ exists and is continuous for Re z ≥ −β 2 and is analytic for Re z > −β 2 .Due to (3.5) and (3.6) so we can apply Proposition 3.4 to get (3.13).We see that det[R ∞ + z Ŷ (z)] is non-zero at z = 0. From the continuity of Ŷ (z) at zero there exists an open neighbourhood centred at zero on which det[R ∞ + z Ŷ (z)] = 0. Also det[zI + P ] is non-zero except at zero in an open neighbourhood centred at zero with radius less than one.Choose α > 0 such that det[R ∞ + z Ŷ (z)] and det[zI + P ] are non zero for 0 < |Re z| < α.Define the function B as follows