This paper considers the asymptotic behaviour of solutions of the scalar

linear convolution integro-differential equation with delay

x0(t) = −

n Xi=1

aix(t − i) + Z t

0

k(t − s)x(s) ds, t > 0,

x(t) = (t), −  t  0,

where  = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1),

i  0, ai > 0 and  is a continuous function on [−, 0]. The kernel k is subexponential

in the sense that limt!1 k(t)(t)−1 > 0 where  is a positive subexponential function. A

consequence of this is that k(t)et ! 1 as t ! 1 for every  > 0

Appleby, John A.D.

Győri, István

Reynolds, David W.

English

DCU Online Research Access Service

FUNCTIONAL VOLUME ?DIFFERENTIAL 2003, NO ?-?EQUATIONS PP. ??– ??SUBEXPONENTIAL SOLUTIONS OF SCALAR LINEARINTEGRO– DIFFERENTIAL EQUATIONS WITH DELAYJ. A. D. APPLEBY ∗, I. GYO˝RI† AND D. W. REYNOLDS ‡Abstract. This paper considers the asymptotic behaviour of solutions of the scalarlinear convolution integro-differential equation with delayx′(t) = −n∑i=1aix(t− τi) +∫ t0k(t− s)x(s) ds, t > 0,x(t) = φ(t), −τ ≤ t ≤ 0,where τ = max1≤i≤n τi. In this problem, k is a non-negative function in L1(0,∞)∩C[0,∞),τi ≥ 0, ai > 0 and φ is a continuous function on [−τ, 0]. The kernel k is subexponentialin the sense that limt→∞ k(t)α(t)−1 > 0 where α is a positive subexponential function. Aconsequence of this is that k(t)et →∞ as t→∞ for every  > 0.Key Words. Volterra integro–differential equations, subexponential function, expo-nential asymptotic stability.AMS(MOS) subject classification. 34K20, 34K25, 34K06, 45D05, 45J051. Introduction and Results. This paper examines the asymptoticbehaviour of solutions of the scalar linear integrodifferential equation withdelayx′(t) = −n∑i=1aix(t− τi) +∫ t0k(t− s)x(s) ds, t > 0,(1)∗ Centre for Modelling with Differential Equations, Dublin City University, Dublin 9,Ireland.† Department of Mathematics and Computing, University of Veszpre´m, Veszpre´m,Hungary.‡ Centre for Modelling with Differential Equations, Dublin City University, Dublin 9,Ireland.12 J. A. D. APPLEBY, I. GYO˝RI AND D. W. REYNOLDSsubject to the initial conditionx(t) = φ(t), −τ ≤ t ≤ 0.(2)The following hypotheses are postulated.(H1) τi ≥ 0, ai ≥ 0, ∑ni=1 ai > 0 and τ = max1≤i≤n τi.(H2) The characteristic equationp =n∑i=1aieτip,has a real root.(H3) The kernel k is a non-trivial function in L1(0,∞)∩C[0,∞) with k(t) ≥ 0for all t ≥ 0.(H4)∫∞0 k(s) ds <∑ni=1 ai.(H5) limt→∞ k(t)/α(t) > 0 for some positive subexponential function α.(H6) The initial function φ is in C[−τ, 0].The significance of (H2), and sufficient conditions for it to hold, arediscussed in Section 2. The definition of positive subexponential functionsand some of their important properties are also reviewed there. The mainresult of this paper is the following theorem.Theorem 1. Suppose that (H1)–(H6) hold. Then the solution of (1)and (2) satisfieslimt→∞x(t)k(t)=φ(0)−∑ni=1 ai ∫ 0−τi φ(s) ds(∑ni=1 ai −∫∞0 k(s) ds)2 ,(3)limt→∞x′(t)k(t)= 0.(4)The decay rate given in (3) can also be written aslimt→∞x(t)k(t)=∫∞0 x(s) ds∑ni=1 ai −∫∞0 k(s) ds,It is shown in [1, 2] that the decay rate of two classes of stochastic Volterra in-tegrodifferential equations with subexponential kernels, can also be expressedin this form.In order to prove Theorem 1 we introduce the resolvent for (1), which isthe solution of the equationr′(t) = −n∑i=1air(t− τi) +∫ t0k(t− s)r(s) ds, t > 0,(5)SUBEXPONENTIAL SOLUTIONS 3which satisfies the initial conditionr(t) ={1, t = 0,0, −τ ≤ t < 0.(6)The significance of r is that the solution of (1) which obeys (2) is given bythe variation of parameters formulax(t) = φ(0)r(t) +∫ t0r(t− s)φ˜(s) ds, t ≥ 0,(7)whereφ˜(t) = −n∑i=1aiφ(t− τi)χ[0,τi](t), t ≥ 0,(8)and χJ denotes the indicator function of a set J . The asymptotic behaviourof the resolvent is described in the next theorem.Theorem 2. Suppose that (H1)–(H4) hold. Then the resolvent r, definedby (5) and (6), is in L1(0,∞), r(t) > 0 for all t ≥ 0 and r(t)→ 0 as t→∞.If, in addition, (H5) holds,limt→∞r(t)k(t)=( n∑i=1ai −∫ ∞0k(s) ds)−2, limt→∞r′(t)k(t)= 0.(9)Moreoverlimt→∞∫ t0 r(t− s)r(s) dsr(t)= 2∫ ∞0r(s) ds.(10)Theorems 1 and 2 are generalisations of [3, Theorem 6.2], which concernslinear scalar convolution integro-differential equations with subexponentialkernels but without delays. Theorem 2 has the following converse, which isan extension of [3, Theorem 6.4].Theorem 3. Suppose that (H1)–(H4) hold, and that k(t) > 0 for allt ≥ 0. If the resolvent r satisfies (9), then k is a positive subexponentialfunction and (10) is true.2. Mathematical Preliminaries. The convolution of two appropriatefunctions f and g defined on [0,∞) is denoted, as usual, by(f ∗ g)(t) =∫ t0f(t− s)g(s) ds, t ≥ 0.4 J. A. D. APPLEBY, I. GYO˝RI AND D. W. REYNOLDSWe recall a definition from [3], based on the hypotheses of Theorem 3 of[5].Definition 1. A positive subexponential function is a continuous in-tegrable function α : [0,∞)→ (0,∞) satisfyinglimt→∞(α ∗ α)(t)α(t)= 2∫ ∞0α(s) ds,(11)limt→∞ sup0≤s≤A∣∣∣∣α(t− s)α(t) − 1∣∣∣∣ = 0, for all A > 0.(12)It is noted in [3] that the class of positive subexponential functions in-cludes all positive, continuous, integrable functions which are regularly vary-ing at infinity. It is known that (12) implies for every  > 0 thatα(t)et →∞ as t→∞,(13)(cf., e.g., [4, Lemma 2.2]), and hence by (H5) that k(t)et →∞ as t→∞ forevery  > 0.If α is a positive subexponential function and f is a function on (0,∞)such that limt→∞ f(t)/α(t) exists, we defineLαf = limt→∞f(t)α(t).An important result is the following lemma. It is essentially Theorem 4.1 of[3]. Perusal of the proof of this theorem shows that the hypotheses that f/αand g/α be bounded continuous functions on [0,∞) are redundant, and aretherefore omitted here.Lemma 1. Suppose that α is a positive subexponential function. Let fand g be integrable functions on (0,∞) for which Lαf and Lαg both exist.Then Lα(f ∗ g) exists and is given byLα(f ∗ g) = Lαf∫ ∞0g(s) ds+ Lαg∫ ∞0f(s) ds.(14)Next we introduce the resolvent z associated with the purely point delaypart of (1). It satisfies the equationz′(t) = −n∑i=1aiz(t− τi), t > 0,(15)SUBEXPONENTIAL SOLUTIONS 5and the initial conditionz(t) ={1, t = 0,0, −τ ≤ t < 0.(16)We collect in a lemma some salient properties of z.Lemma 2. Suppose that (H1) and (H2) hold. Then z(t) > 0 for t ≥ 0and ∫ ∞0z(t) dt =1∑ni=1 ai,(17)z(t)→ 0 exponentially as t→∞.(18)Thus if α is a subexponential functionLαz = 0.(19)The positivity of z and (17) are part of Proposition 2.1 of [7]; (18) followsfrom the same proposition and Lemma 6.5.3 of [8]; (19) is a consequence of(13) and (18).It is shown in [9] that a necessary condition for (H2) to be true is∑ni=1 aiτi ≤ e−1, and that τ∑ni=1 ai ≤ e−1 is a sufficient condition. In the caseof a single delay with n = 1, a1 = a > 0, τ1 = τ , a necessary and sufficientcondition for (H2) to hold is aτ ≤ e−1.The following yields a representation of the solution ofy′(t) = −n∑i=1aiy(t− τi) + f(t), t > 0,(20)y(t) ={1, t = 0,0, −τ ≤ t < 0.(21)Lemma 3. Let f be in C[0,∞). Then the solution of (20) and (21) canbe represented as y(t) = z(t) + (z ∗ f)(t), t ≥ 0.3. Proofs. Proof. (Theorem 2) The resolvent r of (1) satisfies (5) and(6). It is a consequence of Lemma 3 that r satisfiesr = z + z ∗ (k ∗ r) = z + h ∗ r,(22)where h = z∗k. Due to (H3) and Lemma 2, h(t) ≥ 0 for all t ≥ 0. A standardargument shows that r(t) > 0 for all t ≥ 0. By taking the convolution ofeach term in (22) with k, we see thatρ = h+ h ∗ ρ,6 J. A. D. APPLEBY, I. GYO˝RI AND D. W. REYNOLDSwhere ρ = k ∗ r. Since by (17) and (H4),∫ ∞0h(s) ds =∫ ∞0z(s) ds∫ ∞0k(s) ds =1∑ni=1 ai∫ ∞0k(s) ds < 1,it can be deduced from a theorem of Paley and Wiener (cf., e.g, [6, Theo-rem 2.4.1]) that ρ is in L1(0,∞). It is then an immediate consequence ofLemma 2 andr = z + ρ ∗ z,that r is in L1(0,∞). It then follows from this equation that r(t) → 0 ast→∞, since z is a bounded continuous function obeying (18).Integration of (5) gives−1 = −n∑i=1ai∫ ∞0r(t− τi) dt+∫ ∞0r(s) ds∫ ∞0k(s) ds,which the aid of (6) leads to∫ ∞0r(s) ds =1∑ni=1 ai −∫∞0 k(s) ds.Also we can deduce from Lemma 1, (17) and (19) thatLαh = Lα(z ∗ k) = Lαz∫ ∞0k(s) ds+ Lαk∫ ∞0z(s) ds =Lαk∑ni=1 ai.Suppose now that Lαr exists. Then we can infer from Lemma 1, (19),(22) and the above formulae thatLαr = Lαz + Lαh∫ ∞0r(s) ds+ Lαr∫ ∞0h(s) ds=Lαk∑ni=1 ai(∑ni=1 ai −∫∞0 k(s) ds)+ Lαr1∑ni=1 ai∫ ∞0k(s) ds.Rearranging yields the first formula in (9). To obtain the second, note that(12) implies Lαr(· − τi) = Lαr, so then, by applying Lα to (5) and usingLemma 1, we get thatLαr′ = −Lαr( n∑i=1ai −∫ ∞0k(s) ds)+ Lαk∫ ∞0r(s) ds = 0.We note that by Lemma 1Lα(r ∗ r) = 2Lαr∫ ∞0r(s) ds,SUBEXPONENTIAL SOLUTIONS 7which immediately implies (10).To complete the proof, it only remains to show that Lαr exists. For thesake of brevity a proof is indicated here under the additional (and unneces-sary) assumption that k(t) > 0 for all t ≥ 0. It follows then that h(t) > 0for all t > 0. Then by Lemma 4.3 of [3], h is a subexponential function. Byapplying Theorem 5.2 of [3] to (22), we conclude that Lhr exists and henceLαr.Proof. (Theorem 1) First, we observe from (8) that Lαφ˜ = 0 and∫ ∞0φ˜(t) dt = −n∑i=1ai∫ 0−τiφ(s) ds.Then, by applying Lα to (7) and using Lemma 1, we obtainLαx = φ(0)Lαr + Lαr∫ ∞0φ˜(s) ds+ Lαφ˜∫ ∞0r(s) ds.Therefore we can conclude thatLαx = Lαkφ(0)−∑ni=1 ai ∫ 0−τi φ(s) ds(∑ni=1 ai −∫∞0 k(s) ds)2 ,and hence that (3) holds.It also follows easily from (7) that∫ ∞0x(t) dt =φ(0)−∑ni=1 ai ∫ 0−τi φ(s) ds∑ni=1 ai −∫∞0 k(s) ds.By applying Lemma 1 to (1), we then see thatLαx′ = −n∑i=1aiLαx(· − τi) + Lαx∫ ∞0k(s) ds+ Lαk∫ ∞0x(s) ds=(∫ ∞0k(s) ds−n∑i=1ai)Lαx+ Lαk∫ ∞0x(s) ds= −Lαkφ(0)−∑ni=1 ai ∫ 0−τi φ(s) ds∑ni=1 ai −∫∞0 k(s) ds+ Lαk∫ ∞0x(s) ds = 0.Therefore (4) is true.Proof. (Theorem 3) For convenience we introduceη =n∑i=1ai −∫ ∞0k(s) ds.8 J. A. D. APPLEBY, I. GYO˝RI AND D. W. REYNOLDSDividing (5) by k(t), we see thatr′(t)k(t)= −n∑i=1air(t− τi)k(t)+(k ∗ r)(t)k(t).By letting t→∞ and using (9),limt→∞(k ∗ r)(t)k(t)=1η2n∑i=1ai.Hence (k ∗ r)(t)/r(t) → ∑ni=1 ai as t → ∞. Since r and k are positive andlimt→∞ k(t)/r(t) > 0, Lemma 3.8 of [3] applies. We can conclude from itthat k satisfies (12) andlimt→∞(k ∗ k)(t)k(t)=n∑i=1ai +∫ ∞0k(s) ds− η2∫ ∞0r(s) ds = 2∫ ∞0k(s) ds.Thus k satisfies (11), finishing the proof.REFERENCES[1] J. A. D. Appleby, Subexponential solutions of linear Itoˆ-Volterra equations with adamped perturbation, Functional Differential Equations, Forthcoming.[2] J. A. D. Appleby, Almost sure subexponential decay rates of scalar Itoˆ-Volterraequations, Electron. J. Qual. Theory Differ. Equ., Forthcoming.[3] J. A. D. Appleby and D. W. Reynolds, Subexponential solutions of linear Volterraintegro-differential equations and transient renewal equations, Proc. Roy. Soc.Edinburgh. Sect. A, 132A (2002), 521–543.[4] J. A. D. Appleby and D. W. Reynolds, Subexponential solutions of linear integro–differential equations, Proceedings of Dynamic Systems and Applications, 4(2004), submitted.[5] J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Analyse.Math. 26 (1972), 255–302.[6] G. Gripenberg, S.-O Londen and O. Staffans, Volterra integral and functional equa-tions, Encyclopedia of Mathematics and its Applications, Cambridge UniversityPress, 1990.[7] I. Gyo˝ri, Interaction between oscillations and global asymptotic stabilty in delaydifferential equations, Differential Integral Equations, 3(1990), 181–200.[8] J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations,Applied Mathematical Sciences, Springer-Verlag, 1993.[9] G. Ladas and I. P. Stavroulakis, Oscillations caused by several retarded and advancedarguments, J. Differential Equations, 44(1982), 134–152.

Subexponential solutions of scalar linear integro-differential equations with delay

This paper considers the asymptotic behaviour of solutions of the scalar
linear convolution integro-differential equation with delay
x0(t) = −
n Xi=1
aix(t − i) + Z t
0
k(t − s)x(s) ds, t > 0,
x(t) = (t), −  t  0,
where  = max1in i. In this problem, k is a non-negative function in L1(0,1)C[0,1),
i  0, ai > 0 and  is a continuous function on [−, 0]. The kernel k is subexponential
in the sense that limt!1 k(t)(t)−1 > 0 where  is a positive subexponential function. A
consequence of this is that k(t)et ! 1 as t ! 1 for every  > 0

Irish Universities

This paper considers the asymptotic behaviour of solutions of the scalar\ud
linear convolution integro-differential equation with delay\ud
x0(t) = −\ud
n Xi=1\ud
aix(t − i) + Z t\ud
0\ud
k(t − s)x(s) ds, t > 0,\ud
x(t) = (t), −  t  0,\ud
where  = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1),\ud
i  0, ai > 0 and  is a continuous function on [−, 0]. The kernel k is subexponential\ud
in the sense that limt!1 k(t)(t)−1 > 0 where  is a positive subexponential function. A\ud
consequence of this is that k(t)et ! 1 as t ! 1 for every  > 0

Name not available

http://doras.dcu.ie/18/1/integro-differential_equations.pdf

Subexponential solutions of scalar linear integro-differential equations with delay

Abstract

Similar works

Full text

Available Versions

DCU Online Research Access Service

Irish Universities

Name not available