THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AIND THE ARTS Department of Mathematics Technical Report No. 10 PERIODIC SOLUTIONS IN A THIN CYLINDER OF WEAKLY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Lamberto Cesari ORA Project 02416 submitted for: UNITED STATES AIR FORCE' AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1970

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PERIODIC SOLUTIONS IN A THIN CYLINDER OF WEAKLY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS* Lamberto Cesari 1. INTRODUCTION We consider here partial differential equations L(t,z,u) = 0, z = (z,...,zv), u = (u1,..,Un) (1) which are defined in some cylinder r = [- r < t < + c, zl < a], for which L(t + T,z,u) = L(t,z,u), and for which the Cauchy problem u(o,z) = u (Z), Izl < a, makes sense. By this we mean that given u (z), Izi < a, in a suitable class0', there are numbers s and b, o < s$ < 1, b > o, and a solution u(t,z) to the Cauchy problem in some set r' = [o t < b,= zI < s a], where s may be rather small. Actually, we limit ourselves, to that large class of Cauchy problems, which, as proved in J. G. Petrovsky ([3],pp. 16-17), can be reduced by suitable substitutions to Cauchy problems for.. firs't order partial differential equations. The problem of the periodic solutions of (1) can then be posed in a form which is similar to the one which is traditional for ordinary differential equations. Indeed, we may first ask (a) whether we can take b = T for a suitable class.. of initial data u (z),lzl < a, and then we may ask (b) whether we can choose u in i so that u(T,z) = u (z) for all Izl < s a, so that the periodicity * 0 Research partially supported by US-AFOSR research grant 69-1662 at The University of Michigan. This paper has been prepared for the Symposium in Nonlinear Oscillations, Marseille, September 14-18, 1970. 1

of L guarantees the existence of the solution u(t,z) in the whole thin cylinder r [- o <ot < +, lzl I< s]. soG Problem (a) actually belongs to the general theory of partial differential. equations. Most methods for the treatment of the Cauchy problem yield criteria under which we can take b = T or b large. Each method concerns of course different classes of partial differential equations, different smoothness hypotheses for the equations, different smoothness requirements for the solutions (usual, or generalized solutions, and so on). Problem (b) is a problem of functional analysis, since the mapping u (') + u(T is a mapping froma class 5 of functions u (z), lzil < a, into aclass,4v' of functions u(T,z), |zl < so, with the complication that s may be smaller than one, and thus we are dealing with functions, u (.) and u(T,.), defined in two different sets. In the present paper we discuss the argument recently proposed by L. V. Ovcyannikov [2] for proving the convergence - in a suitably reduced domain — of the usual method of successive approximations, when L is assumed to be continuous with respect to t and holomorphic in z, and the solution u is required to be of class C1 in t and holomorphic in z. We show first (Nos. 2,3) that the same Ovcyannikov's argument —based on functional analysis-can be applied to nonlinear partial differential equations, and we obtain existence theorems for the Cauchy problem in a nonlinear and very general setting. We then give criteria in order that we can take b large, in particular b = T for L periodic of period T. Finally, we obtain (No. 4) criteria 2

for periodic solutions of partial differential equations by the use of a recent implicit function theorem in functional analysis and we consider a few example s. 2. A NONLINEAR FORM OF OVCYANNIKOV'S THEOREM. For every s, o < s < 1, let X denote a Banach space of elements x with norm |xli such that s (a) X D X for all o < s' < s < 1; (b) the inclusion operation j,: X - X, has norm < 1, where o < s' < s K 1. Thus, X D X D X1 for all o < s < 1. Let I denote an interval o < t < a if t o s - - a - - is real, or a disc Iti < a if t is complex. Let x be any given element of X1. Let f(t), t C I, be a bounded continuous function of t, valued in X1, and, if t complex, we assume also that f is holomorphic in t in the open disc I. For every t c I let A(t) be an operator-nonnecessarily linear-with the following properties: (c) A(t):X + X,, for every t C I and o < s' < s < 1, and A(t) maps the zero element of X into the zero element of X. s s! (d) There is a constant C > o such that for all t C I, xl,x2 E X o < s' < s < 1, we have i A(t)x2 - A(t)x1II+ K C(s-s') I 2-xaI;

(c) A(t) is a continuous function of t, that is, given ~ > o, o < s' < s < 1, there is a b = b(E,s,s') such that for all tl,t2 C I, Itl - t2J < 8, a x E X we have IIA(t2)x - A(tl)Xll St < ~ Note that A(t) may not map X into itself. We consider the Cauchy problem x'(t) = A(t) x(t) + f(t), t C I, (2) x(o) = x Note that the second part of requirement (c) is not restrictive since, in the contrary case, we can, write (2) in the form x'(t) = [A(t)x - A(t)(o)] + [A(t)(o) + f(t)] and now the operator B(t) defined by B(t)x = A(t)x - A(t)(o) has property (c). Theorem 1. Under the hypotheses (abcde), and for every number b such that o < b < a, Ceb < 1, there exists a continuous function x(t), t C Ib, with values in X satisfying (2) for every o < t < b, o < s < 1 - Ceb. Proof. Let us take x_l(t) = o, and let xk(t), t E I, k = o,1,2,... be the sequence defined by t xk(t) = x /0 [A(T)xkl(T) + f(T)]dT, t E I ) Let us prove that xk(t) E X for all o < s < 1, t c I,k = o,1,2,.... First, t x(t) - x + f(T)dT c X1 c X o < s < 1 t c I by the definitions of x and f. Also,

t xi(t) = x + f [A(r)xo(T) + f(T)]dT, hence A(T)x (T) E X for all o < s < 1, t c I, and xi(t) c X for all o < s < 1, o s a s t c I. From (3) by induction argument we conclude that xk(t) c X for all o < s < 1, t c I,a k = o,l,2,.... Now we have t x (t) x (t) = x + S f(T)dT XO -_1 O O and we take t M = max jlx + J f(T)dT||i (4) 0O0 where the maximum is taken for t c Ib. Then 1lx (t)-x_ (t)ll, < IlXo(t)-x 1(t)lil < M, o < s < 1, t E I a From (3) we obtain t xk(t) - Xk_ (t) = J [A(T)x k-1(T)-A(T)Xk (T)]dT k = 1,2... and we shall prove that lXk(t) - Xk- (t)s <M[(-s)Cet] (6) o <s < 1, t E I, k = o,l,2,... a As shown by (5) this relation (6) is true for k = o. For k = 1 we have, by the use of (c), 5

t T Iljx(t) - x (t)ll = ii A(T) [x fo f(o)do]dT s -1 < C(l-s) f M du o - M(l-s) Citl < M(1-s) Celtf, and again (6) is proved for k = 1. We assume that (6) is true for k-l and we prove it for k. Indeed, for all s,r, o < s < s + n < 1, from (6) for k-l and by the use of (c),we obtain -1t tIxk(t)-xk (t)lls < CTI fi Xk (T)-xk (T)l dT _ t k-1 < C-1 j M[(l-s-r)- CeTl dt (7) -- 0 MCktke k-1 -l1(l-s- )-(k-i) -1 If we take q = k (l-s), then s + n < s + (l-s) = 1 since now k > 2, and (7) becomes k k k- -k -(k-l) = MCktk ek (l-s) - (l+( k-l) ) kMC k s)-k < MC t e (l-s)and (6) is proved. For o < t < b, o < s < 1- Ceb < 1, we have 1-s > Ceb, and (l-s) Celt| < < (l-s)YCeb < 1.

We conclude that the series 00 x(t) = x + 7 [xk(t) - Xk_ (t)] k=l converges for all o < s < 1 - Ceb, t C I, that ||xk(t)||, 1x(t)|| < M [l-(l-s)-=Cet]-l, (8) ||~x(t)-x t < M[(l-s) Cet] [!-(l-s)- Cet] (9) [Ix (t)-xk(t) J[ s _ (9) for all o < s < 1, t E Ib, k = 1,2,..., and thus x(t) c X for all o < s < 1-Ceb, t e Ib. The theorem above is thereby proved. Remark. In the reasoning above the assumption that the norm of A is < C(s-s') plays an essential role. The argument could not be repeated with the exponent -1 replaced by any other exponent < -1. Conditions (cde) will turn out to be too restrictive in applications, and we modify them, therefore, as follows. For a given T > o let X' c X be the set X' = [x(x c X, |x| < T], and S S S S S -- let us assume (instead of (cde)): (c') A(t): X' + X for every t e I and o < s' < s < 1; s s a - (d'),(e') The same as (d), (e) for the elements x C X' only. s Theorem 2. Under hypotheses (abc'd'e'), and constants T, b, M, C satisfying o < b < a, Ceb < 1, M < T [1-Ceb], where M is defined by (3), there is a continuous function x(t), t E Ib with values in X' satisfying (2) for every o < s < 1 - Ceb.

The proof is the same as for Theorem 1, where now we note that Ix (t)|l t = ix + f f(T)dT |I < M, hence x (t) E X'. Analogously, each successive approximation xk(t) belongs to X' because of (8), and so does x(t). k s 3. THE CAUCHY PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS The local Cauchy problem (I. G. Petrovsky [ 3 ], p. 14) is known to be reducible to an analogous problem for systems of first order partial differential equations (I. G. Petrovsky [ 3 ], pp. 14-17). Let t and z = (zl,...,z ) the v + 1 independent variables, t either real v or complex, all z. complex. Let u(t,z) = (ul,...,un) be the n unknown functions, 1 n let ut = (aui/at, i = l,...,n) denote the system of n first order partial derivatives with respect to t, and let uz = (aui/kzj, i = 1,...,n, j = 1,...,v) be the system of nv first order partial derivatives with respect to zl,..,z V We shall consider a Cauchy problem of the form n v aui/at = Z h. (tozu)auj /az + gi(tzu) +fi(tZ), (10) j=l 1=1 u.(o,z) = ui.(z), i =n, 1 1. or in vector form Ut = h(t,z,u)ux + g(t,z,u) + f(t,z), u(o,z) = u (z), where h is an n x nv matrix, and g and f are n-vectors. For any n-vector u = (ul,...,Un) we define lu| by taking ul = maxjgjl. For any matrix h = (hij) we define Ihj by taking Ihi = maxi Ej hijl, so that!hut < hul. 8

For the sake of simplicity let us consider only the case where t is real. Let J denote a real interval o < t < a, and by B any polydisc B = [zj|zil <a, i,...,v] in the complex z-space C. We shall denote by u and v also any complex n nv n- or nv-vector variable, and by B, Bh corresponding polydiscs in C or C We shall now consider the following assumptions: (hl) Let a, a, T be positive numbers. Let us assume that the functions h(t,z,u), g(t,z,u), f(t,z) are continuous in J x B x B and J x B a a T a respectively, and that for every t c J they are holomorphic with respect to (z,u) or z in the open set B x B, or B respectively. o (h2) There are constants N, N1, N2 > o such that for all t c I, z c B, u,u1,u2 c B we have lh(t,z,u)l < N, 0 Ih(t,z,ul) - h(t,zu2) I< N1IU1-U21, g(t,z,ul) - g(t,z,u2) < N21U1-U21 For every s, o < s < 1, we denote by H the space of all functions g(z), z (z',...,z ), with values in C1, which are continuous in B,holomorphic in B, equipped with the maximum norm topology in B SG SOG Lemma 1. For every o < s' < s < 1, g(z) E H Swe have 6g/kz. c HS, and for all z c B we have also s CY 6ig(z)/azj < a (s-s') max jg(), j = l,...,v, (11) where max is taken for 5 E B. so 9

Proof. For z c B we have S' ag(z)/6z, = (2itY) V (11-zi) (_2(-Z2)1..( z -Z )'g(G)di1...d~ where we can take for r the oriented manifold r = [%1 |i-zi = (s-s')n, i = l,...,v], and c B. Analogous relations hold for the other first order partial derivatives. By taking absolute values we then have (11) for all z E B o< s' < s < 1. Lemma 1 can be interpreted as a property of the linear operators 6/6zj Namely, for all o < s' < s < 1 we have /6z ~:H + H j S st and the operational norm of /3zj. is < a (s-s'). We shall denote by Hn the space of n-vector functions u(z) = (ul,...,u), S z E B with ui E H i 1...,n, and norm iu|| = maxi u=II. We shall denote Tn n n by H the subspace of H made up of all u E H n with ||ull < Tr.'n Then for every t E J and u E H the following function U = U(t,z) = A(t)u a s is defined U(t,z) = (A(t)u)(t,z) = h(t,z,u)u (t,z) + g(t,z,u(t,z)) = g(t,zo), (12) with t c J,z c B,o < s' < s <l. a s' - a Lemma 2. Under the hypotheses (hl), (h2), the operator A(t) defined by (12) satisfies A(t) H' n and S s 10

IIA(t)u1-A(t)u2 1 < [Na + N1(Tc + 1)]11-U211 for all ul, u2 c H', o < t < a, o < s' < s < 1. S The first part is a consequence of Lemma 1. For the second part we have IIA(t)A(t)u1-A(t)u211 _, i [h(t,z,ul)-h(t,z,u2)]ulzll + |Ih(t,z,u2)(u 1u2z)1 + + IJg(t,z,u1)-g(tz u2)11 (15) < Nj1u1-u211,au (s-s') u1 llu + N a (s-s') Tu'-uUl2K + N21 Ui-U211S < [N2 + (N +Nl)a ( s-s')]) u-u2ll If we take C = N1t (No+Nl), (14) then (13) becomes?,HA(t)ul-A(t)u2l, < C|ul-u211 (s-s') since N1 -+ (No+NjT)a (s-s')' < [N2 + (NO+NjT)C-](s-s'- = C(s-s') Let M denote the constant t M max lu 0(z)t+f [g(T,z,o) + f(T,z)]dTj (15) 0 o10 where the maximum is taken for o < t < a. From Theorem 2 we deduce now 11

Theorem 3 (Cauchy-Kovalevsky). Under hypotheses (hl),(h2), and C, M defined by (14) and (15), let us assume that M < T. Then for all numbers b, s such that M < z. o < b < a, Ceb < 1, o < s < 1 - Ceb, (16) M[1-(l-s)-1Ceb]-< (17) there is a solution u(t,z) = (ul,...,u ) to problem (10) which is continuous in Jb x B,and for every t c Jb holomorphic with respect to z = (z1,...,z ) in the open set B. so Proof. First note that M < T by hypothesis, and now we can always choose numbers b, s satisfying relations (16) and (17). Indeed, first we can take b > o as small as we want, and then we can take s as close to zero as we want. The left hand member of relation (17) can now be made as close to M as we want and thus < T. Note that with the definition of A(t) in (12), the Cauchy problem (10) takes the form ut = A(t)u + [g(t,z,o) + f(t,z), u(o,z) = u (z) and now we apply Theorem 2. The method of successive approximations mentioned in No. 2 for the abstract formulation reduces now to the usual method 12

u (t,z) = o _-t t u (t,z) = u (z) + J [g(T,zo) + f(T,z)]dT, o uk(t,z) = u (z) + jt [h(T~zyu (T,z)]uk (Tz)+g(T zUk (T,z)+f(T,z)]dT, k 0 k-1 k-l,z 1 kk = 1,2,... For problems of the form n v aU/6t = Z Z= a (t, z)u /-zu + gi(t,z,u) + f(t,z), u.(o,z) = U i(z), i = 1,.,.,n, we take constants N N. such that lij~, 21 Ihij (t,z)l < Nij Igi(t,z,ul) - gi(t,ZU2) < N2i 1 U1-U21 for all t c J, t e B, u, u, u2 E B, and n v N = maxi Z Z ij. N1 N= max N, 0 1 iij._ 21 j=1 i=l -1 C = N2 + N 0 t t M = max luoi(z) + f [gi(t,z,o) + f.(T,z)]d where the last maximum is taken for all i = l,...,n, t E I, z c B Iar3n a Remark. In Theorem 3 we can take b = a whenever M < T, Cea < 1, and then u(t,z) is continuous in J x B and for every t E J holomorphic with respect a so a to z in B for any s& such that o < s < 1 - Cea, M[1-(l-s) Cea] < T. 13

For linear problems n V n aui/at = Z Z hi (t,z)auj/z + E g..(t,z)u +f.(t,z), j=l ~=l j=l 18 (18) u(o,z) = u o(z), i = l,...,n, 0o Theorem 3 yields Corollary. For linear problems (18) where the functions hij (t,z), gij(t,z), f.(t,z) are continuous in J x B and for every t E J holomorphic with respect ~1'~a a to z in B, and the functions u i(Z) are holomorphic in B, there is a unique a 01 CT, there is a unique solution u(t,z) which is continuous in J x B and for every t c J holomorphic a so a in B for all s, o < s < exp(-Cea). so Proof. We have here Ih ij(t,z)l < N ij I gij (t,z)l < N2i for suitable -ij~ 21j constants N.ij~, N and all t E J, z E B. We then take n v N =max. Z Z N = o 0 1 jZl 1=l 113. n N2 = maxi, C = N2 +N o i j=l 2ij 0 t M = max lu o(z) + f i(Tz)dTJ, where the last maximum is taken for all t E J, z E B, i = l,...,n. Let. > 1 be an integer such that, for b = a/p we have Ceb < 1, that is,p. > Cea, for (N2 + N o )ea < p. Then the solution u(t,z) is continuous in [o,b] x B, and for every o < t < b holomorphic with respect to z in B~ for o < s < 1 - Ceb, so say for all o < s < 1 - Ceb - r, where o < q < 1 - Ceb is arbitrary. We can now repeat the argument in the interval b < t < 2b, where now u (z) is replaced

by u(b,z) and t = o by t = b. Then u(t,z) can be continued in [b,2b] x B for So all s with o < s < (l-Ceb-v)2, and so on. By repeating this argument p times, we conclude that the solution u(t,z) exists and is continuous in [o,a] x B SG and for every t E [o,a] is holomorphic with respect to z in B~ for all s with Su o < s < (l-Ceb-j)4. Since Ti is arbitrary we see that the statement is true for every o < s < (1-Ceatl )4 where p. is any integer p. > Cea. As p - + +o we see that the statement is true for every o < s < exp(-Cea). 4. APPLICATION TO PERIODIC SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS We consider now a partial differential system as (10), or (15), where the second members are periodic in t of some period T = 2t/w. The question of the periodic solutions of period T of such systems in a thin cylinder E1 x B for so s, o < s < 1, sufficiently small, can be reworded as the problem of determining a function u (z) holomorphic in B~ such that the solution u(t,z) with u(o,z) = = u (z) can be continued in a cylinder [o,T] x B such that 0 s Sa u(T,z) = u(o,z) = uo(z), z C B. (19) Then, by periodicity, u(t,z) can be extended to the whole of E1 x B so For linear systems (16) we have seen that the first requirement is always satisfied. Note that u(t,z) depends functionally on u (z), and we shall denote it by u(t,z;u ). Thus, relation (19) becomes u(T,; u) = u, (20) 15

where u in the first member is a holomorphic function in B, or u c H1, while u in the second member is only a holomorphic function in B, or u e H o S 0 S o < s < s < 1, and suitable s 0- 0 - A situation where problem (20) can be handled easily in the case where system (10), or (18), contains a small parameter, and for ~ = o the reduced system is known to possess periodic solutions of period T. In two of the examples below we use an implicit function theorem in functional analysis we have proved elsewhere [lb]. Example 1. Let us consider the problem of periodic solutions of period T = 2t of the equation ut = EUz + sin t, (21) where ~ is a small parameter, v = 1, n = 1. If we take an arbitrary function u (z), z E B, u C H1, then the method of successive approximations t Ul(t,z) = o, uk(t,z) = u (z) + f [~u (T,z) + sin T]dT, k = 1,2,..., yields u (t,z) = uo(z) + 1 - cos t ul(t,z) = U (Z) + Etu'(Z) + 1 - Cos t, (') = d/dt, o 0 U2(t,z) = u (z) + ~tu'(z)+ 2 82t2u (z) + 1- cos t, 0 0 and hence uk(tz) = 1 - cos t + u (z) + 8tu'(z) + (E2t2/2')U"(z)+...+(Ektk/k')u(k )/z). 16

By the previous analysis we know that the series u(t,z) = 1-cos t + X (ktk/k)u (k)z) k=o is uniformly convergent in [o,2t] x B for all s, o < s < s, and e, I~ < Eo and suitable s > o Eo > o. The problem of determining u (z) in such a way that u(2A,z) = u (z) in B reduces here to 0 50 k k (k) (E t /k.)u (z) = o, z B k=l for all E in absolute value sufficiently small. The only solution is uo(z) = o, or u (z) = C, a constant. Thus, all periodic solutions of (21) for ~ in absolute value sufficiently small are of the form u(t,z) = C + 1 - cos t. Example 2. Let us consider the problem of periodic solutions of period T = 2ri of the equation t = E(u + u ) + sin t, where ~ is a small parameter. If we take an arbitrary u (z), z c Ba, u e H1, then the method of successive approximations t u (t,z) = o, u (t,z) = u (z) + [( (,+u (,z)) + sin T] dT, = [Uk o k-ioy i+k-ezz k = o,1,2,..., yields uo(t,z) = uo(z) + 1 - cos t 17

ul(t,z) = u (z) + Etu (z) + Etu'(z) + ~(t-sin t) + 1 - cos t u2(t,z) = u (z) + Etu (z) + (~2t2/2)u (z) + etu'(z) + ~2t2U'(z) + + (62t2/2)u"(z) + 1 - cos t + ~(t - sin t), 0 where (') = d/dz. At the limit as k -+ o we have u(t,z,~) = 1 - cos t + E(t - sin t) + u (z) + Et(u (z) + u'(z)) + o(E2) for all (t,z) c [o,2T] x B, and all s, E, o < s < s < 1, IEI < ~, and suitable s and E positive. o o We now discuss the question as to whether u (z) exists such that u(T;z,E) = = u (z), Izi < s'a, at least for all Izl < s'o, o < s' < s < 1, and s' sufficiently small. If u (o) = C, an arbitrary constant, we see that a necessary condition is that u'(o) = - C. We shall now write u in the form u(t,z,E; 0 uo(0), I|I < o) to emphasize that u depends functionally on u (z). Also, we take v(z) = u'(z) as a new unknown function, and we write jt for v(o) = u'(o). o 0 Then the functional equation u(2T,z,e) = u (z) takes the form 0 W(z,;V(O), | 5j < a) = E [U(2t,Z,,~,UZ (), i K < aC) - u (z)] z = u(2T,z,E, c + f v(a)da, [~l < o) - c - f v(a)da = o, IZl < sa, o o where W is holomorphic in z, where W(o,o,-) reduces to c + i = o, and the functional determinant aW(o,o, )/6a = 1 / o. By an implicit function theorem similar to the ones in [lb], we know that there are numbers s', o and a function u (z;~), jz < a, |E~ <, O < S' < S < 1, 8 > o, such that u(2,z,~E)-u (z,E) = o 18

for all zl < s'Y, |1J < ~. Also, U0(ZE) = C - CZ + O(~) + 0(Z2) u(t,z,8) = uo(z,e) + Et(u (z,e) + u'(z,~)) +1-cos t + E(t - sin t) + O(). Example 3. Let us consider the problem of periodic solutions of period T = 2T of the equation utt + = ~(lu2)ut + uXj= cos(wt + Ca) + 8 P u, (22) where v = 1, n = 1, T = 2Tr/a, c > o, p, p are real constant, and ~ is a small real parameter. For p = o, equation (22) reduces to the usual van der Pol equation with a forcing term ([19], (8.5.26), p. 133). The usual transformation Yl = iWu + ut, Y2 = iWu - ut yields the system of partial differential equations Ylt = iLy1 + ~f, Y2t = -iy)y2 - Ef 7 _l[1 + (2w22 f = 2 [1 + (2u) (Yl+Y2)2] (Yl-Y2) + pw cos(wt + C)+(2i) - (y + y Finally, the transformation Y1 = e Y1, Y2 = e Y2, yields - it iLot, Ylt e F, Y = e Fit 2t -1 _ 2 iLt -iLt 2 iCt -iLt F(t,z,Y) = 2 [1+(2o) (e l + e Y)2] (e Y1 - e t2) - 1 iLt -iot + ro cos(ot + o) + (2io) 5(e Y1 + e Y1), 19

where Y = (Y1,Y2). If Y (z), Y (z), IzI < a, denote arbitrary functions, the method of successive approximations ( __1) (-1), - 0, Y2 O, (k) t -ict (k-i) Y1 (t,z) Y (z) + E f e F(T,z,Y (T,z))dT 10 (k) -f; iw~t F(~,zY~k-l)t (i) Y2 (t,z) = Y (Z) - a i e F(Tz,Y (T,z))dT 20 O k = o,l,2,..., yields first (o) -i ic- -i 2iwt-isa -is Yi (t z) = Y (z) + Ex [2 t e - (4iu) (e -ei) 10 Y (t,z) - EP [2 t eia + (4i)-l(e2i( t + i-e ia)] Y2 (tyz) = 20 - ap L Also we have (1) -(z) -1 -2iot Y1 (t,z) = Y (z) + {2 [t Y (z-2iu) e Y () + 2 (2) [(2iu) e Y 3(z) +tYl(z)2o( - (-2i) -1e Y (z)Y 2(z) 10 10 20 -1 -4it 3( - (-14i) e Y (z) ] 20 -1 i-1 -2iWt-ct -iCY + rxi [2 -t e - (4iic)-l (e -e + (2iw)-l [t Y (z) + (-2iw)- e Y2Z (Z)] + *** 10Z 2oz (1) (z) - a (2- [(2iw) -1 2iit 2 Y2 (t,z) = Y20 (z) - 2 [(2iw) e Y (z) - t Y (z)] 20 10 20 + 2 (2)- [2(4iw) -1e ~ Y (z) + (2iw) e Y (z) Y (z) 2)1 -2iwt 3 - t y (z) y 2(z) (-2i) e Y (Z)] 20

-'~ 2i~flt, + ia ia -1~ xO [(4iw) (e -e ) + 2 t e ] l 2ie t + (2iXw)- [(2i2i) e Y + t Y (z) ] +... ), 1OZ 2OZ 0 0 where we have written only the terms in E and E. Only these same terms in E (k) (k) and ~1 are in all successive approximations Y1 (t,z), Y2 (t,z), and thus in the limits Yl(t,z), Y2(t,z) as k + oo The equations E [Y1(T,z) - Y10 (z) ] = o ~ [Y2(T,z) = Y0(z)] = o yield now _2 2 i 1 Y(z) + (2w) Ylo(z)Y2o(z) + +powe + (iw)-1 Y + O(E) = 0o Y (z) + (2 Y) Ylo ( z) - Dwe -(iW) O(E) = O 20 10 20 20Z If we take Y1(z) = k(z)e,(z) Y (z) = -(z)e (i), A, 0 real, then = i$ -iO IY = (k + ik$ )e, Y = (- + ikO )e, and we obtain the real equations X 3 _ -2=x _ 4pw3 cos (C-0) - I0_ + 0(E) = O, (23) pw sin (a-o) - O1 5 k + 0(E) = o For E = o, =- O the equations reduce to the usual ones for periodic solutions of the van der Pol equation with forcing term ([la]), (8.5.27), p. 133). Note that the equation A3 - 4o2k - 4ps3 = o has certainly a simple positive root X o If 5 is small, we shall think 0 close to a, sin (a-e) close to zero, cos (a-e) close to 1, and X close to X For every vector A = (A1,A2) in a suitable neighborhood V of (o,o) let us denote by \(A) and 9(A) the roots close to X and a respectively of the equations 21

-3 - 2 4 - hpw3cos (a-0) = 41XA -1 p sin (a-e ) -= X A2 We shall now denote by W = (W1,W2) in vectorial form the functional defined, for Izl < sa, El < E, by the first members of the equations 0 W1(z,a; A(), < ) = 3z42 _ 4p32 cos(a-0)-4wXA1 + o() -1 W2(z,a; A(), <I a) = pw sin (a-0) - Xc A 2 + 0(E), where x = K(A), 0 = 0(A). For z = o, E = o these equations reduce to the same equations without the terms O(E), they can be satisfied by taking k(o,o) = k(A), O(o,o) = O(A), and the functional determinant of W(o,o,-) with respect to A is then (-4cowg(o,o))(-w 5) = 4p2X(A), a number close to 4p2%, and hence f o. By the use of the same implicit function theorems mentioned in example 2 we conclude that there is a solution A1(z,E), A2(z,E) or k(z,E), 0(z,E), to equations (23), for all I z! < s'o, aE~ < 1,. and some s' < < S1 < 1, 0 <, < E. The solutions Y1, ~2 then have the form Yl(t,z) = i(z,E)e + 0(E), Y2(t,z) = -.(z,E)e + 0 (), and hence -1 x(t,E) = w \(z,E) sin [0(z,E) + wct] + 0(E), is a periodic solution of period T = 2T/ow of equation (22), with Izl < s'a, KE < E1 and where Al(o,o) = X (o,o), A2(o,o) = E (o,o), are arbitrary numbers, (A1,A2) C V. 22

REFERENCES [1] L. Cesari (a) Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 2nd Edition, 1963. (b) The implicit function theorem in functional analysis, Duke Math. J. 33, 1966, 417440. [2] L. V. Ovcyannikov, A singular operator in a scale of Banach spaces, Soviet Math. Doklady, 163, 1965, 1025-1028. [35 I. G. Petrovsky, Lectures on partial differehtial equations. Interscience 1954. [4] F. Treves, Ovcyannikov theorem and hyperdifferential operators. Inst. Nat. Pura Appl., Rio de Janeiro, 1968. 23

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