proceedings of the american mathematical society Volume 73, Number 3, March 1979 AN Lx REMAINDER THEOREM FOR AN INTEGRODIFFERENTIAL EQUATION WITH ASYMPTOTICALLY PERIODIC SOLUTION KENNETH B. HANNSGEN Abstract. For a certain integrodifferential equation of Volterra type on (0, oo ), with piecewise linear convolution kernel, it is shown that the solution is u(t) = a cos Bt + p(i), with p e L'(0, oo) and a and ß constant; u' is represented similarly. 1. Introduction. Let c > 0, and suppose a(t) is nonnegative, nonincreasing, and convex on (0, oo) with a E L'(0, 1) and a(co) = 0. (1.1) It was proved in [3] that if u'(t)+ f'[c + a(t - s)]u(s)ds=0 (t > 0), m(0) = 1, (1.2) 'o (' = d/dt) then u(t) -> 0 (/ -» oo), except in the special case where a has the piecewise linear form with t0 > 0 and / minff, kt0\ \»-.?,*('-** ) (U) 8k>0, 0 < a(0) =f 8k ~ 8 < oo, (1.4) A=l u =\(8 + c) =2tTJ/t0 for some integer j. (1.5) If (1.3),(1.4), and (1.5) hold, we have 2 ux(t) = u(t)-cosu/-»0 (f-> ) (1-6) with y = (38 + 2c)/(8 + c). Results of S. I. Grossman and R. K. Miller [2] (for a E Lx(0, oo)) and of D. F. Shea and S. Wainger [7] show that u, u' E Lx(0, oo) if a has no representation (1.3) satisfying (1.4) and (1.5). Here we establish that Jo M0 + «i(0 <ft< o (1.7) Received by the editors May 1, 1978 and, in revised form, June 26, 1978. AMS (MOS) subject classifications (1970). Primary 45D05, 45J05, 45M05. 1979 American Mathematical Society 0002-9939/79/0000-0108/S02.75 331
332 K. B. HANNSGEN in the asymptotically periodic case if I a(t) dt <oo, (1.8) 'o and we demonstrate that (1.7) need not hold if (1.8) fails to hold. Theorem 1.1. Let c > 0 and let (1.3), (1.4), (1.5), and (1.8) hold. Let u be the solution of (1.2), and define w, as in (1.6). Then ux satisfies (1.7). Miller [6] and Jordan and Wheeler [5] give remainder estimates like (1.7) for various classes of integral and integrodifferential equations. We use special properties of kernels satisfying (1.1) to avoid moment conditions assumed for analogous problems in [5], [6]. In particular, (1.1) and (1.8) imply that the Fourier transform â(r) - i""\(t)e~m dt is twice continuously differentiable. The key to the proof is an estimate for the difference between 5(t) and its linear approximation at r = ±u. Since this result may have further applications, we prove it in a slightly generalized form in Lemma 3.1. The author thanks R. L. Wheeler for a number of corrections. 2. Preliminaries. In this section we summarize some results from [3], valid under the assumptions of Theorem 1.1. Here and below we assume that t0 = 1 ; this may be accomplished by rescaling. The Laplace transform a*(z)=f e~zta(t) dt -f + -l2!(*"**-1) (2.1) Z k= 1 is analytic for Re z > 0 and continuous for Re z > 0. Moreover, ux(t) -> 0 (t -» oo), and Note that ux*(z) = * -i* (Re z > 0). (2.2) z + c/z + a*(z) y(z2 + to2) v > \! u*(i) = U*(z) (2.3) (~ = complex conjugate), u* has a continuous extension to {Re z > 0, z i= ± io)}. A little rearranging shows that c ( i 8(z + iu>)\ z + C- +a*(z) = y(z - to) - 11 + ^v j(z - to)2 + z~2 2 T [^-*(z"'") -! + *(*- to)]. (2.4)
AN LX REMAINDER THEOREM 333 By considering the cases where \k(z iw) > 1, < 1, one sees that e-ku-iu) _ j + k^z _ /wj k(z - iu)2 <3k (1 < k < oo, Re z > 0, z = ia>). Since c j 2 k8k = foa(t) dt < oo, we may use dominated convergence to see that lim (z - iw)"2 2 -r [e-*(z-'u) - 1 + *(z - ico)l = f a(/) <#. z nu '->" *=i * 'o Rez>0 By (2.2), (2.3), and (2.4), u* is continuous and bounded in (Re z > 0}. Let We shall show that/is /(t) = M*(/t) (-oo<t<oo). (2.5) absolutely continuous with C /'(r) dr< oo. (2.6) '-oo To complete the proof that ux E Lx, we adapt an argument from [7], based on the estimate / oo \P (0 1 /-OO / t dt<2-fjfit)ldt for functions F of Hardy class Hx in the lower half-plane. (F = f in our case; v denotes the inverse Fourier transform.) The proof for u'x involves a similar argument. 3. A linear approximation. In this section we consider kernels of the general type (1.1). Lemma 3.1. Let a(t) satisfy (1.1) and (1.8). Let p(r) = q(r) + â(r) on an interval ( t v\ < e) with e < \v\, where q E C2 and q" is Lipschitz continuous on [v e, v + e]. Assume that p(v) = 0, p'(v) ^ 0, and p(t) = 0 (0 < t - v\ < e). Let P(v) = p"(v)/2[p'(v)]2, P(r)=[p'(v)(T - v)yx-[p(t)yx (T ^,). (3.1) Then P is absolutely continuous on {\t v\ < e). Proof. As is shown in [1, Proof of Lemma 4.1], under the conditions of the lemma we have Jf oo (\-ht-e-"')da'(t) (t>0). (3.2) o (By symmetry, we may and shall assume that v > 0.) Here a' is redefined where necessary so as to be continuous from the left on (0, oo). Integration by
334 K. B. HANNSGEN parts, together with (1.1) and (1.8), shows that fœt2da'(t) < oo. 'o (3.3) Therefore â E C2(0, oo), and it follows easily that P is continuous for t - v\ < and twice continuously differentiable for 0 < t - v\ < e. Thus we need only show that P' is of class L1 on the interval. We have p'(v)p'(t) - (T - v) V(t) P (T)-77TT7\- P'(v)P2(t) (T * ") But (t - v)-\p'(v) - p(r)(t - v)~x) -* - \p"(v) (T ^ V), so (3.4) p'(v)p'(r) - (t - v)~2p2(t) -/»'(") p'(r) + /,'(") - T y /(") />(') T P = /(") />'(t) +p'(p)- 2/,(r) + 0(t-»')2 (x^r). (3.5) By (3.2), (3.3), and the hypothesis on q, p(r) = p~2b(r) + h(r), where h E C2[v - e, v + e], h"(r) - h"(v) - 0(t - y)(t-» r), and (3.6),(T) = /""(I - fo - e-,t') /a'(0- 'o Then by Taylor's Theorem, there exist t,, t2 between t and v such that ä(t) - h(v) h'(r) + h'(v) - 2 T_pyJ - A'(*) - h'(v) - h"(tx)(r - v) = [h"(r2)-h"(rx)](t-v) = 0(r-p)2 (t^v). Since >'(t) = fuie1** - l)da'(t), \b'(r) + b'(v) - 2(t - v)-\b(t) - b(v))\ (3.7) i e_m + e "] + 2,- p~m J t(r (r-v) - tda'(t). The function J(t, t) inside the absolute value signs in this integral can be written J(r, t) = ie~iv'k((7 - v)t), where K(x) is a C2 function with K(0) = K'(0) = 0, K"(0) = - ±, and AT(je) < M < oo for all x. We choose n > 0 such that AT(x) < x2 if x < p. Then by Fubini's Theorem and (3.3) (with the change of variable a = ±(t - v), and with M = E^t) = {í t - p <
M}, ; = {ft</ T-H<oo}), AN L1 REMAINDER THEOREM 335 f + \r - v)~2\b(r) + b(v) - 2(t - v)~\b{t) - b(v))\ dr < f+e(t -p)~2 ( (t - v)2t3da'(t) + MÍ t da'(t) Jp-e Jeu Je; dr <2( t3 f do + tmíxa-2da Jo Jo Jp/t da'(t) = 2(/i + Mfi~x) í t2da'(t) < oo. '0 From (3.5), (3.6) and (3.7) (recall thatp(v) = 0) we see, therefore, that f + )p'(p)p'(r) - (t - v)-2p2(t)\(t - r)"2i/r< oo, so /*' E Lx(v e,v + e), and our assertion follows. 4. Proof of Theorem 1.1. From the proof of Lemma 3.1, we see that ô(t) C2 for t =?*= 0. In [7, Lemma 1] it is shown (from (3.2)) under weaker hypotheses than ours that Then by (1.8), â'(t) < 40 CMta(t) dt< 40a(0)/ r 2 (t = 0). (4.1) '0 f ' o'(t) í/t< 80 [ y-2 (yta(t) dt dy = 80 [C ta(t)fx y~2dydt J0 -/max{l,i} From (2.5), (2.2), (4.1), (4.2) and the continuity of/(t) 0 < e < u, [ /'(t) dr < oo. < oo. (4.2) it follows that if (4.3) Setp(r) = it + c(it)"1 + â(r), and note (from (2.4)) that Then /(*)- 1 ^(w) = 0, 1 p(r) p'(u)(r - a) />'(«) = /y. (4.4) + r(r) (0 < t «I < e), (4.5) where r E C '[co - e, u> + c]. Thus Lemma 3.1 (with v = w), the fact that p(-r) = p(r), and (4.3) imply that/ is absolutely continuous and satisfies (2.6). By [4, Lemma 3.1] (which merely summarizes a procedure used in [7]), there is a function w E L'(0, oo) such that ux*(z) - w*(z) (Re z > 0). By uniqueness of Laplace transforms, u, E L'(0, oo).
336 K. B. HANNSGEN A similar argument applies to u\. By (1.2) and (1.8), u\ is bounded and < 'w ^ -a*{.z)-c/z 2<o2 (ux)*(z) = - + z + c/z + a*(z) Y(z2 + w2) = (^-l) + zuf(z) (Rez>0). Thus (u'x)*(z) is analytic for Re z > 0 and continuous and bounded for Re z > 0 with â(r) + z'c/t 2co2 (u'x)*(ir) = g(r) = P(t) y(co2 - T2) Then = 0(T-') ( T ^00). = iâ(r)- irâ'(t) + 2c/r 4<A P\r) y(w2 - t2)2 (4.6) (4.7) From (4.7), (4.1), and (4.2) it is clear that with 0 < e < to we have Near t = w, (4.4) and (4.6) show that f \g'(r)\dr<oo. (4.8) g(r) =[- í(t) + íc/t] P(t) p'(u)(t - co) â(r) iy(r â(u>) - co) + R(r), where R(t) E C2[u c, «+ e]. Since â E C2, an application of Lemma 3.1, together with (2.3) and (4.8) gives /" J g'(r)\dt < oo, and we finish as before. This proves (1.7). 5. An example. In Theorem 1.1 our assumption that a E L\0, oo) cannot be dropped. For example, let c = 0, 8k = k~3/1. If ux were in L'(0, oo), the function [p(r)]~x - [iy(r - co)]"1 would be continuous at t = co (see (4.5)); then P(r) - í'y(t-co)= O(t-co)2 (t^co). (5.1) But by the first part of (2.4) (valid here), there is an m < oo with 2 â:-5/2(cosâ:(t-co)- 1) < m\p(r) iy(r - co) +0(r - co)2 (t-*co). Each term in the sum is negative, so if we let (t co) = tr/2n and neglect all
AN LX REMAINDER THEOREM 337 terms with - tr/2 < k(r - w) < tt/2 (mod 2it), we see that 2 Lx\ is at least i \ S k-"2> i \ n-3'2 = const (t - <o)3/2; k = n i this contradicts (5.1). References 1. R. W. Carr and K. B. Hannsgen, A nonhomogeneous integrodifferential equation in Hilbert space, SIAM J. Math. Anal, (to appear). 2. S. I. Grossman and R. K. Miller, Nonlinear Volterra integrodifferential systems with L1- kernels, J. Differential Equations 13 (1973), 551-566. 3. K. B. Hannsgen, Indirect abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc. 142 (1969), 539-555. 4._, Uniform boundedness in a class of Volterra equations, SIAM J. Math. Anal. 6 (1975), 689-697. 5. G. S. Jordan and R. L. Wheeler, Structure of resolvents of Volterra integral and integrodifferential systems (to appear). 6. R. K. Miller, Structure of solutions of unstable linear Volterra integrodifferential equations, J. Differential Equations 15 (1974), 129-157. 7. D. F. Shea and S. Wainger, Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97 (1975), 312-343. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061