THE UN I VE RS IT Y OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report No. 11 A CRITERION FOR THE EXISTENCE IN A STRIP OF PERIODIC SOLUTIONS OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS Lamberto Cesari ORA Project 05304 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP-57 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1964

,9 r i-1, ~<,$ LE- - ) \1 - - 1 < \#l'i

A CRITERION FOR THE EXISTENCE IN A STRIP OF PERIODIC SOLUTIONS OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS By Lamberto Cesari As in a previous paper1 we take into consideration a canonical system of hyperbolic partial differential equations (with fixed characteristics) of the form Utx = f(t,x,UUt,Ux), - 00 < t < *+,o - a < x < a, (0.1) or Uitx = fi(t,x,U,Ut,Ux), i = 1,...,n, (0.2) where U = (U1,...,Un) is an unknown vector function of the two independent variables t, x, where UtUxUtx are the vector functions of the partial derivatives with respect to t, to x, and to t,x, where f(tx,z,p,q) = (fi,...,fn) denotes a continuous vector function of its arguments for (t,x) in a strip A = [- o < t < + o, - a < x < a], and (z,p,q)EE3n, and f is periodic in t of some period T: f(t+T,x,z,p,q) = f(t,x,z,p,q), (t,x)EA, (z,p,q)EE3n. (0.3) Let u(t) = (ul,...,un), - 00 < t < + a, be a continuous function of t, periodic in t of period T, and v(x), - a < x < a, a.continuous function of x, both with values in En, and u(t+IT) = u(t), - 00 < t < + 00, v(o) = 0. In the present paper we give criteria (Section 2) for the existence of a periodic solution 0(t,x) = (01,-,fn) of period T in x for the Darboux problem iii

4tx = f(t,x,1,0tix), (t,x)EA, 0(t,o) = u(t), - oo < t < +, (0.5) 0(t+T,x) = O(t,x), (t,x)EA, 0(o,x) = O(T,x) = u(o)+v(x), - a < x < a Namely, given u(t), the criteria assure the existence of a number a > 0 sufficiently small, of a function v(x), - a < x < a, as above, and of the corresponding solution O(t,x) of the Darboux problem kO.5) in A, O(t,x) being periodic in t of period T. To obtain these criteria we make use of previous results in Ref. 1 concerning a modified Darboux problem. Namely, in Ref. 1 we gave existence, uniquenesses, and continous dependence theorems in order that, given u(t), v(x) as above, there exists a pair of vector functions 0(t,x) = (l, —.,~n), m(x) = (ml,...,mn) such that ~tx = f(t,x,$,Ot,~x)-m(x), (t,x)cA, 0(t,o) = u(t), - < t < + oo (0.6) 0(t+T,x) = 0(t,x), (t,x)EA, O(o,x) = O(T,x) = u(o) + v(x), - a < x < a. In Section 1 of the present paper we restate and improve the theorems proved in Ref. 1 for the problem (0.6). In Section 2 we prove the criteria for the existence of a solution $ tothe problem (O.5). These criteria show that we can choose v(x), - a < x < a, in such a way that m(x) = 0, - a < x < a. In Section 2 we shall make use of an implicit function theorem of functional analysis we proved in a previous paper (Ref. 2). iv

SECTION 1. THE MODIFIED DARBOUX PROBLEM 1. WE PROVED IN REF. 1: THEOREM I (Existence theorem for the modified Darboux problem (0.6). If a, T > 0, and N. N1, N2, L, M, b, M1, M2, M3 > 0 are constants, if A and R are the sets A = [O < t < T, - a < x < a], R = [o < t < T, - a < x < a, ZI M1, IPl < M2, I q < M3, z,p,qEEn], if M1 > N + 2-N1T + N2a + LTa, (1.1) M2 > N1 + 2La, (1.2) M3 > N2 + LT, (1.3) if u(t), o < t < T, v(x), - a < x < a, are vector functions which are continuous with u'(t), v'(x), if f(t,x,z,p,q), (t,x,z,p,q)ER, is continuous in R, and u(T) = u(o), lu() l < N, Iu(tl)-u(t2)I < Nlltl-t21, (1.4) v(o) = O, Iv(xl) - v(x2)I < N2_xl-x21, (1.5) f(T,x,z,p,q) = f(o,x,z,p,q), If(t,x,z,p,q)l < L, (1.6) If(t,x,z,pl,ql) - f(t,x,z,p2,q2)| < Mlpl-p21 + blql-q2l (1.7) 2Tb < 1, Ma < i, (1.8) then there exists a vector function ~(t,x), (t,x)EA, continuous in A together with ~t, ~x, ~tx, and a continuous vector function m(x), -a < x < a, such that 1

0(o,x) = O(T,x) u(o) + v(x), (1.10) T m(x) = T-1 f(t,x,~(t,x),tt>xit(t,,xt, ))dt (1.11) o Otx = f(t,x,O(t,x),Ot(t,x),Ox(t,x)) - m(x) (1.12) for all 0 < t < T, -a < x < a. Thus, by extending both O(t,x) and f(t,x,z,p,q) for all - 00 < t < + 0o, Ixl < a, Izl < M1, IPI < M2, Iql < M3, by means of the periodicity of period T in t, Equation (1.12) is satisfied in the strip - 00 < t < + oo, - a < x < a. 2. In the proof of Theorem I:(Ref. 1), we denoted by wj(a), U2(p), L3(7) continuous monotone functions in o < a < 0, o < < < 0, o < Y < 0, such that w1(o) = 0a(o) = W3(o), and If(tl,x,z,p,q) - f(t2,x,z,p,q)l _ Wl(ltl-t21)!f(t,xl,z,p,q) - f(t,x2,z,p,q)l < C02(Ixl-x21) If(t,x,zl,p,q) - f(t,x,z2,p,q)Jl < o3( Zl-z2l) for all o < t, tl,t2 < T. -a < x, xl,x2 < a, IZI,IZ,1 I21 < M1, IPI M2,!ql ) M3, z,zl,z2,P,qEEn. Analogously, we denote by L4(a), 0 < < 00, s5(p), o < o _ 0, continuous monotone functions such that c4(o) = o-5(o) = O, and lu'(tl) - u'(t2)1 < c4(1tl-t21), IV'(X1)-V'(x2)I <_ 5(lxl-x21), for all 0 < tl,t2 T, -a < xl,x2 < a. Finally, we take (11p) = (1-2Tb)- [a5(P)+2TW2(P)+2TI3(M3) + 4LMT], (1.13) 2(Ca) = (l-aM)-l[04(ca)+aml(a)+aw3(M2a) + 2Laba], (1.14) TV3([) = U2([) + 03(M3) + 2LM. + bnl([), (1.15) Both r1(a), o < a <,'j2(B), o _ f < 0, are continuous monotone functions with r1(o) = T2(0) = 0. 2

We proved in Ref. 1 that the functions Y,m of Theorem I satisfy the following relations It(tl,x) - Bt(t2,x)l < ~2(ltl-t21), (1.16) JIt(t,xl) - ~t(t,x2)I < 2LIxl-x21, (1.17) 1Ix(t1,x) - $x(t2,x)I < 2LItl-t21, (1.18) IOx(t,xl) - Ox(t,x2)I1 < 1(Ixl-x21)* (1.19) I$(tl,xl) - (tl,x2) - $(t2,xl) + 0(t2,x2)1 < 2LItl-t2l Ixl-x21, (1.20) I(t,x) <_ M1, IJ(tl,x) - $(t2,x)I < M21tl-t21, (1.21) 1I(t,xl) - $(t,x2)1 < M31x1-X21, (1.22) I|t(t,x)l < M2, Ix(t,x)l M3, (1.23) Im(x)I < L, 1m(x1) - m(x2)l < PT3(lXl-X21) * (1.24) 3. Under the conditions of Theorem III, if f,u,u',v,v' are Lipschitz functions in their arguments, then Oytyxytxm are all Lipschitzian with constants depending only on the Lipschitz constants of f,u,v,u',v', and the constants listed in Theorem III. Indeed, if 4l(~c) = kl1, 0)2(p) = k2a, s3(Y) = k37, w4(a) = k4a, C5(5) = k55, then rll(D) = (1-2Tb )-1(k5+2Tk2+2Tk3M3+4L MT) ) = hl, (1.25) I2(Ua) = (l-aM) l(k4+akl+aM2k3+2Lab)a = h2a, (1.26) 13(P) = [k2+M3k3+2LM+b(1-2Tb)-l(k5+2Tk2+2Tk3M3+4LT) ] = h3, (1.27) and the other relations of the Remark 1 show that n,t,, xy txym are all Lipschitzian.

4. THEOREM II (Uniqueness theorem for the modified Darboux problem (0.6)) Under the hypotheses of Theorem I, and W3(7) = k37 for some constant k3 > 0, there is only one vector function $(t,x) continuous in A with $t,~x, otx and one vector function m(x) continuous in [-a,a] satisfying (0.6). THEOREM III (Continuous dependence upon the initial data for the modified Darboux problem (0.6)) Under the conditions of Theorems I and II (XI3(7) = k37), the unique solution 0(t,x), m(x) of (0.6) depends continuously on u(t), u'(t), v(t), v'(t). In other words, if (ul(t), vl(x)), (u2(t),v2(x)) are initial data, and (l1(t,x), ml(x)), (02(t,x),m2(x)) the corresponding solutions, and = maxlul(t)-u2(t)l + maxluj(t)-uh(t)l, x = maxlvl(x)-v2(x)l + maxlvi(x)-v(x)l, E = X+l, c= = maxl~i(t,x) - 2(t,x)I, (1.28) P = maxl!lt(t,x) - ~2t(t,x)l,; = max 1lx(tx) - 02x(t,x)1, 6 = maxImM(x) - m2(x)l, then there is a constant K depending only on the constants a,T,N,N1,N2,L,M, b,M1,M2,M3 such that a,1,y,) < Ke. (1.29) 5. In the proof of Theorem III given in Ref. 1, we proved this theorem for a strip Ao = [o < t < T, o < x < c] with c = a/P, P integer sufficiently large. The resoning has to be repeated for the remaining strips'.[o <t < T,(i-l)c < x < ic], i = 0,1,...,P. The same reasoning holds for -a < x < O. Let gi = ic, i = O,l,...,k, (50 = 0, Cp = a). If we take

k = (1-2Tb)-1, k' = 1 + 2kTb, and c < (4Mk')-1, c < (4Tk3k')-l (1+4k'Ma) 1 We proved in Ref. 1 that, for the strip A, we have ao < 4(l+Tbc+TMk'c(1+2bc))e = QE, o < 2(+2bc)e + 4k3k'cac < (2(1+2bc) + (4k3k'c)Ql)e = Q2E, yo < kc + 2kTk3a + 2kTM <_ (k+2kTk3Ql+2kTMQ2)e = Q3E, 60o k3a + MP + by = (k3+MQ2+bQ3)e = Q4C where a Acoyon 6 are the numers o,ey,6 relative to the strip AO only. Let us denote by Q the largest of the numbers Ql,Q2,Q3,Q4. In the strip A1 we can take the same number X, but q = io, E = eo have to be replaced by rll = maxlBL (t,i) - 2(t,1l)1 + maxlIlt(t,ri) - U2t(t,2)1 <_ C+B, 1 = +X < + +X Hence, the corresponding numbers, say a,, p1, a2, P2 become a1,P,l,l _ < Qci _< Q(a+l+X) < Q(2Q(X+q)+X): (Q+2Q2)X + 2Q2 ~ In the strip A2 we take the same X but T1i, E1 are replaced by 2 < i + i, E2 = a2 + X < 1 + 1 + X, and then we have 5

< Q[2(Q+2Q2)X + 4Q2T + X] = (Q+2Q+22Q3)X + 4Q3T After P of these operations we have ap, pP, p, p = (Q+2Q2+...+2PQP+l)x + 2PQP+1. Thus we have only to determine c = a/P with P > (4Mk')a, P > (4Tk3k')(1+4k'Ma)a, and take 2 3 P P+l K = Q + 2Q + 2 Q +...+2 Q Then we have a,V,f,6 < K(x+r) = KE. (1.-o) This remark completes the proof of Theorem III given in Ref. 1. 6. We shall now improve formula (1.29) concerning the continuous dependence on the initial values u,u',v,v' so that a much sharper estimate can be deduced when u,u' do not vary, that is,,then Tr = 0. Indeed, if we denote by ac1,c,7c,6c the same numbers (1.28) relative to the strip [o < t < T, -c < x < c], (o < c < a), then, for a convenient constant K," we have K( c), < K(l+Xc) c K"( T+X' (. (1. l) We shall show that the continuity of v' can be relaxed by allowing an arbitrary set [5] of P points of discontinuity of the first type for v' (x) in [-a,a]. The number a of Theorems I, II, III and all other constants, in particular K" in formula (1.51), can be chosen independently of [e] and P. 6

A form of Theorems I,II,III answering the question above reads as follows for discontinuous v' and Lipschitz data. A Modified Form of Theorems I,II,III: If a, T > 0, and N, N1, N2, L, M, b, M1, M2, M3, kl, k2, k3, k4, k5, are constants, if A and R are the sets A = [o < t < T, -a < x < a], R = [o < t < T, -a < x < a, IZI < M1, IPI < M2, Iql < M3,z,p,qeEn], if M1 > N+2- N1T + N2a + 2LTa + k5a, (1.32) M2 > N1 + 4La, (1.33) M3 > N2 + LT, (1.34) if u(t), o < t < T, v(x), -a < x < a, v(o) = 0, are vector functions, if u(t)is continuous together with u'(t), if v(t) is continuous and v'(t) sectionally continuous, if u(T) = u(o), lu(o)l < N, lu(tl) - u(t2)1 < Nlltl-t2l, (1.35) v(o) = 0, Iv(xl) - v(x2) < N2lxl-x2, (1.36) f(T,x,z,p,q) = f(o,x,z,p,q), If(t,x,z,p,q) I < L, (1.37) If(t,x,z,pl,ql) - f(t,x,z,p2,q2)l < Mlpl-p2l + blql-q21, (1.38) If(tl,xl,zJ,p,q) - f( t2,x2,,pq)I < klltl-t21+k21xl-x21+k31 Z1Z21, (1.39) 2Tb < 1, Ma < 1, if lu(t)u1(t)-u(t2)l < k41t1-t21, o < tl, t2 < T, (1.40) 7

if -a < tj < 52 <...< <Q < a are the points of discontinuity of v(x) in [-a,a], and |v'(xl)-v'(x2)1 < k51x1-x21, (1.41) for all ji < x1,x2 < Ci+l' i = O,l,...,Q (~O = -a,, Q+l = a), then there is a vector function 0(t,x), (t,x)eA, Lipschitzian in A with partial derivatives Ot(t,x) again Lipschitzian in A and partial derivatives Ox(t,x), otx(t,x) also Lipschitzian in each strip Ai = [o < t < T, 5i < x _< i+l], i = 0,1,...,Q satisfying (1.6)-(1.9) of Theorem I. The functions ( and m are uniquely determined by the initial data u(t), o < t < T, v(x), - a < x < a. If (uj(t),v1(x)), (u2(t),v2(x)) are two sets of initial data, and A,X,E e X,3, 5 are defined as in Theorem III (with sup replacing max where needed), then there is a constant K' depending only on the constants a,t,...,k5 listed above such that a,,y,6 <_ K'E. Finally, if c is any number o < c < a, and if acGcc,yc,5c are defined as in (1.32) for the strip o < t < T, o < x < c, [or -c < x < 0], then there is a constant K" > 1 (depending only on a,T,...,ks such that xc03c < K"(l+Xc)j, Pc>c < K(7+X1. (1.42) 7. PROOF OF THE STATEMENT OF NO. 6 As we have mentioned, it is not restrictive to consider only the strip A' = [o < t < T, o < x < a], the points of discontinuity say Q < tj <...<'p < a, of v'(t) in (o,a), and the strips Ai = [o < t < T, Is _ x < Is+l], s = O,1,...,P(o0 = o, = P+1 = a). Also, it is not restrictive to assume v'(x) continuous at every x f 51,...p, and verifying v'(gs-O))4 v'(9s) = v'(9s+0), s = 1,...,P, at the points of discontinuity. We shall apply Theorem I to each strip As, s = O,1,...,P in succession, where a is replaced by t2, a2-91,-.,,P -gP-1, a-9p respectively. The initial conditions for the strip Ao, 0(t,o) = u(t), Ot(t,o) = u'(t), o < t < T, is then replaced for the strip Al by ~(t,~l) = us(t) = ~(t,~1-O), t(t,.i) = uj(t) = ~t(t,Si-O), o K t < T, 8

and, in general 0(tSs) = us(t) = 0(t,)s-O), Pt(t,ls) = Us(t) = Ot(t,5s-O), 0 < t < T, (1.43) for the strip As, s = 1,...,P. Obviously, the initial condition 0(o,x) = U(o) + V(X), X(O,X) = V'(X), o < X < for the strip Ao is replaced by 0(o,x) = u(o) + v(x) = u(o) + v(Ss) + [v(x)-v(s)] = us(o)+[v(x)-v()] (1.44) $x(O,X) = v'(x), ES < X <I s+1, (1.45) for the strip As, s = 1,...,P, since US(O) = O(o,5s) = O(o,As-O) = Us_-(o) + [v(Ss)-V(~s-1)1 Us-2(0) + [v( s)-v(ts-2)] =... = u(o) + v(s). (1.46) Obviously, we can apply Theorem I to the strip Ao since all relations (1.321.42) imply the analogous relations (1.1-1.8). Assume that we can apply Theorem I to As_'l, and prove that we can apply it to As. By (1.46), (1.4), and (1.42) we have Es Es Us(o) u= U-l(o) + v'(x)dx = u(o) + v'(x)dx, s-1 0s lui(o)| < lu(o) I + l Iv(t) dt < N + ak5. By (1.43), (1.20) we have 9

lus(tl) - US(t2)1 < lus-l(tl)-us-l(t2)l + 2LItl-t21(Ms-5S-l) < lUs_2(tl)-Us_-2(t2) + 2LJtl-t2j1(s-s_-2). By repeating this argument and by force of (. 31) we have lus(tl) - us(t2) < lu(ti) - u(t2)1 + 2Lltl-t2(iS < (Nl+2La)Itl-t2l Thus we can replace N and N1 by N+aN2, N1+2La respectively in each strip Ai. Let us prove that we can replace k4 by k4s = (1-sM)-l [k4+s( kl+M2k3+2Lb)] (1.47) in the strip As, i = O,l,...,P. Again this is possible in Ao since k40 > k4. Assume that this is possible in As-1 and let us prove that this is possible in As. By force of (1.43), (1.16), 1.26), and (1.47) we have Ius(tI) - US(t2) = It(tl,ts-O) - Ot(t2,tS-O) < r12(ltl-t21) = (l-(sts-s_)M)-l[k4,sl-+(ts-5s-1)(kl+M2k3+2Lb)] Itl-t2( = (1-(5 s- s1)M)Y ((1-t sM_lM) [ k4+t s-l(kl+M2k3+2Lb) ] +(ts-ts-l)(kl+M 2k3+ 2 LB) ] Itl-t21. (1.48) Since, for all a,p > 0 with a+P < 1 we have (1-a)-1(l1-)-1 < (1-cz-B)-1, we have (l-(ts- s-l)M) -l(l-ts-lM)-1 < (1-tsM)1. (1.49) Since 0 < (ts-ts-l)M < tsM < 1, we have in succession (l-S-M) -S-)+(-(S-S-l s-S.1) < s(l-sM)- s- - (1.50) ~_~ sM)-~< -~ )H-~( ~ <(]-~sM10~s

By using (1.49) and (1.50), relation (1.48) becomes 4us(tl) - Us(t2)I < (1-5sM) -l[k4++ s(kl+M2k3+2Lb)]jtl-t21 = k4s5tl-t21, and we have proved the contention that k4 can be replaced by k4s in As. The numbers k4s are of course all smaller than 1 k4 = (l-aM)- l[k4 + a(kl+M2k3+2Lb)]. Now the relations (1.32), (1.33) of Theorem I, are obviously satisfied in each strip As, s = O,1,...,P, since the relations (1.32), (1.33) in the hypotheses of Theorem III are obtained by relations (1.1), (1.2) by replacing N. N1, k4 by N+aks5 Nl+2La, k4 respectively. All other constants and relations are the same. By induction we conclude that Theorem I can be applied to each strip As, s = O,l,...,P, in succession, and thus the existence part of the theorem is proved The uniqueness follows from Theorem II applied to the strips Ao,A1,...,Ap in succession. The continuous dependence upon the initial data follows from III applied to the strips Ao,A1,...,Ap in succession and where K' is a constant analogous to K obtained by replacing N, N1, k4 by N+ak5, N2+La, k4 in the definition of K. Let us prove now relation (1.42) and, therefore, the continuous dependence of the solution Em on the initial data u,u',v,v'. First we have Ojtx(t,x) = f(txjijtjx) - mj(x), t x Oj(t,x) = uj(t) + vj(x) + [f(t,x,.j,)jt,/jx) - mj(x)]dt dx, o o x Oit(tx) = uj(t) + f [f(t,xYj,1jtj jx) - mj(x)]dx, (1.51) To t Ojx(t,x) = vj(x) + [f(t,xij, jt,ljx) - mj(x)]dt, o T mj(x) = T-1 j f(txijIjtjx )dt, 11

for all o < t < T, - a < x < a, j = 1,2, and where the usual conventions hold for Ojt,'jx,~jtx at the lines of discontinuity [x=ts, o < t < T], s=l,...,P. We have taken here as set [ts,s=l,...,P] the union of the two sets of points of discontinuity of vj and v2. Now, for all c, o < c < a, and o < x < c, we have vi(o) = v2(o) = 0, Ivl(x)-v2(x)l < x, fx Ivl(x)-v2(x) | | |[v'(x)-v2 (x)]dxI < Xx < Xc, Bul(t)-u2(t)l, lur(t)-u2(t)l < n. By subtracting corresponding relations (1.51) for j = 1,2, and standard estimates, we obtain, as in the proof of Theorem III in Ref. 1, Ii(t,x) - $2(tx)I < n + xc + Tc(k3a+MP+by+6), 1lt (t,x) - "t,x)l < + c(k3a+MP+by+8), (1.52) I2x(t,x) - 02x(t,x)l < X +T(k3a+Mp+by+8), Imi(x) - m2(x) I < k3a + M3 + by, where Clplyl are computed as in Theorem III for the sole strip o < t < T, o < x < c (and thus are.< the numbers C0c,c,7c,YSc of No. 6). For some t and x, relations (1.52) yield Oc < r + Xc + Tc(k3a+Mp+by+b), f < il + c(k3a+Mp+b7y+), 7 < x + T(k3a+Mp+by+b), 6 < k3a + MP + b7y By elimination of 6 we have O K< r + Xc + 2Tc(k3ca+M+by), < ~ + 2c(k3a+M+b 7), 12

Y < X + 2T(k3c+MP+by) For k = (1-2Tb), k' = 1+2kTb, we have Y < kX + 2kk3TC + 2kMTI, a < ~ + (1+2kTb)Xc + 2k'k3Tac + 2k'MTPc, P < n + (2kb)Xc + 2k'k3%c + 2k'Mc' The same restrictions already used on c, o < c < a,;namely 2k'Mc < 1/2, 2k'k3T(1+4k'kMc)c < 1/2, (1.53) give _ < 2rI + (4kb)Xc + (4k'k3)ac, and finally C < 2(1+4k'MTc)I + 2(1+2kTb+8kk'MTbc) Xc = Q1q + Q1Xc, _ < 2(1+2k'k3Qic)n + 4(kb+k'k3Qlc)Xc = Q2r + Q2Xc, y < 2kT(k3Q1+MQ2)h + k(l+2Tc(k3Q'+MQ2 )X Q3rl + Q3x, 6 < (k3Q1+MQ2+bQ3) + (k3Qlc+MQc+bQ)X = Q4 + QX. Let co, o < co < a, be any number for which relations (1.53) are satisfied. Let Q be the largest of the numbers Qj,Qj, j = 1,2,3,4, above computed for c = co. Then, for every o < c < co, we have c,c -< Q(0+Xcc),, 6c -< Q(nr+X). This proves (1.42) for o < c < co, K" = Q. 13

We shall now take c = a/P with P integer sufficiently large so that c < cO. Let us consider the P strips As = [o < t < T, (s-l)c < x < sc], s = 1,2,...,P. We shall denote by css7,ys,bs the numbers ca,y,6 relative to the strip As. We have proved above that for the strip Ao we have ac,1 Pi< Q(+xc) y1,8 < Q(1+X) In the strip A2 we can take the same number X1 = X, but v = o must be replaced by 1 = max| i(t, ) - 02(t, t) + maxI$lt(t,t) - 2t(tP0) < ai + P1 < 2Q(n+Xc). The reasoning above implies (X2,p2 < Q(il+Xc) < Q[2Q(Tq+Xc) + Xc] 2Q2n + (Q+2Q2)Xc, 2,a52 <_ Q(n+X) < Q[2Q(r+Xc) + X] 2Q2n + (Q+2Q2c)X For the strip A3 we have a2 < a2 + P2 < 4Q22 + (2Q+4Q)Xc, X2 = X, and hence QC3, 3 < Q(r2+Xc) = Q[4QQ2 + (2Q+4Q2)Xc + Xc] = 4Q3n + (Q+2Q2+4Q3)Xc 7snb <_ Q(r2+X) = Q[L4Q2T + (2Q+4Q2)Xc + X] = 4Q3q + (Q+2Q2C+4Q3C)X.

By this process we obtain successively as,,ys,s = 1,...,P. Since a = max as, and analogous relations hold for P,y,6, we have c, < ( 2PQP+l)~ + (Q+2Q2+...+2PQP+l)Xc y7, < (2PQP+1)r + (Q+2Q2C+...+2PQP+lc)X Let Q' be the largest of the numbers in parenthesis for c = a/P < co. Then we have a,B < Q'(n+Xco), y,6 < Q'(n+X). Now for any c, co < c < a, we have ac <, c <, c < 7c, c < 5, and finally ac,ypc < Q'(n+Xc), yc,65 < Q'(n+X) This proves relations (1.42) for co < c < a, K" = Q'. Then relations (1.42) hold for every c by taking K" = max(Q,Q'). 8. A DIFFERENTIAL RELATION FOR m We shall assume now that the components of f(t,x,z,p,q) = (fl,*..,fn) possess continuous partial derivatives fiqj with respect to the components ql,.-.,qn of q in R. Let u(t), v(x) be given initial data, and let 0(t,x), m(x) be the corresponding solution. Now let x be any given number, say o < x < a. Since we can introduce discontinuities in v'(x) (of the first kind), we may think to change suddenly the value of v' at x, assigning a new value y = v'(x), provided we remain within the limitations listed in No. 6, namely lyl < N2. This change does not alter 0(t,t), m(t) for o _< < x, but we get a new function 0(t,x) = q(t) and a new number m(x) = m, which are the solutions of the ordinary differential problem dq/dt = F(t,q) - m, o < t < T, q(o) = y T m = T-1 F(t,q)dt, (1.54) where F(t,q) = f(t,x,(t,x), t(t,x),q), 15

and, for emphasis, we have treated x as a constant. We proved in Ref. 1 that q(t) and m exist, are uniquely determined by y, and depend continuously on y. We already proved that the continuous dependence of q(t) and m upon y is uniform with respect to x and with respect to v,v' for o < x < a, and for v,v' satisfying the limitations listed in Theorems I-III (or No. 6). We shall now assume that the components fi of f(t,x,y,z,p,q) = (fl,...,fn) possess continuous partial derivatives with respect to the components ql*,...,qn of q. Then the components Fi of F = (F1,...,Fn) have continuous partial derivatives with respect to ql,'..,qn Fiq (t,q) = fiqj(t,x,J(t,x),$t(t,x),q) Then, as proved in Ref. 1, all components of f(t) and m (here q,m denotes a solution of the problem (1.54)) have continuous partial derivatives with respect to the components yj of the arbitrary initial value y = (Yl,.'',Yn)' In particular aij = ymi/ayj = - F. d exp. dt i-ep L l X [l dt iexp oneThis relrvativeon was actuallyhe provedof in Ref. 1same for n = 1 and then there was onlyt reone derivative dm/dy). The proof is the same for any n, and we shall not repeat it here, since it is based on usual theorems of continuous dependence and differentiability with respect to parameters and initial values for ordinary differential systems (as in Ref. 4, pp. 155 and 161). In Ref. 1 we assumed F to be of class C' (both in t and q) but the existence and continuity of Ft was never used. Thus (1.55) stands under the condition just stated that the partial derivatives fiq exist and are continuous in R. Here we only add that the continuity of the partial derivatives aij is proved uniformly with respect to x and with respect to v,v' with the usual conventions. 9. THE FUNCTIONAL G(x,y,v') We shall now assume u(t), o < t < T,as fixed, and study the dependence of m(x) on v(x), v'(x). Obviously, v'(x) determine v(x) since v(o) = 0. Thus, for each x, say o < x < a, m(x) = (ml,...,mn) depends on the values taken by v'(e) for o < t < x. At t = x we can always take an arbitrary value y = v' (x) for v'(x) introducing a new point of discontinuity, and the dependence of m(x) on y has been studied in No. 8. Thus m(x) is precisely a vector valued functional depending on v'(t), o < ~ < x, on y = v'(x), and x itself: 16

m(x) = G(x,y;v'(5), o < < < x), o < x < a. For the sake of brevity we shall write sometimes G(x,y;v' ). It is understood that all the variables are assumed to satisfy the limitations listed in No. 6. We shall prove for G certain properties G1234. Gl. There is an M > 0 such that IG(x,y;v'(~), o < g < x) - G(x,y,v2(g), o < g < x)l < MXx, where Xx = suplvl(~) - v2()1i for all o < g < x. This is only a different form of the statement 6 < K"X of No. 6. G2. There is a constant M such that for anytwoxl,x2E[o,a] and y we have lG(X1,y,v'(t), o < t < x1) - G(x2,y,v'(0), o < < x2)l < Mlxl-x21. Indeed we have seen in No. 3 that m(x) satisfies a Lipschitz condition with constant M which depends only on the constants a,M,...,k5 Therefore, if v'(g) denotes the function which is equal to v'(g) for o < g < x, and is constant and equal to y for x <. < a, then IG(X1,y,VT) - G(X2,y,V')I < Mlxi-x21, IG(xi,y,vl) - G(X2,y,v')l < MIxl-x21, G(x1,y,v') = G(x,y,v'). The last relation is trivial since V'(g) = v' () for all o < _ < xl. Finally, we have IG(X1,Y,vt) - G(x2,yv') | 2MIX1-X21. G3. For x = O, G does not depend on v', and G = (Gi,...,Gn) is only a vector valued function of the vector y = (Yl,...,Yn). We shall write G(o,y, ). 17

G4. For each t, o < t < a,and continuous function v'(t), o < t < a, with values Iv'(t)l < N2, the components Gi of G are real valued continuots functions of y with continuous first order partial derivatives aij(t,y,z) = aGi/6Yj, i,j = 1,...,n. This is only a rewording of statements proved in No. 8. Note that for t = 0, all aij are functions of y only, and we may write aij(o,y, ). Actually, G1 can be modified as follows: G5. There is a constant M > 0 and that, if c, o < c < a, is sufficiently small, then y,vk(0),v2(0) as in No. 6 and o < x < c imply JG(x,y,v1(~), o < < x) - G(x,y, v(~), o < e < x) < McXx. To prove this, let us observe that r = 0, and that, by No. 6, Acc,fc < KcXx, hence, if $l,ml,a2,m2 are the solutions corresponding to vl and v2,we have l1(t,x) -2(t,x), Ij lt(tx) - $2t(tx)I l KcXx. Then we have to solve the two differential problems dq/dt = f(t,x,Oi(t,x),$it(t,x),q) - m, q(o) = y, m = T 1 T f(tVx,Pi(t,'X),it(tx), q(t))dt. With the same initial value y. If ql(t), ml, q2(t), m2 are the two solutions, we have, with obvious notations, t lql(t)-q2(t)1 = I (fl-f2)dt + (ml-m2)tI < (k3T+MT)(KcXx) + Iml-m2 T, Iml-m21 < (k3+M)(KcXx) and hence Iml-m21 < K(M+k3)cXx, jql(t)-q2(t)| < 2K(M+k3)cXx The first of these relation obviously proves G5. 18

SECTION 2. CRITERION FOR THE EXISTENCE OF PERIODIC SOLUTIONS TO THE ORIGINAL PROBLEM 10. AN IMPLICIT FUNCTION THEOREM OF FUNCTIONAL ANALYSIS In Ref. 2 we have proved an implicit function theorem, a corollary of which will be stated here and applied in No. 11. Let H(t,y,z( ), o < ~ < t) be a functional, with values in En, depending on the real variable t, the real vector yEEn, and the values in En taken by a function z( ) in the variable interval o < e < t. We proved, under the hypotheses below, that there is a continuous function v(t) with values in En such that H(t,r(t),3r(~), o < ~ < t) = 0. (2.1) We state now, with precision both hypotheses and contentions. Let I be the interval I = [o < t < a] for some a > O, let i be a point of En and Yo the sphere Yo= [yeEnI ly-~I < D] for some D > 0. Let Zo be the family of all continuous functions z( ), o < < K a, with values'in Yo, or z:I -+ Yo. Then H:IxYoxZo + En. For the sake of simplicity we shall write H in the form H(t,y,z). We shall take in Zo the topology of the uniform convergence in [o,a]. We shall assume that, for t = 0, H does not depend on z, and then we may write H(o,y,.). As in the implicit function theorem we assume H(o,t,.) = 0. The following statement holds (Ref. 1, Cor. 1, Section 5, No. 2). (A) If H(t,y,z) = (H1,...,Hn) is uniformly bounded and continous in t,z, if all Hi have partial derivatives with respect to i,...,Yn, say aij(t,y,z) = aHi/6yj, i,j = l,...,n, which are bounded and continuous in t,y,z, if H(t,ii,.) = 0, and det Ao $ O, with Ao = [aij(o,=, )], then there is some ao,0o, o < ao < a, o < K <., and a continuous function *(t), o < t < ao, such that V(o) = l, k(t)-il <~ o,. and f(t,4r(t),4) = 0 for all o < t < ao. Also we proved in Ref. 2 that, if there are numbers a',', o < a' < a, o < P' < P, such that o < t < a', Y-PI <', [z1(t)-i1, Iz2(t)-[l < 1', o <t < a', imply IG(t,y,zl) - G(t,y,z2)| < (1/2M) max Iz1(5)-z2(5)1 (2.2) o< <t 19

with lA-1II < M, then the function v(t), o < t < ao < a', of statement (A) is unique. Finally, if G depends uniformly and continuously on some parameter a describing a topological space 2, and the conditions above hold uniformly with respect to a (in particular, IIAolll < M for some constant M independent of a),then the unique function lia(t), o < t < a', above depends continuously on a in Q. 11. AN EXISTENCE THEOREM FOR THE ORIGINAL PROBLEM (0.5). We shall now use again the notations of Section 1. THEOREM IV Given constants a,T,...,k5, and functions u and f as in No. 6 (thus, satisfying all relations but (1.36), (1.41)) let us assume that, for some IEEn, I41 < N2, we have T m(o;) = T-l[q(T)-q(o)] = T-1 f f(t,o,u(t),u'(t),q(t))dt = O, where q(t) is the solution of the initial ordinary differential problem dq/dt = f(t,o,u(t),u'(t),q), o < t < T, q(o) = ~. (2.3) Assume also that det A 1 O, where A = [aij] and T ~ig ~ flpT e x d Leai / TKfiqj [ f iq ) dt (2.4) fiqj = fiqj(t,o,u(t),u'(t),q(t)), o < t < T. Then there is some ao, o < ao < a, a function v(x), -ao < x < ao, continuous in [-ao,ao] with v(o) = O, v'(o) =., satisfying (1.36), (1.41), and a function 0(t,x) continuous in the strip [-a < t < + a, - ao < x < ao] with ~t,'x',txx satisfying 0tx(tx) = f(tx,j(t,x)',t(t,x),0x(ttx)), (t,x)cA, ~(t,o) = u(t), - X < t < + a, i(o,x) = $(T,x) = v(x), - ao < x < ao, 20

~(t+T,x) = 0(t,x), (t,x)eA. In addition 0(t,x) together with t(t,(t,x), tx(tx), tx(tx),, v(x), v'(x) are uniquely defined by u(t), and depends continuously on u(t). Proof of Theorem IV. We have first to prove that the function G of No. 9 verifies the hypotheses of statement (A) of No. 10. Indeed, G12 assure that G is uniformly bounded and continuous in t and z, and G34 assure that G satisfies the remaining hypotheses. Thus', by applying (A) to both intervals [o,a] and [o,-a] we conclude that there is some ao > 0, Po > 0, and a continuous function v'(t), - ao < t < ao, with v'(o) = A, Iv'(t)-11l < Po, such that G(t,v'(t),vt(5), o < 5 < t) = 0, o < t < ao, G(t,v'(t),v'(t), o > t > t) = 0, 0 > t > -ao. Since I iL < N2, by reducing ao if needed, we can satisfy Iv'(t)l < N2 in [-ao,ao]. If v(t) = ft v' ()d5, - ao < t < ao, then v(o) = 0, and, again by reducing ao if needed, v(t) satisfies (1.35), (1.41). Finally, if 0(t,x), m(x) is the solution of problem (0.6) corresponding to u(t), v(t), we have m(x) O, -ao < x < ao, and 0(t,x) satisfies (0.6 ). The uniqueness of the function v'(x) so determined, and therefore of tv,.,0t,ox are a consequence of the remark following statement (A) and property G5 proved in No. 9. Then,v,v't,,it,0x are continuous functions of u and u', that is, the difference between any two corresponding elements are small when ~ is small. 12. EXAMPLE OF A DIFFERENTIAL EQUATION (0.1) WITH PERIODIC SOLUTION, Take T =2i,n=1, f = sin t+Xq, A; O. Then Equation (0.1) becomes utx = sin t + XUx (2.5) Let u(t) be a periodic function of period T, continuous in (-oo,+oo) together with u'(t), and let v(x) be a continuous function in some [-a,a) with v'(x), v(o) = O. Then the relations 0(o,x). = v(x), 0(t,o) = u(t), Atx = sin t+%Ox yield ~t(t,x) - (t(t,o) = x sin t + k?(t,x) - EI(t,o), $x(t,x) - k((t,x) = x sin t + u'(t) - Au(t), 21

By integration we obtain O(t,x) e x) + [u'(T)-XU(T)+ X sin T]e dT 0 and, by manipulations, also O(t,x) = u(t) + v(x)eXX + (1+%2) lx[eXX - X sin t - cos t] Thus, 0(t,o) = u(t), O(o,x) = u(o) + v(x), 0(2T,x) = u(2ic) + v(x)e2"X + (l+?2)-lx(e2 -l) and hence 0(2r,x) = O(o,x) if and only if v(x) = _(l+X2)-1 x. With this function v(x), then $(t,x) is periodic in t of period 2A, for all x, -oo < x < -o. Also we have v'(o) = -(l+k2)-. If we apply Theorem IV to Equation (2.5), we see that we have to consider first the ordinary differential problem dq/dt = sin t + Xq, q(o) =,. Hence t q(t) = [ekt + eXt e-T sin T dT = [i + (l+%2)-l]eXt + (l+%2)-Yl( sin t - cos t), and m(opl) = O if and only if,u = -(1+X2). Then a = dm/d = [1-e -oT ] [fe t dt ]O = 22

The conditions of Theorem IV are then satisfied, and the existence of a solution 0(t,x) periodic in t of period 2T in a some strip A can be deduced from Theorem IV, with 0(t,o) = u(t), Ox(o,o) = Ox(2it,o) = p = -(l+x2)-1. 13. EXAMPLE OF A DIFFERENTIAL EQUATION (O.1) WITH PERIOD SOLUTION IN A STRIP Take T = 2r,n=l,f = sin t+%q+xL(txz,zp,q), where L is an arbitrary function, and f satisfies the general hypotheses of No. 6. Then Equation (0.1) becomes Utx = sin t + kux + xL(t,x,u,ut,ux). (2.6) For x = O, both f and fq are the same as those of No. 12 Hence, the conditions of Theorem IV are satisfied with. = -(1+X2)-1, and (2.6) has a solution 0(t,x) periodic in t of period T, with 0(t,o) = u(t), for any periodic function u(t) of class C', in the sense of No. 6 and Theorem IV, and therefore in a strip A sufficiently narrow. 14. EXAMPLE OF A DIFFER-ENTIAL EQUATION (0.1) WHICH HAS A PERIODIC SOLUTION IN A STRIP A BUT NOT IN THE WHOLE PLANE Let us take T = 2it, n=l, f = x+(l-x)(sin t+Xq). For x = O f, fq reduce to the ones of No. 4, and an application of Theorem IV assures the existence of a periodic solution in some strip A. On the other hand for x = 1, the equation reduces to utx = 1, hence ux(2n,l) 4 ux(o,l), and the strip A cannot include x = 1. An equation as (0.1) may have no solution periodic in t in any strip as for example utx = 1, (t,x)EE2. 15. AN HYPERBOLIC EQUATION REDUCIBLE TO (0.1). We shall consider the differential equation A(t,x)utt + utx = f(t,x,u,ut,ux), (2.7) where both A and f are periodic in t of period T. Here one set of characteristic lines is x = constant. The other set can be obtained by solving the differential equation 23

dt/dx = A(t,x), - a < x < a, (2.8) and we shall assume A(t,x) > 0. Let us assume A(t,x) continuous in (t,x) and Lipschitzian in x in the strip A = [-oo < t < +, -a < x < a]. For every real T let t = X(x,T) be the solution of (2.7) with X(a,T) = T. Then, for some constant H > 0 we have IX(x,T) - T <_ Hixi, -a < x < a, Xx(x,T) = A(X(X,T),X), Ixx(x,T)I < H. If 7(T) denotes the line t = X(x,T), then 7(T) cuts the line x constant at a point (t,x) with t = X(X,T). On the other hand, each point (t,x)eA belongs to one and only one line 7(T) for some T = c(t,x). We have defined, therefore, a change of coordinates T = D(t,x), x = x, or t = X(x,T), X = x. From (2.8) we deduce XTx(X,T) = (6/ax)XT(x,T) = At[X(xI),X]XT(x,I), and since X(O,T) = T, we have XT(O,T) = 1. Hence, XT(xT) = exp 0 A(X(tT),)d Xx(x,T) = A(X(x,T),xX) On the other hand, from t = X(xT T), = 0(t,x), we deduce x = X(x,A(t,x), hence 1 = XTT, 0 = Xx + XTOX, and finally A~t + Ox = 0, ~t=; = X1 = exp LA(X(,),)d],

Ox = - X -1X A(X(X,T),T) exp A(X(-,T),,)dS 0 If U(T,X) is obtained from u(t,x) by means of the change of coordinates above, we have U(T,X) = U(X(X,T),X), u(t,x) = U(~(t,X),X) and Ut = UTt U =u UTx + UX, Utt =UtT + Ut UTTTtX + U U + UTt Equation (2.7) is then changed into UTX = g(T,x,U,Ux,UT) (2.9) with g = f(X,x,UUTX;1T -UTX-XXX+UX)XT - UTXTXXT, gq = fq(X,x,U,UTXT1, -UTXT1X + UX)XT. For x = 0 then X(o,T) = T, XT(O,T) = 1, XX(O,T) = A(T,O), XXT(O,T) = AT(T,O) g( T,o,U,UT, q) = f(T,o,U,UT, -AUT+UX) - ATUT, gq(T,O,U,UTq) = fq(T,OIUUT, -AUT+Ux) Relations (2.3),(2.4) become, with U(T,o) = U(T), dq/dT = f(T,o,U,u' -Au'+q) - ATU, q(o) = 1 (2.10) T m(o) = q(T)-q(o) = T-1 (f(T,o,u,u',-Au'+q) - ATU')dT = 0, a = p- e p f d exp q d.dT ~ 0 (2.11) 25

As an example, if we assume A(t,x) = xB(t,x), hence A(t,o) = 0, then Equations (2.10), (2.11) are the same as those of Theorem IV. In particular, the equation xB(t,x) utt + Utx = sin t + \ux + xL(t,x,u,ut,ux) has a periodic solution 0(t,x) in some strip A with ~(t,o) = u(t), u(t) arbitrary of class C', and ox(o,o) = $x(2o,o) = -(l+X2) ~ ~~2

REFERENCES 1. L. Cesari, Periodic solutions of hyperbolic partial differential equations. Intern. Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (Colorado Springs, 1961), Academic Press 1963, 35-57; Intern. Symposium on Nonlinear Oscillations (Kiev, 1961), Izdat, Akad. Nauk SSSR, 2, 1963, 440-457. 2. L. Cesari, Remarks on the implicit function theorem in functional analysis and applications. To appear. 3. B. A. Fleischman and F. A. Ficken, Initial value and time periodic solutions for a nonlinear wave equation. Comm. Pure Appl. Math. 10, 1957, 331-356. 4. E. Kanke, Differentialgleichungen reeller Funktionen, Akad. Verlag Leipzig, 1930, xiv + 436. 5. G. Prodi, Soluzioni periodiche di equaziorialle derivate parziali di tipo iperbolico nonlineari. Annali Mat. Pura Appl. 42, 1956, 25-49. 6. 0. Vejvoda, Nonlinear boundary-value problems for differential equations, Proc. Conference Differential Equations and their Applications (Prague 1962). Publishing House Czechosl. Acad. Sci. Prague 1963, 199-215. 27

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