THE UNIVERSIT Y OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No, 16 FUNCTIONAL ANALYSIS AND DIFFERENTIAL EQUATIONS Lamberto Cesari ORA Project 07576 under contract with: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH CONTRACT NO. AF-AFOSR-942-65 WASHINGTON, D. C administered through: OFFICE OF RESEARCH ADMINI STRATION AINN ARBOR September 1968

LSg*!_ r, ki' V:

TABLE OF CONTENTS Page 1. THE SOLUTIONS OF AN EQUATION AS FIXED POINTS OF A SUITABLE MAP 1 24 THE RELAXED EQUATION AND THE DETERMINING EQUATION 3 3. A CONNECTION WITH GALERKINv S APPROXIMATIONS 6 44 THE MAP T AS A CONTRACTION 7 5. A CONNECTION WITH THE LERAY-SCHAUDER FORMALISM 9 6. A FEW EXAMPLES 10 7. PERTURBATION PROBLEMS FOR PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 11 8, SQUARE NORM AND UNIFORM NORM 12 9* APPLICATIONS 13 10. EXTENSIONS OF THE PREVIOUS CONSIDERATIONS 14 11. KNOBLOCH'S EXISTENCE THEOREMS FOR PERIODIC SOLUTIONS 15 12. PERIODIC SOLUTIONS OF NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS 18 REFERENCES 22 ii

1. THE SOLUTIONS OF AN EQUATION AS FIXED POINTS OF A SUITABLE MAP We are concerning ourselves with an equation Kz O (1) whose solutions are to be found in a Banach space S (or in a subset G of S), and we consider a projection. operator P in S. If So is the range of P, and. S1 the nullspace of P, then S = So+S1, and every element zcS has the representation z = (x,y), or z = x+y, with x = Pz, y = (I-P)z, I the identity operator in S. Obviously, equation (1) decomposes into the system of the two equations PKz =0, (I-P)Kz = 0o (2) If Do denotes the projection of G on So, Do = PG c So, then we are seeking pairs z = (x,y) with xEDo and z EP -x, such that z = (x,y)EG, and. z satisfies (2). We shall denote by 11 Zl the norm of z in S. As in [ld] we shall assume K = E-N, where E: SE - S is a linear operator nonnecessarily bounded. with domain SE c S, N: SN + S an operator nonnecessarily linear with domain SN c S, so that K: SE n SN + S. Then, the given equation (1) takes the form Ez Nz, and. the equivalent system (2) 4the form PEz = PNz, (I-P)Ez = (I-P)Nz We may think of E as a linear differential operator and of SE as the set of those elements zcS sufficiently smooth so that E can be applied.. We shall assume t;hat a linear operator H = S, - S, is known to exist, for which we require for a moment that 1

H(I-P)Ez (I-P)z for all zESE In other words, we require here that H: S1 + S1 is a partial left inverse of Eo Then, for any solution z of Kz = 0, or Ez = Nz, we have H(IDP)Ez = H(I-P)Nz, (I-P)z = H(I-P)Nz, or, finally, z Pz + H(I-P)Nz. () In other words, any solution z of (1) is a fixed point z = Tz of the map T: SN + S defined by T = P+H(I-P)N, or T = P+F with F = H(I-P)N. Here the operator T has a relevant property, indeed T maps each fiber of P into itself, precisely PTz Pz for every zcSN, or T: (P-lx) n SN - P-Ix Indeed, for z~SN, we have PTz = PPz + PH(INP)Nz Pz, since P is idenpotent, and H(INP)NzES1, the nullspace of P.

2. THE RELAXED EQUATION AND THE DETERMINING EQUATION To simplify the exposition, let us assume that G = Do x D1, where both D. c So and. D1 c S1 are spheres, and that G c SN. If it happens that T maps each set P-Lx n G into itself, and we know that P-1x n G is compact (or that P-1x n G is complete, and T(P-lx n G) is compact), then by Schauder's theorem there is at least one fixed. point z = Tze P-lx n G for every xeDo0 Actually, by a suit;able choice of P it is possible to make T a contraction in the sense that IITz-Tzli < kliz-z'1| for any two points z, z'eP-lx n G of the same fiber of P, with k < 1 and x c Do (see [ld], or no. 4 below). It should. be noted t-hat T: P-lx n G + S is a contraction if and only if F is contraction. Indeed, if z, z'e P_-x n G, then Tz - Pz+Fz, Tz' = Pz'+Fz', Pz Pz' = x, and hence IlTz-Tz'1i = IIFz-Fz9ll Whenever T: P- x 0 G + PV x n G is a contraction and into for every x e Do, then there is one and only one fixed point z = Tzc P'lx n G for each x c Do~ and we may well consider the map`: Do + G which maps every x c Do into the unique fixed element z = Tz of T in P x n G. The graph of -C is a cross section of the fibers of P, and Z itself can be -thought of as a "lifting" operation from So to S. namely, from Do to Go. The map r: Do -+ G defined above is continuous. Indeed, for every two points x, x' e Do and corresponding points z = Tz = x, z' = Tz' = x', we have z C P 1x, z' C P-x 1, and!lFz-Fz'l? ITz-Tz'1J < kllz-z'I, k < 1 with T = P+F, and hence il Z- sz'l ITz-Tz' < IPz-Pz1Pz + II Fz-Fz'11 < 1 x-x'j1 + kllz-z'j| and. 3

itx- mx'Il =!z-z, 11 < (l-k)'-lx-x'I[. (5) Let us assume for a moment that every fixed element z = Tz of T satisfies z C SE PEz - EPz, EH(I-P)Nz = (I-P)Nz, (6) in other words, any such element is reasonably smooth, and H has also the property to be a right partial inverse of E as stated by (7). Then every fixed. point z = Tz of T satisfies the equation (I-P)Kz = O, that is, the second of the two equations (2). Indeed (I-P)Kz = (I-P)(E-N)z = = Ez - PEz - (I-P)Nz (7) - Ez - EPz - EH(I-P)Nz = E[z-Tz] = 0 Thus, in case T: P-lx n G + P-'x n G is a contraction, then for every x E Do the element z ='x of G satisfies the relaxed. equation Kz = PKz, or Kz = PK'x. (8) Indeed Kz = PKz + (I-P)Kz = PKz = PKjrx In other words, for every x E Do there is an element z = rx of G. namely z = T = = zx E P-ix n G, satisfying Kz = PKz. Then, z = ~x satisfies Kz = O if and only if the determining equation PKez = O0 (9) is satisfied. This shows that the search for solutions z E G of Kz = 0 is reduced to the search for elements x c Do c So such that (9) is satisfied. Equa

tion (9) was first encountered. in perturbation problems for periodic solutions, or cycles, of ordinary differential equations, where it gave a new interpretation of Poincare's bifurcation equation. Equation (9) is thus a functional analysis extension of Poincarg's bifurcation equation. Often So is one dimensional and. Do is an interval, and. then the equation PKCx - 0 has certainly a solution in Do as soon as we know that, say PKt'x has opposite signs at the end. points of Do. Often So is a finite dimensional space, and Do is a cell. Then the equation PK'x = 0 has certainly a solution in Do as soon as we know that the topological degree d(PKZ,D, 0 ) is nonzero. Even if So is infinite dimensional, as it does occur, the study of the determining equation has led. to simple criteria for the existence of solutions to the original equation. The method briefly summarized. above first appeared in [ld.,e] with details and applications, and was then reported. in [3]1 and. [5b]. 5

35 A CONNECTION WITH GALERKIN'S APPROXIMATIONS If So is finite dimensional, and [~1,...,rm] is a base for So, then PK-x = cl(x)~l +..+ Cm(X))rm, where the coefficients cl,...,cm depend on x c Do, and the determining equation PKtx = 0 can be written in the form of the m equations C1(x) = O,.o., Cm(X) 0, XE Do (10) These are similar to the equations for an mth Galerkin approximate solution, with the difference that equations (10) yield an exact solution z ='x to the given equation Kz = O. Actually we may well expect that the exact,o1ution z = rx is "close" to an mth Galerkin approximation x(m) E So, or x = Y71(x)l+...+ym(x)M. If we write PKx(m) = rl(x)~l +...+ rm(X)jm, then x(m) is determined by the m Galerkin equations rl(x) = O,.., rm(x) = 0, x E Do. A closer inspection [ld.,e] shows that indeed. this is the case, and. error bounds for ||z-x(m)li (z ='tx, x E Do, PK-x = 0) are obtained by the analysis of the determining equationo 6

4-. THE MAP T AS A CONTRACTION It is typical of the method described in nos. 1 and 2 that the operator T can be made to be a contraction on each fiber of P as a consequence of spectral properties of the linear operator E, even if the nonlinear operator N is only Lipschitzian and nonnecessarily smooth. To see this let us assume here that S is a Hilbert space with inner product (z,z') and norm izlz = (z,z)l/2. We shall also assume that the linear operator E possesses eigenvalues Xi and eigenfunctions ki, say Eii+ki~i = O, i = 1,2,..., such that 1Xll < Ix21 < _.. IXil +0 as i oo+, and [1,~2,.. ] is a complete orthonormal system in S. Then, every element z e S has a Fourier series z = Cll1 + C22 +..-, =ith oi (Z,~i), i = 1,2,..., lzll 2 1/2 with ci= (...,i) and we define P by taking Pz = C.ll +. —+Cmm Then [Al,... am] is a basis in So and [m++,m+2,m+,-. ] is a complete orthonormal system in S1. If we take m sufficiently large, then Xi 0 for all i > m+l. For every z E S1 we have z = i = m+l cii, and we may define H: S1 +S by taking Hz C _ H = m+ 1 i 1i We showed in [Id] that this operator H possesses the properties required in nos. 1 and 2. In addition I|Hz =l. 1= II i = m+i +1 and thus H is a Lipschitzian operator in S1, or IIHz-Hz'11 < kollz-z'1, zz E S 3,

with constant ko = Ikm+l-1 which can be made as small as we wish by taking m sufficiently large. Finally, if we assume that the restriction offNon G, say N: G+S, is Lipschitzian of constant L, say IINz-NZ'1I < Lllz-z'11, z,' e G, then we have, for z,z' e P- x n G, x C Do, ITz-Tz'II = IFz-Fz'll = IIH(I-P)(Nz-Nz')11 < Im+ |1LmIz-!z'1 ~ Thus, T is a Lipschitzian operator on P-Ix n G for every x E Do of constant k = X m+ll-1L, and we can always take m sufficiently large so as to have k < 1. Actually it turns out that k is already < 1 for rather small values of m, in a number of applications.

5. A CONNECTION WITH THE LERAY-SCHAUDER FORMALISM Once the structure mentioned. above is well defined., that is, K, E, N, are given, G = Do x D1, and T: P-lx n G + P-lx n G is a contraction for every x E Don so that C: Do - G is defined by z = Tz = tex, then S. A, Williams has shown [14] that it is possible to establish a theoretical connection with the LeraySchauder theory. To do this he has introduced the maps W: G + S and. W' = G + S d.efined by Wz Pz - PKCPz + H(I-P)Nz, z c G, W'z = WPz, z E G. As a consequence of the properties of the maps E, N, T, A,, Williams has shown that W' is completely continuous, that the fixed points of W' are the solutions of the original problem, and that the topological degree of (PK, Do, 0) coincid.es with the Leray-Schauder degree in Bcanach spaces dLS(I-W',G,O). Thus, the present approach gives rise to a situation where the Leray-Schaud.er theory theoretically applies through the map W' which in turn is defined in terms of the maps K, P, T, a.n This result parallels in the present situation, and hence even in the absence of smoothness properties for the operators (cf. no. 4) an analogous result established by Jo Cronin in connection with her approach. Williams has further discussed the question as to whether the topological degree of (PK-, Do, O) is invariant for suitable changes of P, say, if we replace P: S + So by another projector operator P': S - S' whose range So is a linear subspace of S containing So, say So c So c S. The answer is affirmative, at least when So is already sufficiently large. This answer is in harmony with Leray-Schauder theory where the topological index dLS is actually defined. by considering suitably finitely dimensional subspaces So of S, and showing that the corresponding Brouwer topological degree defined in SO does not change by enlarging So, once SO is already sufficiently large.

6. A FEW EXAMPLES In [ld] we considered. the boundary value problem x" +X +3 t, o< t<, x(o) - 1, x(l + ) and we *took m dim S. 1. Then for h 1, we found kQ 0o 044. For o - 1/2 a first Galerkin approximation is x(l)(t) = - 0.11721 sin (2.0288t), 0 < t < 1 By applying the cornsiderations of the previous numbers we proved that the problem above has an exact solution X(t), 0 < t < 1, with IIX-x I < 0. 0038 In [le] we considered the boundary value problem X + X = sint, x(o) = x(2i), X(o)= x(2c), and we took m = dim So = 2. We found ko = 0.04, and a second Galerkin approximation x(2)(t) = 1.434 sin t 0.124 sin 3t, 0 < t < 2T. By applying the considerations of the previous numbers we proved that the problem above has an exact solution X(t), 0 < t < 2n, wi-th IX5x(2)I < 0.124. In [lj] we considered the boundary value problem (of nonlinear potential) u + u = g(x,y,u) for (x,y) c A [x2+y2 <l] xx Y-Y~ (11) u= O:for (x,y) E A [x2+y2 1], where g is a given function continuous in'u, measurable in x, y, and. bounded. for u bounded.. We'took m = dim So = 1, and we found ko = 0.069. By applying the considerations of the previous rnumbrters we proved the existence of solutions u(x,y) to problem (11) whi.ch are continuous in A U 3A, with first order partial derivatives continuous in A and whose Laplacian Au = uxx+uyy (in the sense of the'theory of distribution) is a measurable function in A. Essentially, we proved. in [lj].the existence of at least one such solution provided g is not too large for u = 0 and does not grow too rapidly with l u as lul -+ o. For instance, for the case uxx+uyy = f(x,y) lu+h(x,y), f, g measurable If(x,y)l < a, |h(x,y)l < <, 3, 5 finite, we proved that a solution exists for every f, g as above with O<3 02 and any 5. In these problems a first approximation solution of the form uo(x,y) = YJo(kol) < p2 - x2+y2 1, is singled. out, where kol is the first positivre zero of Jo and y is a suitable constant. Error bounds for such a solution, that is, upper bounds for the norm IlU-UolI of the difference between exact and. approximate solutiorns have been given by C. D. Stocking in [13]. 10

7. PERTURBATION PROBLEMS FOR PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS The ideas of the previous numbers can be already traced. in the papers by Cesari, Jo K. Hale, Ro A. Gambill, W. R. Fuller, H. R. Bailey in the years 1952-60 (see reports and. summaries in Fla,b,c] and [5b]) on periodic solutions of perturbation problems for ordinary differential equations and systems (periodic or autonomous) dz/dt = Az + eq(t,z), (12) n c a small parameter, A an n x n constant matrix, z = (z,..,z ), q(t,z) periodic in t (or independent of t), measurable in t, and. continuous, or Lipschitzian in z. It was shown in [le] that the method used in those early papers is a particular case of the process described, in nos. 1 and. 2 above, and we can choose m = 1 since the numbers k, ko are now replaced by numbers [ c!EI, koEeI which can be made less than one by taking [EI sufficiently small. For problems (12), periodic or autonomous, extensive existence theorems of periodic solutions, or cycles, have been proved., together with conditions for asymptotic, or asymptotic orbital stability. In particular J. K. Hale discovered by the method above simple criteria for the existence of families of given dimension k, 0 < k < n, of periodic solutions [5a, la,b,c]. Under conditions of continuity only, the existence of solutions to the determining equations, and hence to the original problem, can be assured [la] by the use of a Co Mirandads statement which gives an equivalent form of: IBrouwer's fixed point theorem. This statement concerns vector-valued continuous functions F(z) = (Fi,,,,Fn) from a cube C = [z = (z1,ooznn ) Izil < Li, i = 1~,n If Fi has constants opposite signs on the sides zi - + LTof C, then there is at least one point z e C where F(z) = 0. (C. Miranda, Boll. Un. Mat. Ital. 3, 1941, 5-7o ) Under Lipschitz conditions the method for these problems of periodic solutionsas described in [la, c] gives rise to a process of successive approximations which is reported in [4, ppo 308-317]o 11

8. SQUARE NORM AND UNIFORM NORM In problems as those of noo 6 it is advantageous to use the square norm because it affords the smallest value of the constants k, or ko (for instance, k, = IXm+lrlunder the hypotheses of no. 4) and thus very small values of m could be used. (m = 1 or m = 2 in no. 6)~ Actually, in these problems both norms, the square norm and the uniform norm, have been used, since N (often a polyrornmial expression) may be Lipschitzian of a certain constant, say L, only if sup I z is not larger than some other constant R. On the other hand, the best values of k or ko may not be essential as, say, in perturbation problems (no. 7) as mentioned, or in association with Knobloch-type arguments as in no. 11 below. Ho Wo Knobloch [6a] has given the necessary estimates for the use of the uniform norm only in questions involving periodic solutions of ordinary differential equations and the use of the method of nos. 1 and 2 above. Recently Jo Mawhin [81 has applied directly the formalism of nos. I and 2, with the use of a uniform norm only and Knobloch estimates, to the same perturbation problems of noo 7. 12

9. APPLICATIONS P.A.T. Christopher [2a,b] has applied the method of the present paper to determine regions in the parameter space corresponding to stable harmonics and subharmonics for the nonhomogeneous Duffing equation: x + bx + cx + c3x3 = Q sin at. A. M. Rodionov [12] has shown that the method of the present paper applies to nonlinear differential equations and systems with a fixed. lag X: x = g(t,x(t),x(t-x)) C. Perello [10] has initiated the same type of analysis for functional differential equations x = g(t,xt), where g depends on t and on all values of x(T) in an interval t - X < T < t. 13

10. EXTENSIONS OF THE PREVIOUS CONSIDERATIONS The formalism of nos. 1 and 2 is based on certain properties of the operators, or axioms, and Jo Locker [7] has proved, that these properties are satisfied, for instance, for usual boundary value problems of nonlinear ordinary differential equations, when the underlying linear problem is selfadjoint. For the corresponding nonselfadjoint problems J. Locker has developed a theory which extends the one of the present paper, and whose axioms are satisfied for boundary value problems of ordinary differential equations. As an example, he gives numerical bounds for a and p in the nonlinear initial value problem + x + C2 = +t, O< t < 2, x(O) = 0, under which the solution exists in [o,2c]o. He also obtained estimates of the growth of the solution and. error bounds. J. K. Hale, So Bancroft, and D. Sweet [5d] also have presented a formalism which extends the present one in a different direction, and established a connecting link with the previous work by Do C. Lewis, H. A. Antosiewicz, Jane Cronin, R. G. Bartle, and Lo Nirenberg.

11. KNOBLOCH'S EXISTENCE-THEOREMS FOR PERIODIC SOLUTIONS For problems of periodic solutions of ordinary differential equations and. systems (not of the perturbation type), say zi = fi(t,z), i = loo,n, z = () fi(oz) = fi(T,z), let us consider the space S of all continuous periodic vector functions with uniform norm. Every element z c S has a Fourier series z(t) -2-mao +.o (ai cos iwrt + bi sin ict), = 2ic/T, T the period, where ao, ai, bi are n-vectors. Let us define P by taking Pz = 2-1ao + Z7m (ai cos ict + bi sin icut), (15) 1z 2-1 (13) so that dim So = n(2m+l). By taking m sufficiently large, the corresponding map T is a contraction and. ~ is defined. (H. W. Knobloch [6b]). Then, for every element x c So (a trigonome-trical polynomial as in (13), we have PKrx = 2-lco + Z. m (ci cos io0t + fi sin iot), where aO, c, i, i = 1,ooom, are functions of x, that is, of the coefficients ao, ai, b1, i = l,,.,m, of the trigometric polynomial x. Then the determining equation PK~_x 0 takes the form =o = i O i = 1.,m (14) namely a system of n(2m+l) equations in n(2m+l) unknowns. H. W. Knobloch has shown that, under hypotheses, a suitable distribution of signs of the n functions fi, or associated functions, imply a typical distribution of signs for each of the n(2m+l) left hand members (components) of the determining equations (14) so that the existence of a solution x c So to the same equations, and hence 15

the existence of a solution z to the original problem, can be d.educed by the same C. Miranda's form of Brouwer's fixed. point theorem mentioned. in noo 7, and. this no matter how large n and. m are. H. W. Knobloch has used. this analysis of the determining equation as a basis toward. a qualitative discussion of the second order nonlinear differential equation x = g(t,x,), g(t,x,y) = g(t+l,x,x), (15) where g is a real-valued, function of t, x, x, which is Lipschitzian in x, x, periodic of period 1, and sectionally continuous in t. Two functions a(t), B(t), 0 < t < 1, (continuous with continuous first derivatives and sectionally cont.inuous second derivatives) are said to be lower and upper solutions respectively provided. a < &, a + g(t,cy,z o) < 0 + g(typy) > 0, for t in [0,1]o If ac,' is such a pair, if Q denotes the region [O < t < 1, a(t) < x < p(t)], and there are two functions ((t,x),'(t,x) continuous and continuously differentiable in.., such that.(tnao) < & < (t,a) D (ty) < < ~ ~(t, ),.(x,OO) = (x,T), T(x,O) = T(x,T), and. such that both expressions x~O + Ot g(t-,x,D), and. Tx~ + ~t - g(t,xj,) have constant signs in 0., then (15) possesses a periodic solution x(t) satisfying a(t) < x(t) < P(O.), (to(t)) < x(t) < t(t,(t)) for all t in [0,1] (H. W. Knobloch [6b]). A corollary of this statement states that if a, p is a pair of periodic lower and upper solutions, and g(tx,x) < C/x|2 for some constant C and all t, x, x with (t,x) EQ and |x| sufficiently large, then again (15) possesses a 16

periodic solution of period 1. H. W. Knobloch has proved. other statements of the same type, based on the process mentioned in the previous numbers and topological considerations [5b]. In successive papers H. W. Knobloch [6c] has also proved, by the same process, comparison and oscillation theorems for second order nonlinear equations (15), which extend Sturm theory for linear equations. Some of the results give general form to statements proved, earlier by M. Carthright for the van d.er Pol equation only. 17

12. PERIODIC SOLUTIONS OF NONLINEAR HYPERBOLIC' PARTIAL DIFFERENTIAL EQUATIONS The question of the periodic solutions of the hyperbolic partial different;ial equation u = g(xyuu,y), (16) xy x y can be posed in two ways, either we assume g periodic in x only and we ask for solutions u periodic in x of the same period, and existing in a strip, say -a < y < a, -oo < x < +oo (problem in a "cylinder"), or we assume g periodic in both x and. y of a given period T and, we ask for solutions u period.ic in both x and. y of the same period., hence existing in the whole xy-plane (problem on a "torus"). To simplify the exposition we limit ourselves to the second. case. The process described in the nos. 1-5 above has been applied. by Cesari [lf,g,h,i] to this problem, taking for S the space of all continuous functions u(x,y) periodic of period T in x and y, and continuous with their first order partial derivatives, and uniform norm, and taking for So the subspace of the functions u of the form u(x,y) = h(x)+k(y)o Then, the projection operator P is defined by taking Pu(x,y) = u(x,O)+u(O,y)-u(O,O), and. S is now infinite dimensional. This choice leads to the following existence statement and. a very simple interpretation of the relaxed. problem: If g is continuous in its arguments, periodic in x and y of period T, and. Lipschitzian in ux, Uy of constants bl, b2 satisfying 2Tbl < 1 2Tb2 < 1, (17) then for every two functions uo(x), vo(y) continuous with uS, vr, periodic of period T, and uo(O) = vo(O), there exists at least one solution u(x,y) periodic in x and. y of period. T continuous with ux, uy, Uxy, satisfying the relaxed. (Darboux type) boundary value problem xy = g(x,y,u,u,u) - m(y) - n(x) - l, xy x y u(xO) = (x) = u(x,T), v(Oy) = v (y) = v(Ty) (See [lh] for detailso) Here m, n are suitable continuous functions and f a 18

constant, m, n, 4 depending on uo, vo, and. of course T T 2 TT m(y) = T-1 fog dx n(x) T-1 fTg dy e, f = T -g dx dy 0 0 0 0 In other words, for every Darboux type boundary condition there is a periodic solu'tion to the relaxed. problem. This relaxed problem is "well posed.". Indeed, if g is Lipschitzian in u, ux, uy of constants bo, bl, b2 satisfying relations analogous to (17), then the solution u(x,y) is unique and depends continuously on u,, vo in the proper uniform topology. The determining equation is now a functional equation (since So is oo-dimensional). Indeed., the solution u(x,y) above of the relaxed problem is actually a solution to the original problem (16) provided, the functions uo, vo are so chosen that m(y) =0, n(x) = 0, 0 = O (determining equation). Very simple criteria can be given (see [lh]) for the existence of a solution to the determining equation (and. hence for the original problem (16)) for perturbation equations of the form uxy = (x,y) + Cu + tl(y)ux + t2(x)uy + eg(x,y,u,ux,uy), (18) or of the form u = E [t(x,y)+Cu+tl(y)U +*2(x)u ] + E2g(x,y,u,u,u ), (19) xy x y x y E a small parameter0 These results, for the relaxed. problem, as well as for the originalproblem, can be completed with smoothness theorems [li], and transferred, to the nonlinear wave equation u - u g(x,y,u,u,u ), (20) xx yy x y by usual transformations lh, i] o J. K. Hale [tc] has studied the same problems for the nonlinear wave equations choosing as operator P the projection of S into the space of the periodic solutions of the homogeneous problem uxx-uyy = 0, and has obtained. corresponding results for the relaxed problem, and. ensuing perspicuous criteria for the exact 19

problem for equations (18) and (19), some of which extend. or parallel previous results of Rabinovitch and 0. Vejvoda. In a sense, analogous results have been proved for periodic solutions in a strip, and Cesari Fig] has shown that criteria for the solutions of the original problem can be obtained by a novel implicit function theorem of the hereditary type based. on functional analysis. Well within the frame of the present method A. K. Aziz (Proc. Amer. Math. Soc. 17, 1966), by more stringent estimates, slightly reduced requirement (17) in the existence theorems for the relaxed problem. Again by the method of the present paper D. Petrovanu [10a] has recently shown that the analysis mentioned above for the periodic solutions of the hyperbolic equation in the plane can be extended to the periodic solution of the equation u = g(X,y,z,u,u,u,uz,), xyz x y z with g also periodic in x, y, z. D. Petrovanu [10b] has proved. also existence theorems for the relaxed problems, both in a strip and on a torus, for the periodic solutions of the Tricomi system of equations u = g(x,y,u,v), v = h(x,y,u,v) ~X ~y u an m-vector and v an n-vector. Very recently A. Naparstek [9] has discussed the nonlinear wave equation u - 2u = Eg(x,y,u) (21) xx yy on a torus by the same method of the present paper. Here e is a small parameter, g is periodic in x and y of period, say, T = 2n, the solutions u, also periodic in x and y of period 2-t, are sought in suitably Sobolev spaces on the torus, and the hypotheses on g are also expressed in terms of measureability and belonging to suitable Sobolev spaces. Above, a > 0 is a fixed number, and two cases must be distinguished: c rational, and a irrational. In the first case the homogeneous equation uxx-a2uyy = 0 has oo- many well known and independent (periodic) solutions on the torus, while for a irrational the same equation has only constant solutions. For a irrational also questions of small divisors occur. If C denotes the class of all positive irrational numbers a such that m2-a2n2 is bounded away from zero when m, n describe all possible pairs of not both zero integers, then C is known to be countable and everywhere dense in [0,oo). For the two cases 1. a rational, 2. a 20

irrational, a E C, the projection operator P is chosen as the projection of the doubly periodic functions into the subspace So of the periodic solutions of the homogeneous equation uxx-a2uyy = O. Thus, So is ou- many dimensional for a rational, and one-dimensional for a irrational, a e C. For both cases A. Naparstek has given existence theorems for the relaxed problems, together with uniqueness and. continuous dependence statements, and has given criteria for the existence of solutions to the original problem. For a rational the determining equation is a functional equation. Two sets of different considerations are used for its discussion. One is based. on the existence of a Gateaux nonzero differential at C = 0, and, the use of implicit furction theorems of functional analysiso Another set of considerations concerns the interpretation of the first; member of the determining equation as an operator, for which conditions are given for it to be monotone in the sense of G. Jo Minty and. F. E Browd.er. From both A. Naparstek deduces criteria for the existence of doubly periodic solutions to the original problem (21)o A. Naparstek also has extended. these results to equations similar to (21) containing the first order partial d.erivatives ux, uy. 21

REFERENCES [1] Lo Cesari: (a) Existence theorems for periodic solutions of nonlinear Lipschitzian differential systems and fixed point theorems. Contributions to the theory of nonlinear oscillations, 5, 1960, 115-172. (b) Existence theorems for periodic solu.tions of nonlinear differential systems. Boletin Soco Mat. Mexicana 1960, 24-41. (c) Asymptotic behavior and stability problems in ordinary differential equations, Springer Verlag, 2nd ed. 1963, viii+271, (d) Functional analysis and Galerkin's method, Michigan Math. J, 11, 1964, 385-418, (e) Functional analysis and periodic solutions of nonlinear differential equations, Contributions to differential equations, Wiley, 1, 1963, 149-187o (g) Periodic solutions of hyperbolic partial differential equations. Nonlinear Differential Equations and: Nonlinear Mechanics, Academic Press 1963, 33-57- (g) A criterion for the existence in a strip of periodic solutions of hyperbolic partial differential equations. Rend. CircoloMato Palermo (2) 14, 1965, 95-118. (h) Existence in the large of periodic solutions of hyperbolic partial differential equations. Archive Rat, Mech. Anal. 20, 1965, 170-190. (i) Smoothness properties of periodic solutions in the large of nonlinear hyperbolic differential systemso Funkcialaj Etvacioj 9, 1966, 325-338, (j) A nonlinear problem in potential -theory. Michigan Math. J., to appear. [2] PoAoT. Christopher: (a) A new class of subharmonic solutions to Duffing's equation, CoA Report Aero 195, 1967, ppo 54. The College of Aeronautics, Cranfield., Bedford., England.. (b) An extended, class of subharmonic solutions to Duffing's equation. Ibid., Report 199, 1967, pp. 48. [31 JO Cronin: Fixed. points and topological degree in nonlinear analysis. Amer. Math. Soc. Math. Surveys no. 11, 1964. [4] A. Halarnay: Differential. equations, Academic Press 1966, xii+528, part.. pp. 308-317. [5] JO K. Hale: (a) Sufficient conditions for the existence of periodic solutions of systems of weakly nonlinear first and second, order differential equations. Journ. Math, Mecho 7, 1958, 163-172. (b) Oscillations in nonlinear systems. McGraw Hill, 1963, ix+180. (c) Periodic solutions of a: class of hyperbolic equations. Archive Rat. Mech. Anal. 23, 1967, 380398. (d) Alternative problems for nonlinear functional equations (with S. Bancroft and D, Sweet), J. Differential Equations 4, 1968, 40-56. 22

REFERENCES (Conclud.ed) [6] H. W. Knobloch: (a) Remarks on a paper of Cesari on functional analysis and. nonlinear differential equations, Michigan Math. J. 10, 1963, 417430. (b) Eine neue Method.e zur Approximation von periodischen Loesungen nicht linear Differentialgleichungen zweiter Ordnung, Math. Zeitschr, 82, 1963, 177-197- (c) Comparison theorems for nonlinear second. ord.er differential equations, J4 Differential Equations 1, 1965, 1-25. [71 J. Locker: An existence analysis for nonlinear equations in Hilbert space. Trans. Amer. Math. Soco 128, 1967, 403-413. [8] J. Mawhin: Application directe de la method.e generale de Cesari a l'etud.e des solutions period.iques de systemes diffe'entials faiblement non line'aires, Bullo Soc. Roy. SciL Liege 36, 1967, 193-210. [9] Ao Naparstek: Periodic solutions of certain weakly nonlinear hyperbolic partial differential equations, A thesis at The University of Michigan 1968. [10] Co Perello: A note on periodic solutions of nonlinear differential equations with time lags, Differential Equations and. Dynamical Systems, Academic Press 1967, 185-188, [11] Do Petrovanu: (a) Solutions periodiques pour certaines equations hyperboliques, Analele Stintifice Iasi, to appear. (b) Periodic solutions of the Tricomi problem, Michigan Math. J, 1969, to appear. [12] A. M. Rodionov: Period.ic solutions of nonlinear Differential Equations with time lag. Trudy Seminar Differential Equations, Lumumba Univo Moscow 2, 1963, 200-207 (Russian)o [13] C. D. Stocking, Nonlinear boundary value problems in a circle and related. questions on Bessel functions. A Ph.D. thesis at The University of Michigan 1968o [14] So Williams: A connection between the Cesari and Leray-Schauder methods; Michigan Math. J. 1968, to appear. 23

UNIVERSITY OF MICHIGAN III3 9015 02844 9513 III 3 9015 02844 9513