THE UNIVERSIT Y OF MICH IGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 11 FUNCTIONAL ANALYSIS AND BOUNDARY VALUE PROBLEMS Lamberta. Cesari.: " ^/., *' ' ' ~ 0-:-;?, ~, *.' *, *o* -, -ORA' Pro.jsec t 02,416; 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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(*) FUNCTIONAL ANALYSIS AND BOUNDARY VALUE PROBLEMS( Lamberto Cesari 1. A Review of Research 1.1 EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS OF PERTURBATION PROBLEMS In the years 1952-60 a great deal of work was done by a group of us (Cesari [5a,b, 1, 6a,b], R. A. Gambill [2, 9a,b,c], J. K. Hale [10, 12a,b,c,d,e,f,g,h,i,j], and others) in relation to periodic solutions of perturbation problems for ordinary differential equations. This work was based on functional analysis, and particularly on fixed point theorems, projector operators, and bifurcation equations in terms of functional analysis. We obtained criteria for the existence of periodic solutions (harmonic, subharmonics, ultraharmonic) of systems of periodic ordinary differential equations of the perturbation type du/dt = Au + ~f(t,u,~), u = (ul,...,un), f(t + T,u,6) = f(t,uE), criteria for the existence of cycles of systems of autonomous equations du/dt = Au + ~f(u,~), u = (ul,...,un), *Research partially supported by US-AFOSR research grant 69-1662 at The University of Michigan. This paper has been read at the Symposium in Differential Equations at Western Michigan University, Kalamazoo, Michigan, April 30-May 2, 1970. 1

and methods for the analysis of the asymptotic stability of periodic solutions, and of asymptotic orbital stability of cycles. For the particular purpose of a new and convergent method of successive approximations was devised. In particular, J. K. Hale proved that nonlinear autonomous perturbation type systems of ordinary differential equations may easily present families of periodic solutions, or cycles, under suitable conditions of symmetry. J. K. Hale gave also simple criteria for this occurrence, and for the determination of the dimension of the family of cycles [12c, 5b]. I presented aspects of this first phase of the research in [5c] and in my book ([5d], pp. 66-78, and pp. 123-135). Halanay reported the method of successive approximations in his book ([11], pp. 308-317) together with the proof of its convergence and applications. J. K. Hale gave a detailed account of aspects of this first phase of the research in his books ([12k], pp. 27-94). This work on periodic solutions of perturbation type problems in ordinary differential equations was then continued by C. Imaz [17], and by J. Mawhin [21a,b,c,d]. C. Imaz discussed particularly the case in which the underlying linear problem has multiple characteristic roots. J. Mawhin, in an extensive and systematic investigation still underway, has improved some of the results of J. K. Hale and R. A. Gambill. In view of numerical analysis C. Banfi [3a,b] and C. Banfi and G. Casadei [4] considered a variant of the method of successive approximations, suited for high speed computers, and in which the actual successive approximations and the analysis and solution of the bifurcation equation are combined. They successfully experimented with the method. 2

1.2 BOUNDARY VALUE PROBLEMS FOR GENERAL NONLINEAR PROBLEMS In my paper [5e] I presented the underlying ideas of the method, in an abstract form, for general boundary value problems for ordinary and partial differential equations, even, strongly nonlinear. Also, I showed connections of the method with Galerkin's approach. In [5e] I essentially considered problems whose underlying linear formulation was selfadjoint. J. Locker [20a,b], and J. K. Hale, S. Bancroft, and D. Sweet [13] gave suitable extensions of the method to "nonselfadjoint" problems. In [5f] I considered the problem of periodic solutions of systems of periodic ordinary differential equations nonnecessarily of the perturbation type, from the view point of the method as presented in [5c]. Also I proved in [5f] that the method contains as a particular case the much simpler approach used earlier for perturbation problems only. My work in [5f] was extended to problems of periodic solutions of ordinary differential equations with time lags by A. M. Rodionov [25]. S. A. Williams [27] showed that the method under consideration, as described in [5c], has a strict theoretical link with the Leray-Schauder approach [29]. Hale, Bancroft, and D. Sweet in their paper [13] emphasized other connections with previous or current work of D. C. Lewis, H. A. Antosiewicz, Jane Cronin, R. G. Bartle, L. Nirewberg, and Y. Sibuya. I presented aspects of this second phase of the research in [51.]. Jane Cronin reported the main idea of the method, as presented in [5f] for periodic solutions only, in her book ([8], pp. 180-185). J. K. Hale gave a detailed account of aspects of this second phase of research in his books ([12k], pp. 96 -99, and [12m], pp. 252-290). 3

For the particular problem of periodic solutions of ordinary differential equations, nonnecessarily of the perturbation type further theoretical work was done by H. W. Knobloch [18a] in connection with the use of the uniform norm, suitable estimates of the Fourier approximations, and a discussion of the bifurcation equations for any order of approximation by topological considerations based on C. Miranda's version of Bro wer's fixed joints theorem, version which had been already used in [5b]. H. W. Knobloch proved then simple existence theorems [18b] for periodic solutions of nonlinear periodic equations yt + g(t,y,y') = o, based only on qualitative properties of g, and this analysis led him to comparison and oscillation theorems for the same equations [18c]. In [5g,i,j] I studied periodic solutions in x and y of the partial differential equation u = g(x,y,u,u, ) xy x y where u is an n-vector, in the frame of the same method as presented in [5e]. By assuming g continuous in all its argument and Lipschitzian in u, u, or in u, ux, uy, I obtained criteria for the existence of periodic solutions u(x,y) continuous in E2 with u, u and u. In [5h] I applied the same approach to x y xy the problem of solutions u(x,y) to the same partial differential equation above, which are periodic with respect to x only and continuous with u, Uy, u in a x' y^ xy thin strip [- o < x < + o, IYI < a]. The results of [5g,i,j] yield analogous results for the nonlinear wave equation tt - u = g(t,x,u,ut,u ) By the use of the same method, based on [5f], and actually the same projection 4

operators, more stringent quantitative estimates have been successively obtained by A. K. Aziz [Proc. Amer. Math. Soc. 17, 1966]. J. K. Hale [122], by the same method applied directly to the nonlinear wave equation, and the use of a different choice of projector operators, obtained perspicuous criteria for doubly periodic solutions in E2. Recently D. Petrovanu [24a] has used the same method in the study of periodic solutions of the equation u = g(x,y,z,u,u,u,u ) xyz x y z and in the study of the periodic solutions [24b] of the Tricomi system of equations u = g(x,y,u,v), v = h(x,y,u,v) where u and v are m- and n-vectors respectively. A. Naparstek [22] has also used the same method for periodic distributional solutions of the nonlinear wave equation - a2u = g(t,x,u) Utt - auxx where g is periodic (of period 2ir) in x and y, and either a is a rational number, or an irrational one of a known class of real numbers badly approximated by rationals and everywhere dense on the positive real line. The result extend to the problem of periodic solutions of equations of the form utt - a2u = ~g(tx,,UtU x) and required a great deal of mathematical investigation. The bifurcation equation is studied in this work in terms of the theory of monotone operators. 5

In the same line W. S. Hall [14] has recently proved the existence of doubly periodic solutions of partial differential equations of the form utt+ (-1) D u = Ef(t,x,u,ut,ux,...,DP u) in suitable Banach spaces of periodic functions and distributions. The deep analysis which was needed for this work includes a discussion of the bifurcation in the lines of Minty and Browder's theory on monotone operators, a discussion of suitable boundary conditions for the problem under consideration, and smoothness properties of the solutions. E. M. Landesman and A. C. Lazer [19] have recently proved a general existence theorem for the Dirichlet problem for nonlinear partial differential equations of the form Lu + au + g(u) = h(x), where L is a general second order selfadjoint uniformly elliptic operator in a domain G, with bounded measurable coefficients, where a is a positive constant, h is a given function in L2(G), and z is a monotone continuous real valued bounded functions on El. The very subtle proof makes use of a technique which has points of contact with the one of the method under consideration. Cesari [5k] by the use of the process described in [5f], has discussed the existence of solutions u(x,y) for the boundary value problem Au = g(x,y,u) for (x,y) e A, u = o for (x,y) E A, where A is the unit circle in the xy-plane, 6A its boundary, g is measurable in 6

in x,y for every u, and Lipschitzian in u for every x,y, and where the solutions u(x,y) are required to be continuous in the closed circle A U aA, with continuous u, u in the open circle A, and Au = 62u/ax2 + a2u/sy2 — in the sense of the x y theory of distributions-is a measurable function in A. Recently, W. A. Harris, Y. Sibuya, and L. Weinberg [15] have used the same approach under consideration here to obtain new and extremely simple proofs of the classical theorems of Cauchy, Frobenius, Perron, and Lettenmeyer on linear ordinary differential equations and systems in the complex field. These authors are actually concerned with systems of the form s. n z du./dz = a..(z)u., i = l,,n, I j=l1 where z = x + iy, and the coefficients a.i are holomorphic functions of z in a complex neighborhood V = [jzl < &] of the origin z = o. In the regular case (all s. = o) the bifurcation equation is trivial, and the system has a fundamental system of holomorphic solutions in V. In the regular singular case (all s. = 1) the bifurcation equation reduces essentially to the indicial equation, and the system has the expected number of solutions of the forms zPP(z), p complex, P(z) a convergent power series of z. For d = max s. < n, the given system has at least n - d solutions holomorphic in V. 7

2. A Direct Proof of Cauchy-Kovalevsky's Theorem 2.1 FORMULATION OF THE PROBLEM The Cauchy-Kovalevsky theorem for partial differential equations in the complex field is usually proved by the method of majorants, which, for the problem under investigation, leads to a first order ordinary differential equation with separable variables, whose solutions are then proved to be majorants of the solutions of the given system of partial differential equations. All this is avoided in the following simple proof (No. (2.5) below), based on functional analysis. Let S denote the class of all functions u of t and z =(zl,...,z ), which V are holomorphic in some complex neighborhood of the origin (t = o, z = o), and I ml mv thus possess power series expansion u(t,z) = Z~ Z u, t Z1...z, ~ m m v m = (mi,...,m ), which is convergent in some neighborhood of the origin. For the sake of simplicity we limit ourselves to the linear Cauchy problem, and we know that it is not restrictive (see I. G. Petrovsky [32], pp. 16 -17) to formulate the linear Cauchy problem as follows: Given elements a.ih, bij, ci E \ determine elements ui E C Ssuch that n v n Uit = Z Z a ijhuj + Z b. + c (1) j=l h=l ijh zh j=l j ' u(o,z...,z ) = o, i = l,...,n 1 V in some complex neighborhood of the origin, and where uit, u. denote partial It UJZh derivatives with respect to t and zh, h = l,...,v. If U(t, z) = (ul,...,u ), V 8

and U = [u., i = l,...,n, h = l,...,v], then (1) takes the form U = F(t,z,U,U ), U(o,z) = o U Z The solution U = (ul,...,u ) e n of problem (1) is uniquely determined by n i ml "mv the usual argument: If e ui t zl...z, i = l,...,n, are power series Zm iu tz z,lm,.n v satisfying formually Ut = F(t,z,U,U ), and the coefficients u. are known, then t z lorn all remaining coefficients can be determined by induction. Indeed, each power m mi m t zl...z appears in the first member of (1) with coefficients (1+l)u., and in the second member with coefficients E which are finite linear combinailm tions of coefficients uj., s = (s,...,s ), j = l,...,n, with o < k < ~. For jhs v - - a solution U E n of problem (1) all u. are zero. lom A proof of the following theorem will be given in (2.5). (2.i) (Cauchy-Kovalevsky). If all aijhbij, ci are elements of, then there is a unique element u e S' satisfying (1) in a complex neighborhood of the origin t = o, z = o. 2.2 BANACH SPACES OF HOLOMORPHIC FUNCTIONS For 5 > 0, 0 < a < 1, and k > 0 integer, let Sk = S k be the class of all elements u E S"with coefficients um for which ullk Sup, m m u:m.'ml!...m vk'I((+ml+...+m +k):) ~ml+' '+m V < o Then Sk is a Banach space with norm lul|k = l||ul k. Actually, we shall need below only S and Si 9

If u E Sk, then ut, uz E Sk+ with Ilutilj (k+l)()k+l -1l, Ilu | k+1 < (k+l)-1 Iu|llk, h = 1,...,v. Conversely, if v E Sk+l and u(t,z) = f v(T,z)dT (formal integration), then u e Sk and I|u| = (k+l) (c5)||v|kl 0 K k k+1 -k-1 If 0 < 6 < a, u E Sb5, v E S, then uv E Sck and l|uvl|lok < (1-5/a) II UI.ll I vl I ". Note that S k1 S ak and that u e Sk implies |lull > lullSk+ Also, for 0 < a < p < 1, we have Sak Ssk and l||u||b < |ull|. For U(t,z) = (ul,...,u ) E (S5 )n, that is, each u E S5, we take n O t. ce take n 1/2 |ullak = ( Z ||ui2 ) /. Note that if u E Sk then u has the majorant OUsk. i kuiKl ) i=l |ullk(l-5-(t/cz + zl+...+zv)) -, the same kind of majorant used in the classical proof of Cauchy-Kovalevsky theorem (see I. G. Petrovsky [32], pp. 18-24). 2.3 A PRIORI ESTIMATE Let a > o be such that aijh, bij, c. S o, and take A = Max[llaih o, lbij 1lo]. Then, for every o < 5 < a, o < a < 1, we have IlaiJhlI, llbijllo < A. If C denotes the n-vector C = [ci, i = l,...,n], then we have also |ICII| ollICIlaIo Let us choose 5 and a so that o < 5 < 2-a, o < a < 1, 4An2(6 + v)a < 1. Let us prove that for any solution U S = (S ) of (1), if any exists, we have O 6Oo 1|U|l < a(1 - 4n2A(v + )a)-1 llCo = lCll~. (2) Indeed, if U E S then Ut, U E S = (S h = l,...,v. On the other n Zn t hand, U(t,z) = /U ((,z)dT, hence l||Ul = caS||Ull. Thus, o T 0 t 1 UIi0 = cbl||ltl = al||F(t,z,U,U )ll 1 1 < c [n2v(l-/a)-z2A (MaxUlll i + + n2(l-C/o)-zAiUI1 |c|], h i 10

where 1 - -/a > '2l, lUl < 6'lull, |U|I < Iu, IICI_ < aiCi We obtain where1 -r 5/zh 1 ~ 1 - 1 I|U| < a6[4n2vAS-11|1ul + 4n2AI||UI + I||c| ], which yields (2). Note that relation (2) yields another proof that there can be at most one solution U E 5to problem (1). Indeed, any two solutions U1, U2 E 5jmust belong to the same space S~ for 6,a chosen as above, and then U1 - U2 is a solution sao of (1) with C = o. By (2) we have then I11U-U211 = o, or U1 = U2. 2.4 PROOF OF THE EXISTENCE OF A SOLUTION TO PROBLEM (1) Let us denote by H the operator of integration with respect to t from o to t already considered in No. 3. Then, problem (1) is equivalent to U(t,z) = H F(t,z,U,U ). z Thus, if T: Sin + J'n denotes the operator defined by, V = TU = H F(t,z,U,U ) all we have to do is to find the fixed points of T. Let a, A, 6, a be chosen as in No. 4 so that k = 4n2A(v+5)a < 1. Note n n n n that, if U E S, then U S and F(t,z,U,U ) H F E S and T n n n is actually a map T: S S. If U, U S, then 0 0 0 ||TU1 - TU211 = a&l TU1 - TU211 = 0 l = cx||F(tzUU1) - F(tzU) - F(t,z,U2,U) 1 11

< a[n2v (1-/ar) A (maxll U - U I ) Zh 2Zh + n2(16/a)-2 AllU1 - U211 ] 1 < 4n2A (v+6) o|lU1 - U2j1 = k 1|U1 - U211 00 Sn n n Since k < 1, T: S S is a contraction in the norm || || of S, and possesses, therefore, a unique fixed point U e Sn n 0 2.5 PROJECTION OPERATORS For fixed integers N > 1, M > 1, let us denote by PN PM:, the projection operators defined by t le Ml N ml m P_.(,2Lu. t z"I..z' -- Z Zu t z1...z N"' MUm t v 0,=o m m v z m. mV M M t ml mv Pt( z u t zi...z = Z.. t M mvm v m1= o m =o m v v so that we have also PN,P: Sk > Sk. For fixed N and M, both P S and PS are subspaces of Sk, actually Banach spaces in the same norm Ilull of Sk. k k For any fixed M > 1 we may consider the Cauchy problem analogous to (1): Ut = PM F(t,z,U,Uz), U(O,z) = (5) Z n for which we seek solutions U e (PM S ). The argument of No. 2 leading to the uniqueness of a solution U, if any exists, repeats in the present situation. The same holds for the argument of No. 4 leading to an a priori estimate. Thus, and solution U E (pZ S )o of problem (3), if any exists, is uniquely determined and satisfies relation (2). 12

Analogously, for fixed N, M > 1 we may consider the Cauchy problem analogous to (1) and (3): Ut = t PZ F(t,z,U,U), U(O,z) = 0 (4) t N-i M for which we seek solutions U e (PN PM S ) (polynomials in t and z). Both the N M o argument leading to the uniqueness of a solution, and the argument leading to the a priori estimate hold in the present situation. But now U is a polynomial, and thus we conclude that a polynomial solution U E (PN PM S ) of problem (4) exists and U satisfies (2). 2.6 AN ALTERNATE PROOF OF THE EXISTENCE OF A SOLUTION TO PROBLEM (1) For every M = 1,2,..., let us take N = M in problem (4), and let UM(t,z) tz n M be the polynomials so obtained, l E (PN P S ). Let u be the coefficient N M o i m of the polynomials UI. All these U are elements of the unique Banach space S, and satisfy the a priori estimate J||Ju < p, or (2), as proved. Let u. be the coefficients of the formal power series expansions corresponding to problem (1), as mentioned in No. 2. Then for every fixed i and m = (ml,...,m ) and V M all M large enough, we have u = u. Hence, also u are the coefficients of an element U e S and ||U| < I. Thus U e S c <j, and U is a solution to 0 0 0 problem (1). 2.7 ANOTHER PROOF Let g(t,z) = (g,...,gn) be any n-vector polynomial, or element g E (Pt PM S )n, with g(O, z) = 0, and let giem, 0 < ~ < N, 0 < mh < M, denote Nthe coefficients of gi all i nd all im > N or m > M, being zero). the coefficients of g. (all gio.,m and all g with i > N or mh > M, being zero). l lomn i~m 13

n For every U(t,z) = (ul,...,u ) E S, let ui denote the coefficients of the power series of ui, and let H: S - S denote the operation of integration with respect to t as in No. 3, or HU = V(t,z) = (V1,...,V ) with 00 Ml ^+1 m^ V.(t,z) = Z Z ug (g+i) t z...z, i = l,...,n. Vi(t,z) U _ U lm (t+1) 1 l+l =N m t (NI, and also Hn (P )n Then, (6/6t)HU = U - PN1U, ||HU| < (N+l) ||U|, and also H: (P ) (Pq ). Finally, let T: (P S )n (PZS )n be the operator defined by taking V = TU = g + (I-PM) F(t,z,U,U). Note that, for U e (P ) we have Uh E (P So), and |IU zh| < M ||ul h = l,...,v. Then, for U, V E (P.S ) we have also, as in No. 4, I|TU - TVII < (N+l) (2AMn2v + 2An2) ||U - V|| By choosing N = N(M) sufficiently large, T is a contraction. The fixed point U = TU E (PM S ) is an element U = TU = Z g which depends on g, and which Mo satisfies U = g + H PM F(t,z,U,U ) with ui. = gim for o < i < N. By difM z iYM iim ferentiation we obtain U = PZ F(t,z,U,U ) + A, (5) t M z where A = g - PN P F. Thus, U = TU =' g is a solution of (4) if and only t N- 1 M if the determining (or bifurcation) equation t gt = PN- F(t,z,U,U ) (6) is satisfied. Here ui. = gi. for all K < N and gi =0, and the determination i am iam - orn 14

of a polynomial g satisfying (6) is a problem similar to (4). Thus g can be uniquely determined so as to satisfy (6), and correspondingly U = TU =Zg satisfies problem (4). If we denote by U e (PM S ) the element so determined, and we take M = 1,2,..., we have a sequence [U ] as in No. (2.6) and a solution U of the original problem (1) can be derived. 15

3. A Direct Proof of an Hormander's Theorem 3.1 STATEMENT OF THE PROBLEM The proof and the remaining considerations of No. 2 extend to a more general statement (Hormander [28], Th. 5.1., p. 116), from which Hormander deduces as corollaries the Cauchy-Kovalevsky theorem, a theorem by Darboux, Goursat, and Bendon, and a number of other statements ([28], pp. 118-126). For the sake of simplicity we limit ourselves to partial differential equations with one unknown function. The extension to systems is easy. Let denote the class of all holomorphic functions u(z), z = (zl,...,z ), V which are holomorphic in some complex neighborhood of the origin z = o, and m ml my thus possesses power series expansion u z, or u z...z, m mm m o m v (ml,...,m), |m| = ml+...+m. Hormander's problem is as follows: Given any multiindex p = (PD,..., ), I|f > o, and elements f, a, cp E S o < I|| < I|J, determine an element u E.5 such that in a neighborhood of z = o we have D u = \ZQ a(z) Dau + f(z), and (1) k D (u-cp) = o when z = o if o < k <. j =,...,v. (2) J j - A proof of the following theorem will be given in No. 3.5 below. As usual D u denotes 1 u/zil....z V 16

(3.1.i) (Hormander, [28] Th.5.1.1, p. 116). If |la (o)I is smaller I al =1 I than a positive number depending only on |1p, then problem (1), (2) has one and only one solution u E. 3.2 THE CLASSICAL UNIQUENESS ARGUMENT The substitution u = v + cp reduces the proof to the case cp = o so we may assume that cp = o from the beginning. Thus, we replace (2) by k D.u = o when z = o if o < k <, j = 1,...,v. (3) J J 3 If a,P are any two multiindices a = (a,...,a ), P = (l,..., ) we shall say V V that a < P [or a < P] provided a. < Pi [or a. < Pi] for all i = l,...,v. If a < P we shall write as usual (P) = (1)... (P) with (Hi) = I ae 'i a^7 '^ ^ ^-av): V 1 1 1 1 Given a = (3i,... i ) and any integer i> |i|, we shall denote by Ni the number of integral solutions of the equations I 1 = X, 2 > P, or 21 +...+ = X, - V 2i > i. i = l,...,v. Also, we shall denote by P the number of solutions of the equations Ial = IlP, or al + —. av = P1 +...+ p V V The solution u e J& of problem (1-3) is uniquely determined. This can be proved by a simple modification of the usual argument for the Cauchy-Kovalevsky theorem (see [5], p. 19). Indeed, if u E S satisfies (1-3), and d2, A, 2 = (al,..,2 ) denote the derivatives d Du, A = D a z, then all v 2 d z-o z=o derivatives d with at least one 2. < P. are zero. In other words, d = o 2f 1 1 2 for every 2 i. For 2 > P, the derivatives d can be obtained from Leibniz rule by differentiating (1) exactly 2 - 5 lines, that is, applying the operator 17

D- on both sides, and taking z = o. We obtain d= ~ )A Id e (4) and in this sum we may well restrict to only those terms with 2' + a - > (, or 2' > P. Note that, for any given 2 > P we have in (4) 2' < 2, 2 > A, 2' > A, | al < |, and finally 1 +a-p 1= |I('-)-)+al = I '-PI+I|a = | - I -IP+IaI < lQ I 1 21. Thus, for any given \ > o, x > ||I, 121 = X, we may have in (4) j'+a-PI = x if and only if |1'\ = 1|2, \j| = | a| In other words, the N. derivatives d, with I I| = X, 2 > A, satisfy the system of N equations d Z A d+ Ea = An j > 2 (5) where E. denote finite linear combinations of derivatives d with |s| < A, and where A = a (o). In each equation (5) there appear, therefore, besides the Xo unknown d. with coefficient one, at most t other unknowns do+ C- with +a-c = |XI = |I1, with coefficients A whose sum of the absolute values is < Z wo I =1 PI |a (o)|. Thus, if this sum is < 1, the Nk equations (5) have a unique solution ([22] p. 29). 3.35 BANACH SPACES OF HOLOMORPHIC FUNCTIONS For 8 > o and k > o integer, let Sk be the class of all elements u E ok ml mv u = u z1...z, m = (ml,...,m ), for which mm V V - m3l+. +m.. [u||k Supm [Jum m.'...m ' k. ((mj+...+m +k):) 8 ] <cc. (6) k m m vv 18

Then Sk is a Banach space with norm IlullSk. Note that SSk c S,k+1 and that for u e Ssk we have Ilullk > lull,k+ Also, S5 c S c.. c... c J, for u ~ S0 we have Ilull > Ilul8 >.. > I u |. For any given element u E S So S - _1 - - o 0o we can take 8, o < S < a, so small that Il||u| is a close as we want to |u |. If u ~ Ssk then each derivative D u ~ Sk+ with IIDx ull k+ < (k+l)S-'llullSk. Conversely, if v e S and u(z) = D v (formal integration), then u e Ssk S, k+i Xi x and II|u|Sk = (k+l)-1' Ilvii k+l. These are immediate consequences of the definitions. As a consequence, for every multiindex a = (al,...,a ) and u E SSk, we have D U E SS, k+I and llDuSI k+ l1 < + (k + 1).. (k+| a ) 1 ullk. Conversely, if v E S,k+lI and u = D v (formal integration), then u E SSk and II||usk = (k+l)-1...(k+l|a|)lSlaL 81 |lu k+ll. Note that, if o < 8 < a, u E S5k' v E Sak, then uv E S8k and luvllSk < (1 - /a) -k-ll||uk |ilv llk Also note that, if u E SSk, then uml < llUllSk (ml+...+m + k)' (ml'...m.'k.')1 5-ml-. -mv that is, u(z) has the majorant ulk (Z (ml+...+m + k).(ml:...m! k:)- lzml.z l -ml-.. -m or llullsk (1 - (Zl+...+z )5-)-k- -i -k-i Conversely, if u(z) has a majorant pi(l-(zl+...+z )5 - ), then u E S8k and lu11Sk < ' 19

3.4 A PRIORI ESTIMATE We prove here an a priori estimate for the solutions u E \5 of problem (1-5) above (that is, for problem (5.1.1) of [28], p. 116). (3.4.i) Given P, there is a number T = T(f) > o and, for all a E 4 acl < I|P with Z |a (o)| < T, there is another number 5 > o such lal = I| I o that, for any f E SEo, o < < 5, and solution u E 0Sof (1-3) (if any) with u E So, we have also uI6 < 52( )l: I 51I lfll8. (8) 0 0 Proof. Let T = 2. Given a e, |a| < |P|1 let a > o be such that a E S, lac < 1|1, and let A = Max [Ila o, |a| < I|P]. Then, for any r integer we have Iai|l < o aal < A for all c with I|a < |1. If iZ laa(o)I < T, rr - 'ao - I' *|a\ 3 -l then we certainly have 2 |1+ _ | a(o)1 < 2 -1 Let 5 be so chosen that o < < 2, 0 0 -2 Alalll <2 and such that Z1 1 ll < Z Ila(o)l + 2 for every 6, o < < b. Then k = 2I1+1^ Z 6sl-lal + 2'p+l( Z la (o)l + 2-I-4) (9) o Io'aI<II ~ I1=IPI -3 -3 -3 -1 <2 +2 +2 <2 20

Thus, for o < 5 < 6 we have 1-6/a > 2, and if k(6) is the number defined - o0 as in (9) with 6 replacing 6 and i(6) = 1 - k(6), we have o < k(b) < k < 2 -1 -1 1 > () > 1 - 2 = 2 Iff E So and u EJ S, u E So, is any solution of (1-3), then D u E S and u = D- (Du) (formal integration as in No. (5.5 )). Then from (1) and the remarks made in No. (3.3), illl,, = (t1!)O ' 615 |ID U1iET 15 = (II1:)- 1I || Z a DulI | -~lpf 6P < (j1Jfl' ) '1 [(l-o/a)"K <1 Z lPaIl 0 JD u11 + lIf1lI 6,I-] ffi1,-)_- ^^ [21ffi1+l A 0 1l<, (1~1-1A1+1)...(1~1)8 1 11ul,10,1_11 + 21I+ (j1j') 6-I~ ( Z la(o)l + 2 —4) 1llu1o + 11f11] < 211I AZ S 1'^ ' + 2i| ( Z Ia(o) I+2 2 '-)1ull+ (ll i:)pl 11 0 fl o 9 < k(b) II ull + (i1.')- 1 ' 1 or U150 (z(6)) -1 1 llfll, where bt(6) = 1 - k(6) > |j > 2, and (8) is thereby proved. - O 21

3.5 PROOF OF THE EXISTENCE OF A SOLUTION TO PROBLEM (1-3) Let us denote by H the operation of integration H = D already considered in Nos. (3.3) and (3.4). Then, problem (1-3) is equivalent to u(z) = H( Z a.u + f(z)) I d t a If T: G -- vJ' denotes the operator defined by v = Tu = H ( Z aaDu + f(z)), lallpl all we have to do is to find fixed points of T. -1 Let a, A, 8 be chosen as in No.(3.4), so that o < k(6) < k < 2 0O Note that, if u E S80, then D u e S o < Ia1 < p1, and E a a El P, ac'D Ce u + f) E~' I~1 lallPla D u ES,.S1 For f E S,0 we certainly have Halll + f) H S and T is, therefore, a map T: S5, -+ S,. If ul1U2 E So, then 1|Tul - Tu211 = IH ( Z a (ul-u,)) 11 I I I - I II1_<1 I~1 = Il)-811 Z a D (ul-,u)118, I _a~~ (1O1:) ~~1 -8/ )-I 1PI I aIl I1P P 2 11 [21 I 1A z (1 -1 -l)a)... (I B) 1,-, l- I-U2 I1 1 + + 211 +(li1:8-1 1( Z lacI(o)I + 2- 11P-41)1 IU-U211 < (1pl.)-)" i I c4=II I u < 2I+l Z l IPlIcl |+ 2 PI-( Z laa(o)I-+ 2'- )] ul-u2l a|l<lp|l o al=|pl -I = k()||lui-u21, 22

where k(8) < 2. Thus, T: SB -o S is a contraction in the norm |lul| of S, bo 60 6o0 b and possesses, therefore, a unique fixed point u e S8. Theorem (3.1.i) is thereby proved. Considerations analogous to the ones in Nos. (2.6-8) could be repeated here, and are omitted for the sake of brevity. 23

4. References A. REFERENCES ON THE CESARI-HALE METHOD [1] H. R. Bailey and L. Cesari, Boundedness of solutions of linear differential systems with periodic coefficients. Archive Rat. Mech. and Anal. 1, 1958, 246-271. [2] H. R. Bailey and R. A. Gambill, On stability of periodic solutions of weakly nonlinear differential equations. J. Math. Mech. 6, 1957, 655 -668. [3] C. Banfi, (a) Sulla determinazione delle soluzioni periodiche di equazioni non lineari periodiche. Boll. Unione Mat. Italiana, (4) 1, 1968, 608 -619.-(b) Su. un metodo di successive approssimazioni per lo studio delle soluzioni periodiche di sistemi debolmente nonlineari, Atti Acc. Sci. Torino, 100, 1968, 1065-1066. [4] C. Banfi and G. Casadei, Calcolo di soluzioni periodiche di equazioni differenziali nonlineari periodiche. Congresso AICA, Napoli, Sept. 1968. [5] L. Cesari, (a) Sulla stabilita-delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici.Mem. Accad. Italia (6) 11, 1941, 633-695. —(b) Existence theorems for periodic solutions of nonlinear Lipschitzian differential equations and fixed-point theorems. Contributions to the Theory of Nonlinear Oscillations 5, Princeton 1960, pp. 115-172.-(c) Existence theorems for periodic solutions of nonlinear differential systems. Bol. Soc. Mat. Mexicana 5, 1960, 24-41.-(d) Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 2nd ed. Springer 1965. —(e) Functional analysis and Galerkin's method. Michigan Math. J. 11, 1964, 385-418.-(f) Functional analysis and periodic solutions of nonlinear differential equations. Contrib. Differential Equations, Wiley 1, 1963, 149-187. —(g) Periodic solutions of hyperbolic partial differential equations. Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, 1963, 33-57. — (h) A criterion for the existence in a strip of periodic solutions of hyperbolic partial differential equations. Rend. Circ. Mat. Palermo (2) 14, 1965, 95-118. —(i) Existence in the large of periodic solutions of hyperbolic partial differential equations. Arch. Rat. Mech. Anal. 20, 1965, 170-190.-(j) Smoothness properties of periodic solutions in the —large of nonlinear hyperbolic differential systems. Funkcialaj Etvacioj, 19, 1966, 25-338.-(k) A nonlinear problem in potential theory. Michigan Math. J. 16, 1969, 3-20. —(I) Functional analysis and differential equations. SIAM Studies in Applied Mathematics 5, 1969, 143 -155.

[6] L. Cesari and J. K. Hale, (a) Second order linear differential systems with periodic L-integrable coefficients, Riv. Mat. Univ. Parma, 2, 1954, 55-61; 6, 1955, 159. —(b) A new sufficient condition for periodic solutions of weakly nonlinear differential systems. Proc. Amer. Math. Soc. 8, 1957, 757-764. [7] P. A. T. Christopher (a) A new class of subharmonic solutions to Duffing's equation. Co A Rep. 195, The College of Aeronautics. Cranfield, Bedford, England 1967. —(b) An extended class of subharmonic solutions to Duffing's equation. Co A. Rep. 199. Ibid. 1967. —(c) The response of a second order nonlinear system to a step-function disturbance. Co A Report 205, Ibid. 1969. [8] J. Cronin, Fixed points and topological degree in nonlinear analysis. Amer. Math. Society, 1964. [9] R. A. Gambill, (a) Stability criteria for linear differential systems with periodic coefficients. Riv. Mat. Univ. Parma, 5, 1954, 169-181.(b) Criteria for parametric instability for linear differential systems with periodic coefficients. Riv. Mat. Univ. Parma 6, 1955, 37-43.(c) A fundamental system of real solutions for linear differential systems with periodic coefficients. Riv. Mat. Univ. Parma 7, 1956, 311-319. [10] R. A. Gambill and J. K. Hale, Subharmonic and ultraharmonic solutions for weakly nonlinear systems. J. Rat. Mech. Anal. 5, 1956, 353-398. [11] A. Halanay, Differential Equations, Academic Press 1966 (particularly pp. 308-317). [12] J. K. Hale, (a) Evaluations concerning products of exponential and periodic functions. Riv. Mat. Univ. Parma, 5, 1954, 63-81.-(b) On boundedness of the solutions of linear differential systems with periodic coefficients. Riv. Mat. Univ. Parma 5, 1954, 137-167.(c) Periodic solutions of nonlinear systems of differential equations. Riv. Mat. Univ. Parma 5, 1954, 281-311. —(d) On a class of linear differential equations with periodic coefficients. Illinois J. Math. 1, 1957, 98-104. —(e) Linear systems of first and second order differential equations with periodic coefficients. Illinois J. Math. 2, 1958, 586 -591.-(f) Sufficient conditions for the existence of periodic solutions of systems of weakly nonlinear first and second order differential equations. Journ. Math. Mech. 7, 1958, 163-172. —(g) A short proof of a boundedness theorem for linear differential systems with periodic coefficients. Archive Rat. Mech. Anal. 2, 1959, 429-434.-(h) On the behavior of the solutions of linear periodic differential systems near resonance points. Contributions to the theory of nonlinear differential equations, 5, 1960, 55-90.-(i) On the stability of periodic solutions of weakly nonlinear periodic and autonomous differential 25

systems. Contributions to the theory of nonlinear differential equations 5, 1960, 91-114.-(j) On the characteristic exponents of linear periodic differential systems. Boletin Soc. Mat. Mexicana, 1960.(k) Oscillations in Nonlinear Systems. McGraw-Hill 1963. —(1) Periodic solutions of a class of hyperbolic equations, Arch. Rat. Mech. Anal. 23, 1967, 380-398. —(m) Ordinary Differential Equations. Wiley-Interscience 1969. [13] J. K. Hale, S. Bancroft, and D. Sweet, Alternative problems for nonlinear functional equations. J. Diff. Equations 4, 1968, 40-56. [ ] W. S. Hall, Periodic solutions of a class of weakly nonlinear evolution equations. Archive Rat. Mech. Anal. To appear. [15] W. A. Harris, Y. Sibuya, and L. Weinberg, Holomorphic solutions of linear differential systems at singular points. Archive Rat. Mech. Anal. 35, 1969, 245-248. [16] W. A. Harris, Holomorphic solutions of nonlinear differential equations at singular points. SIAM Studies in Applied Mathematics 5, 1969, 184 -187. [17] C. Imaz, Sobre ecuacious differenciales lineales periodicas con un parametro pequeno. Bol. Soc. Mat. Mexicana (2) 6, 1961, 19-51. [18] H. W. Knobloch, (a) Remarks on a paper of Cesari on functional analysis and nonlinear differential equations. Michigan Math. J. 10, 1963, 417 -430.-(b) Eine neue Methode zur Approximation von periodischen Loesungen nicht linear Differentialgleichungen zweiter Ordnung. Math. Zeit. 82, 1963, 177-197. - (c) Comparison theorems for nonlinear second order differential equations. J. Diff. Equations 1, 1965, 1-25. [19] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance. Journ. Math. Mech. 19, 1970, 609-623. [20] J. Locker (a) An existence analysis for nonlinear equations in Hilbert space. Trans. Amer. Math. Soc. 128, 1967, 403-413.-(b) An existence analysis for nonlinear boundary value problems. SIAM J. Appl. Math. 18, 1970. [21] J. Mawhin, (a) Application directe de la methode de Cesari a 'e'tude des solutions periodiques de systemes differentiels faiblement non lineaires, Bull. Soc. Roy. Sci. Liege 36, 1967, 193-210.-(b) Solutions periodiques de systemes differentiels faiblement non lineaires. Ibid., 36, 1967, 491-499.-(c) Familles de solutions periodiques das les syst&mes differentils faiblement non lineaires. Ibid. 36, 1967, 500-509.-(d) Degre topologique et solutions periodiques des systames differentiels non lineaires. Ibid. 38, 1969, 308-398. 26

[22] A. Naparstek, (a) Composition of functions in certain Sobolev spaces. To appear.-(b) Periodic solutions of weakly nonlinear wave equations in Sobolev spaces. To appear.-(c) On the Cesari method and periodic perturbation problem for certain hyperbolic equations. To appear. [23] C. Perello, A note on periodic solutions of nonlinear differential equations with time lags. Differential Equations and Dynamical Systems (J. P. LaSalle,ed.), Academic Press 1967, 185-188. [24] D. Petrovanu (a) Solutions periodiques pour certaines equations hyperboliques, Analele Stintifice Iasi, 14, 1968, 327-357. —(b) Periodic solutions of the Tricomi problem. Michigan Math. J. 16, 1969, 331-348. [25] A. M. Rodionov, Periodic solutions of nonlinear differential equations with time lag. Trudy Seminar Differential Equations Lumumba University, Moscow 2, 1963, 200-207 (Russian). [26] C. D. Stocking, Nonlinear boundary value problems in a circle and related questions on Bessel functions. Thesis, University of Michigan, 1970. [27] S. A. Williams, A connection between the Cesari and Leray-Schauder methods. Michigan Math. J. 15, 1968, 441-448. B. OTHER REFERENCES [28] L. Hormander, Linear Partial Differential Operators, 3rd Ed., Springer 1969. [29] J. Leray and J. Schauder, Topologie et equations fonctionelles, Ann. Sci. Ecole Norm. Sup. 51, 1934, 45-78. [30] L. A. Liusternik and V. J. Sobolev, Elements of functional analysis. Ungar 1961. [31] L. V. Ovcjannikov, A singular operator in a scale of Banach spaces, Soviet Math. Doklady, 163, 1965, 1025-1028. [32] I. G. Petrovsky, Lectures on partial differential equations. Interscience 1954. [33] F. Treves, Ovcjannikov theorem and hyperdifferential operators. Inst. Mat. Pura Appl., Rio de Janeiro 1968. 27

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