THE UNIVERSI TY OF MI C HIGA N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report No. 12 EXISTENCE IN THE LARGE OF PERIODIC SOLUTIONS OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS; Lamb'ert o. Ce ri. ORA Projec't 05504 under contract with: GRANT NO. G-57 WASHINGTON, D.C. administered through:05304 under contract with: NATIONA L SCIENCE FOUNDATION GRANT NO. G- 57 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1964

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EXISTENCE IN THE LARGE OF PERIODIC SOLUTIONS OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS by Lamberto Cesari * The problem of existence of solutions /(x,y) periodic in x and in y of period T for an hyperbolic partial differential system of the form Uxy = f(x,y,u,ux,uy), (1) where u = (ui,...,un) and f = (fl,...,fn) is periodic in x and y of period T, presents a number of difficulties when no damping of any sort is assumed. In this paper we analyze this difficult problem in the line of our previous work on ordinary and partial differential equations. We conclude with criteria of existence for solutions to the problem above. These criteria can then be used for the analogous problem for the equation Uxx - Uyy = g(x,y,U ux,uy) (2) *Research partially supported by NSF Grant G-57 at The University of Michigan, Ann Arbor, Michigan.

1. THE MODIFIED PROBLEM 1. MODIFIED PROBLEM We shall first associate to (1), the analogous weaker problem, or modified problem: Given two periodic functions Uo(x), vo(y) of class C1 in (-0o,+o0) and of period T, uo(x+T) = uo(x), vo(x+T) = vo(x), determine a function T(x,y) continuous with its partial derivatives Ox,$y,$xy, two functions m(y) and n(x) both continuous, and a constant 1i, such that <(x+T,y) = O(x,y) = O(x,y+T), m(x+T) = m(x), n(y+T) = n(y), T m(y)dy = 0, rT JI C (3) n(x)dx = 0, and Oxy == f(xy,yOxy,,y) - m(y) - n(x) -. (4) Fiior this mod.ified problem we shall prove theorems of existence, uniqueness, and continuous dependence on the boundary values and parameters. In (4) we assume f(x+T,y,z,p,q) = f(x,y,z,p,q) = f(x,y+T,z,p,q). Then the function ~ is a periodic solution of the original problem (1) if and only if we can determine Uo(x), Vo(y) in such a way that k[ = 0, m(y) 0, n(x) = 0. Criteria for this occurrence are given in Section 2. 2. THEOREM I (existence theorem for the modified problem) If T > 0, and N,N1,N2,L,Mbl,b2,M1,M2,Ms > 0 are constants, if A and R are the sets

A = [O < x < T, < y < T], R = [O < x < T, 0 < y < T, zl< M1, Ipl< M2, Iql< M], if M1 > N + (N1+N2)T/2 + 5L T2, M2 > N1 + 5L T, M3 > N2 + 3L T, (5) if Uo(x), O < x < T, Vo(y), O < y < T, are vector functions which are continuous with uo' (x), o' (y), if f(x,y,z,p,q), (x,y,z,p,q)eR, is continuous in R, and uo(T) = uo(o), vo(T) = vo(o) = 0, u'(T) = uo'(o), (6) vo'(T) = Vo'(o), luo(o) |< N uo(xl)-uo(x2) N x-x2, I vo(y.) -Vo(y2) I N21 Y-Y2 i, (7) f(T,y,z,p,q) = f(o,y,z,p,q), f(x,T,z,p,q) = f(x,o,z,p,q), (8) If(x,y,z,p,q)j< L, If(x,y,z,pl,ql)-f(x,y,z,p2,q2)|< bjllP1-p21+b2lq-q21 (9) then for 2T bl < 1, 2T b2 < 4 (l) there exist a vector function j(x,y), (x,y)-A, continuous in A together with Ox)y, Oxy, continuous vector functions m(y),-o < y < T,n(x),o < x < T, and a constant,i, such that O(x,o) = O(x,T) = o(), (x),o) = y(x,T), (1l) O(o,y) = ((T,y) = Uo(o) + v(y), &x(o,y) = Ox(T,y), (12) m(o) = m(T), n(o) = n(T), (13) xy(xy) = f(x,y,~(x,y), x'(x,y), Qy(x,y)) - m(y) - n(x) - t, (14) 2

PT % j m(ri)4j = C n0 o o T m(y) = T-1 O0 T T T n(~)d = 0, o (15) f(5,,~(5y),gx(~ y), y(S.qy))d - [Al f(xys,(x,),(xylly~(Xi)yy( XYTI)) - TAY (16) (17) (18) n(x) = T- 0 for all o < x < T, o < y < T. Thus, by extending all functions,(x,y), m(y), n(x), f(x,y,z,p,q) for all -oo < x,y < +0, IZI< M1, IPI M2, Iql< M3, by means of the periodicity of period T in x and y, Equation (14) is satisfied in the whole xy-plane. 3. PROOF OF THEOREM I First let us note that relations (11), (12), (14), (15), imply (16), (17), (18). Indeed, by integration of (14) on A, we deduce (16). Then, by integration of (14) again on o < x < T, or on o < y < T, we deduce (17) and (18) respectively. Note that (8), T11), (12), (17), (18) imply (13), and that (14), (16), (17), (18) imply (15). Let us first prove that every vector function ((x,y), (x,y)eA, satisfying O(x,o) = ~(x,T) = Uo(x), i(o,y) = (T,y) = uo(o) + vo(y), (19) I(x1,yi) - ~(xl,Y2) = ~(x2,y1) + O(x2,y2)1< 6L ix-x21 yi-y21, also satisfies the relations WXYY)l< ml, Ij(x=,y) - ((x2,y) I< M21Xl-x2, 1 i (20) |o(x,yi) - o(x,y2)< M3 yi-Y21l Indeed, we have, for o < x < T, o < y < T, lf(xi,y) - O(x2,y) - ~(x=,o) + $(x2,o) I< 6LI|x-X2|y, where I (Xjyo).- (X2YO) I = Iuo(xl)-uo(x2)l < NlIxl-x2l, 3

and hence lW(xi,y) - O(x2,y)I< (Ni+6Ly)lxi-X21 Analogously, we have l/(x1,y) = (x2,y) < (NI+6L(T-y))Ixi-x.21 Since, either o < y < T/2, or o < T-y < T/2, we have l0(xiy) = O(x2,y) |< (N1+5LT) Ixi-x21 M21Xi-x21. Analogously, we prove that l1(x,yl) = B(xy2) 1< (N2+3LT) Iy-y21< M3 IY-y21 Finally, for 0 < x,y < T/2, Jl(x,y)|I_/(o,o)! + I|(o,y) - B(o,o)l + I/(x,y) - /(o,y) I N + N2 y + (Nl+6Ly)x < N + (N1+N2)T/2 + 3LT2 < Mi. Analogous reasoning holds for (x,y) in the remaining quaadrants of A. Thus I/(x,y)l< M1, (x,y) A. We have proved that relations (19) imply (20). Also, the vector functions ((x,y) satisfying (20) are all Lipschitzian in A, and hence have partial derivatives $xy, y a.e. in A satisfying |Ixl< M2, lyyl< M3 a.e. in A. The vector function f(x,y,z,p,q) is continuous in R. Hence, continuous monotone functions col(cO), wC2(), co3(7) in [o, +oo) such 0)1() = )2(0) = )3(o) = 0, and there are that if(xl,y,z,p,q) - f(x2,y,z,p,q), cu( (lx-x21), f(x,yi,z,p,q) - f(x,y2,z,p,q) 1 C2( Iyi-y21), (21) if(x,y,zi,p,q) - f(x,y,z2,p,q)1< o3(lzl-z2l), for all 0 K x,xl,x2,y,Yl,Y2 < T, Izl, IZ1, [z2l < M1, IPI < M2, Iql < M3. The vector functions uo'(x), Vo'(y) are continuous in [o,T]. there are continuous monotone functions 34(C0), 0)5(P), o < a,P < + 0)4(0) = ()5(0) = o, such that Hence, oo, with luo'(xi) - uo'(x2)I< 0u4(lxil-X21), Ivo' (y) - vo'(Y2)l< o5(lY1-Y21).(22) 4

Let rl(P) = (1-2T b2)l [(D5(P)+2T cu2(P)+2T au3(M3P)+12LT biP], (23) \2(a) = (1-2T bl)'[cW4(a)+2T w1(a)+2T W3(M a)+12LT baa]. (24) Both ri(P) and 2(ca), o < a,0 < + 00, are continuous monotone functions with j1(o) = r2(o) = 0. Let E be the linear space of all vector functions Z(x,y), (x,y)eA, continuous in A together with their partial derivatives yx, zy with norm 1111 = max 1|1 + max ||xl + max IJyl where max is taken in A. Let K be the subset of E made up of all vector functions o(x,y)eE satisfying relations (19) and in addition /x(oy) = Xx(Ty), /y(xo) = $y(x,T), lx(X1y)- x(x2,y) < r2(jX1-X2j ), jX(XiY1) - ^x(x,y2)l< 6LIY1-Y21, I|y(xiy) - yy(x2,y)I < 6LIYi-y21, Iy(x,yl) - By(x,y2)1< l )lyi-y21). (25) Then the vector functions (eK satisfy relations (20) and then I|1< M1, 1 xl< M2, lyl_< M3 everywhere in A. As a consequence the vector function F(x,y) = f(x,y, ~(x,y), Ox(x,y), Oy(x,y)), (x,y)A., (26) is defined everywhere in A and is continuous in A. For OeK the vector functions m(y) and n(x) defined by (17) and (18) are continuous in [o,T]. With pi defined by (16), the vector function x y V(x,y) = uo(x) + vo(y) + [F(,i) - m(n) - n(S) - i] de dar (27) o o is continuous in A together with its partial derivatives y *x(X,y) = U'() + [F(x,n) - m(n) - n(x) - u]dl, (28) x 4y(x,y) = vo'(y) + [F(iyy) - m(y) - n({) - p.]d.. (29) O Thus, relations (16), (17), (18), (26), and (27) define a map ~: $ =~~, 5

or: K -+ E. Let us prove that actually'": K + K. Since Ifl< L, by (16), (17), (18) we deduce ll< L, JIm(y)l, In(x)I< 2L, On the other hand, by (6), (8), (25), (26), (27), (28), the usual convections F(o,y) = F(T,y), F(x,o) = F(T,o), m(o) = m( T T / m r= O. n()d0, 0. o o (29), we have with T), n(o) = n(T), (30) r,(x,o) = V(x,T) = Uo(o) + Vo(y), = Uo(X), *x(oy),y(x,o) =,y(x,T), = *x(T,y), *(o,y) = *(T,y) (31) 't(xi,yr) - V(xl,y2) - *(X2,Y1) + |(X2,Y2) 1 (52) xi yi In other words r = r for OeK satisfies reactions (19) and, hence, relations (20) as proved in No. 3. Also, we have Y2 ljx(X,Yl) - *x(xy2)1 = I [F(x,T) - m(q) - u(x) - ]idri, YL X2 Iy(xl,y) - y(x2,y)I = I [ X1 and hence I|x(xY ) - *(X,,Y2) I < 6LIY1-y21, F(,y) - m(y) - n(S) - I]dS, IVy(x,y) - *y(X2,y) I < 6LIXI-x21. (335) We have further, from (17), 6

T Im(yl) - m(y2)l IT- J f(.,yi,~(,yi),x(,y,)4y (,y )) 0 T - f( yY2(^y2)4x(,Y2), By(,y2))}l< T' [2(I(,i-y21) I+ -)^3(l (Y1)4- ( Y2)I) 0 + bll^x(iyl) - ~x(S,y2)1 + b2lJy((,Y1) - ~y(VY2)l]dS < 2(1Y1l-Y21 ) + G- (M31y-Y21 ) + 6L bllyi-y2l + b2 rll( IY —y21), and analogously In(xi) - n(x2) 1< Wol( Ixl-x21) + 3o(M21Xl-x2 ) + bl 2( Ixi-X21) + 6L b2lxl-x21. We have now, from (29) x ty(xyi) - y(xy2)J = Iv'O(yl)-vl0(y2) + o - m(yl) - f(+,y2, ((,Y2),x(aY2) y(S,Y2))+ T + c2( IYl-y21 ) +c> ( I3(l(,yj)-j (,Y2)1 ) + 0 m(y2) ]dt I< C5( IYl-Y21 ) bll&|(SYl)-OX(yY2)l + b2l|y(SYl) - Xy(tsY2) + Im(y1)-m(y2)I ]d <I 5(lY1-Y21) + 2T (2(IY1-Y21) + 2T 03(M3lY1-Y21 ) + 12LT bi|yl-y>l + 2T b2 r1(IY1-y2l) = (1-2T b2)rl(IY1-2I) + 2T b2 I(JY,-y2l) = 1(lY1-Y21). (34) Analogously, we have l*x(Xiy) - *x(x2,Y)I< r2(Ix1-x21). (35) Relations (31), (32), (33), (34) show that 4 =' 0 for OeK satisfies all relations (19) and (20). Thus r e K, and r: K -> K. The transformation: K - K, K C E, is continuous in K with respect to the norm 11|11 of E. Indeed, for two vector functions je.K, j = 1,2, we have *j =:@ i, Fj = Fj, mj m=n(y), nj = nj(x), j = =j, j 1,2, and from (17) T T - f(t0 0 7

Then from (18) we have T lmi(y)-m2(y)l = |IT' / [f(x,y,~i(xy), ix(x,y), iy(xy)) o - f(x,y,2(x,y),2x(x,y) 2y(xy))]dx + p1 - [21 5 2[ ( 11>Z1-4 )1^1+blll~l-,211+1b211^1-/211] and analogously from (19), nl(x)-n2(x) 1 2 [c( 1 (I1-2 11)+bl || —1- 211+b2ll1 —211 We reduce from (27) x y 1*l(( xy,)-*2(x,Y) I [ F1(,)-ml(T)-nl()-)1-F2(,Tn)+m2(n)+n2() o Uo + 12] dl d|j < 6T2[U3 (II3l1-/211 )+(bl+b2) 111-0211 ] Analogously, from (28) and (29), ix (x y)-*2x (x y)I 6T[Uc3(11||1-211)+(bl+b2)11|1-2113] llry(xy)-2y(xy)l< 6T[cu3(|111-211)+(bl+b2)111i —l21] Thus 11'1-'211 - 0 as 11Y1-211 + 0 uniformly in K. Finally, the set K is obviously convex, closed and compact with respect to the norm I||I| of E. By Schauder's fixed point theorem we conclude that there is an element /(x,y)eK such that o = T ~, or x y O(x,y) = Uo(x)+vo(y)+J [f(,,(~,),x(i,),y(~,T))-m()-()-]d dr, o 0 T m(y) = T f(,y,(,y),x(,y),y(,y))d~ - ~, o Tn n(x) = T-'/ f(x,n,((xl),/x(x,r),(x,7))d -, 0 = O for all o < x,y < T. Obviously ~x, 'y, ~xy exist everywhere in A, are continuous in A, and, everywhere in A, we have 8

xy = f(x,y,,x, y) - m(y) - n(x) - L. Theorem I is thereby proved. 4. REMARK 1 If f is Lipschitzian with respect to all variables x,y,z,p,q in R, and if uo'(x), vo'(y) also are Lipschitzian, then m(y), n(x), as well as >, fx, >ye )xy are Lipschitzian. Indeed, if oi(a) = kla, (02()) = k2a, D3(7) = bo7, 04(o) = k4C0, (3s5() = k5S, then 91(,) = (1-2T b2)-'(k5+2T k2+2T bo M3+12LT bi)P = k6P, T2(a) = (1-2T bl)'l(k4+2T ki+2T bo M2+12LT b2)a = k-7, and then lm(yi)-m(yP) < (k2+boM3+6L bl+b2k6)P = ksp n(xl)-n(x2) 1< (kl+boM2+b1k7+6L b2)c = kg9l. Formulas (33), (34), (35) show that >x, fy are also uniformly Lipschitzian and so is xy = f-m-n-k. 5. REMARK 2 The conditions of Theorem I do not assure uniqueness, as example shows. Take T = 1, uo(x) = 0, vo(y) = O, f = Iz /sin 2t x sin 2ry, for o < x,y < 1, and all yz,p,q. t'ion the following Then the equa uxy - I I1/2 Uxy = I|u|I sin 27x sin 2ry besides the trivial solution il(x,y) = 0, has also the solution 02(x,y) = (16jt4)" sin4rx sin4Jy, o < x,y < 1, and both satisfy the boundary conditions. We have here ml(y) = m2(y) = O, ni(x) = n2(x) = O. Note that we may take N = N1 = N2 = 0, L = 1, Ml = 1, M2 = M3 = 2, bi = b2 = 0. All conditions of Theorem I are satisfied. 6. THE LIPSCHITZIAN CASE We shall assume now that o3(7) = bol7l, so that f is now Lipschitzian in z,y,q with constants bo,bl, b2. In this situation, for given boundary values uo(x), vo(x) and different functions $i, 2 eK we have 9

I l-1_l < (bo+b1+b2) 11l-211, Iml(y)-m2(y) l, nl(x)-n2(x) 1< 2(bo+bl+b2) 11|l-211, l*1(x,y)-*2(x,y) 1< 6T2(bo+bl+b2) 11l-4211, lix(x,y)-sax(x,y ) l, liy(x,y)-2y(x,y ) I < 6T(bobl+b2) ll - 1-4211 Thus 111-r211 = 11 "61-t221I< 6T(T+2) (bo+b1+b2) ll1-0211 If we assume now that 6T(T+2) (bo+bl+b2) < 1, (37) then 5: K + K is a contraction into. This remark yields: 7. THEOREM II (uniqueness) Under the same hypotheses of Theorem I, if D3(y) = bo II, and (37) holds, then ^: K - K is a contraction and problem (11-18) has one and only one solution. The boundary values are represented by the pair of functions w = (uo(x), Vo(y)) of class C1 and satisfying (6) and (7). Therefore, they form a subset 3 of the linear space of all w of class C1 satisfying (6) only, and we take in this linear space the norm I||w1 = maxluo(x)| + maxluo'(x)| + maxlvo(y) + maxlvol(y)|. (38) The solution of the problem (11-18) is actuallythe system W = [~(x,y), m(y), n(x), p]. These quadruples also can be thought of as inbedded in a linear space on which we take the norm I||W| = maxl|j + maxl|xl + maxl|yl + maxlml + maxlnl + 1lj. (39) We shall prove that the solution, or system W, is a continuous function C of the boundary values, or system w, and we write W = ~w, weS3. We shall need the numbers A = (1-6T bl)(l-6T b2) - 36T2bb2, k = 6 T2 bo + 72 T3 bo(bl+b2) (40)

8. THEOREM III (continuous dependence upon the boundary values) Under the conditions of Theorem II, if in addition A > 0, and 0 < k < 1, then the solution W = (U,m,n,|J) of problem (11-18) is a continuous function c^? of the boundary values w = (uovo)e in the topology determined by the norms (38) and (39). 9. PROOF OF THEOREM III Let wl = [uol(x),vol(y)], w2 = [uo2(x),vb2(y)] be a pair of boundary values as in Theorems I and II, and let W1 = [l,,ml,ni,il], W2 = [2,m2,mn2,~2] be the corresponding solutions. Let E = IIwl-w21 = maxluoi(x)-uo2(x) + maxluoi (x)-uo21(x)l + maxjvob(y)-vb2(y)I + maxlvo1' (y)-vo2' (Y) I a = max I|1(x,y) - 2(X,Y)|, y = max I|iy(x,y) - 2y(x,y)l, &' = max Ini(x) - n2(x)I, P = max I|ix(xy) - ~2x(Xy), 6 = max Imi(y) - m2(y)l, 51" = I 1L-421 We shall denote by F1 and F2 the functions F relative to,1 and /2* Then we have f" = T21-22 [FT(x2 y)-F2 (x,y)(x)]dx dyl< boa + biP + b2y, 0 T |ml(y)-m2(y)I = |T- [Fl(x,y)-F2(x,y)]dx - |1 + 121< bo a+ biP + by2 + 6", 0 T |n1(x)-n2(x)| = |T- [Fl(xy)-F2(x,y)]dy-t-l+t2I< boa + bl, + b2y + " 0 Hence 6" < boC + blP + b2y, 6,6' <ba bo b2 + b + b + Analogously, 11

lli(xy)-$2(x,y)l = x y |uoi(x)+vol(y)-uo2(x)-vo2(y) + [Fl(x,y)-ml(y)-nl(x)-il-F2(x,y)+M2(y) o 0 +n2(x)+112]dx dyl< e + T2(boI+blp+b27+b+b'+5"), llx(x,y)-42x(x,y) < e + T(boa+bl+b27+b+b'+&"), lixy(x,y)-2y(xy)_< e + T(bo++bl+b27+y++t+61"), and hence a< ~ + TE(bo+bl+b2y+SPS'+"), P,7 < e + T(boa+b+b+b27+b+P' +6"). (42) Relations (41), (42) yield &" < (boa+bi+b27), 6,a' < 2(boa+bl+b27), a < e + 6T2(boc+blS+b27), (43) P,7 < + 6T(boa+bl1+b27). The last relation can be written in the form = 6T botc1 + 6Tblmip + 6Tb2j17 +, y7 = 6T boSa + 6Tb1lt2 + 6Tb227y + c where o < 1, 2 < 1 are convenient numbers, and then (l-6Tbll)p - 6Tb2~l7 = 6Tbolja + e-,- 6Tb22P + (l-6Tb2~2)7 = 6Tbo2cz+e~2 If A' is the determinant of this system we have A' > A > o, o 6Tbj < 1, j = 1,2, and P = A-l{ (l-6Tb2K2)(6Tbotla+ctl)+(6Tb2l)(6Tbo!+e 2)} < 2Al(e+6Tboa). Analogously, we have < 2A-1(C+6Tboa). Finally, by (43) a < E + 56T3 MoA-1(bl+b2)a + 6T2Moa. Since the number k defined in (40) is o < k < 1, we have a < (l-k)-l(l+12A-lT2(bl+b2)e This proves that,,7,,6&' 6" + 0 as e + 0 uniformly in K. Theorem III is thereby proved. 12

10. A METHOD OF SUCCESSIVE APPROXIMATIONS Under the hypotheses of Theorem III, the usual method of successive approximations defined by Ok+l = Tk,k = 0,1..., converges toward the solution, of problem (11-18), where <o is an arbitrary element of K. It may be convenient to use as first approximation %o = uo(X) + vo(y). Then TnT Fo(x,y) = f(x,y,uo(x)+vo(y), uo'(x), Vo'(y)), ko = T-2/ / Fo(l,i)d dj, o o T T mo(y) = T-l Fo(S,y)d, Uo(x) = T-1 Fo(x,)di, o o x y Si(x,y) = Uo(x)+v0(y) + X [Fo(,~T)-mo(~)-no(~)-io]d dd, o 0 and successively, Fk(x,y) = f(x,y,~k(x,y),Okx(x,y)+ky(xy)), T T T. = T-2 Fk(, f )dd dn, mk(y) = T- Fk(,y)dl, (45) T nk(x) = T-/ Fk(xr)dr, Ok+l(x,Y) = uo(x)+Vb(y) o T T +/ [Fk(,d)-k('n) -nk(,)-(k]di di, k = 1,2,.... o o Then we have Ok -+, Okx + xx, ky + Oy, mk + m, rk+ n, nyk '+ uniformly for o < x < T, o < y < T, and consequently we have also +kxy +,xy as k +* o uniformly. 11. SMOOTHNESS OF THE SOLUTION Two theorems can now be stated concerning the smoothness of the solution (0,m,n,.) of the problem (11-18). (a) Under the conditions of theorems I, II, III, if f(x,y,z,p,q) is of class C1 in R and Uo(x), vo(y) of class C2, then m(y), n(x) are of class C1

and,(x,y) of class C2. This statement was essentially proved in Section 12 of [2a]. A more precise statement is as follows: (P) Under the conditions of Theorems I,, II III, if f(x,y,z,p,q) is of class C1 with Lipschitzian first order partial derivatives, if Uo(x), vo(y) are of class C2 with Lipschitzian second order derivatives, then m(y), n(x) are of class C1 with Lipschitzian first derivatives and ((x,y) of class C2 also with Lipschitzian second order partial derivatives. The proof is the same as for (a). An analogous statement holds: (y) Under the conditions of Theorems I, II, III, if f(x,y,z,p,q) is of class Cl+r in R with Lipschitzian partial derivatives of the order l+r, if Uo(x), vo(y) are of class C2+r with lipschitzian derivatives of the order 2+r, then m(y), n(x) are of class C with Lipschitzian derivatives of order l+r, and ~(x,y) is of class C2+r with Lipschitzian partial derivatives of the order 2+r. 2. CRITERIA FOR THE EXISTENCE OF PERIODIC SOLUTIONS IN THE LARGE OF THE ORIGINAL PROBLEM 12. A DIFFERENTIAL EQUATION CONTAINING A. SMALL PARAMETER Let us consider the differential equation Uxy = f(x,y,u,ux,uy), f = e[BV(x,y)+Cu+rl(y)ux+42(x)uy] + e2g(x,y,u,ux,uy), (46) where e is a small parameter, and j, 1, \r2, g are periodic functions of period T in x and y. The Fourier series of jr, *1, V2 will be denoted by 00 $(x,y) ) (amn cos rnmx cos nuy + bmn cos nxmx sin ncy + cmn sin rnxx cos nuy m, n n + dmn sin mawx sin nwy), 00 8i(Y) eo + (en cos 1 noy + fn sin ncy), 00 82(x) go + (gn cos 1 ncuw + }n sin nwx). 14

If uo(x), vo(y) denote arbitrary boundary values, with Vo(o) = 0, and uo, vb both of class C1 and Uo', vO' Lipschitzian with constants kl, k2 respectively, then it is convenient for us to denote their Fourier series as follows: 00 u~(x) ~ O~ + (cO cos nww - an + Pn sin nax), 1 (47) 00 Vo(y) - (7n cos ncy - 7n + n sin ncy), 1 where both series Zmir, ZYn are absolutely convergent. With this notation we have 00 Uo(o) = a%, vo(o) = 0, T T- 1 uo(x)dx = o - L n, 0 1 T 00 oo T-1 Vo0(y)dy = - no 1 If we apply formally the method of successive approximations of No. 10 to Equation (46) with initial values uo(x), vo(y), we obtain-at the first approximation and preserving only the terms in ~-a quadruple [0o,emoeno,e.o0], with ~o, mo, no, Lo given by ~o(Xy) = uo(x) + vo(y) + e x y 0 0 [Fo(~,T)-mo(e)-no(T)-[o]d~ d), o o T mo(y) = T- Fo(F,y)da - -, 0 T T -o = T-2 / Fo(,y)di dil, 0 0 T no(x) = T- Fo(x,)dr -.i, o (48) Fo(x,y) = *(xy)-Cuo(x)-Cvo(y)-*i(y)uo'(x) - *2 (x) Vo'(Y) If we write 00 mo(y) ~ (Bn cos ncy + Cn sin ncy), 1 00 no(x) 7 (Dn cos nox + En sin nowx), 1 Po = Ao 15

we obtain 00 00.io = Ao = ao + c(C - a - 7YS ) 1 1 00 mo(y) = aOO + i 1(Y) + C(O - s) + CvO(y) + govo'(y) - 1, -I (49) no(x) = aoo + 00 d2(x) + Cuo(x) + eouo (x) + c(- 7s) -, 1 0% In Ao /~~~~T v00 1(y) = T-1 (,y)dS - aoo (aon cos ny + bon sin rny), o 1 00?2(x) = T-1 / (x,y)dy - aoo~ (aon cos rn)x + con sin nrx). o 1 terms of the Fourier constants relations (49) become 00 00 = aoo + C(a- - Ys) B = CYn + n)go 6n + aon, 1 1 (50) (51) Cn = C6n - ngo Yn + bon, Dn = Ca + nogoyn + bon, En = CPn - nceo an + Cno., n = 1,2,... We shall denote by u(x), v(y) the same functions Uo(x), vo(y) up to a constant so as to make them with mean value zero, or 00 00 u(x) = Uo(x) - ao + C as (as cos swx + Ps sin sox), 1 1 (52) 00 00 v(y) = vo(y) + 7 S (7s cos sy + b sin swy). 1 1 If we require o0 = 0, mo(y) 0 O, no(x) 0, then relations (48) reduce to 16

Cv(y) + gov'(y) = - C)(y), Cu(x) + eout(x) For eo = 0, we have u(x) = - C-1 2(x); (535) for eo 7 0, we have u(x) = exp(-eo Cx) [K+eo K = - eo -C(1-exp(-eo-CT)) x exp(eo c), 2(~)d0 ], 0 -1 T 0 (54) / exp0(eo-1C) 2( )d~ 0 exp(e -OCx)dx T where the constant K is determined in such a way that u(x)dx = 0. 0 Analogous relations hold for v(y). This determines all the coefficients n, Pn, 7Yn, bn, n = 1,2,.... Actually, relations (53), (54) are equivalent to those we obtain from (51) by taking Bn = Cn = Dn = En = 0 and solving with respect to an,, Yn, bn: C, = (C2+n22eo2)-1 (-Cano+nLoeocno), 7n = (C2+n 22go2) - 1 (-Caon+ncogobon), Pn = (C2+n22eo2) 1 (-ncueoano-Ccno), &n = (C2+n22go02)-1 (-ncwgo-Cbon), n = 1,2,.... The coefficients C0n, Pn, Yn, bn, n = 1,2,..., being so determined, then equation *o = Ao = 0 yields 00 00 O = - C-laoo + s +, Ys 1 1 provided the series Zas, Z7s converge. hypotheses of the criterion I below. functions Uo(x), Vo(y) so determined, This will be the case under the We shall denote the corresponding say U(x), V(y), or 00 00 U(X) = Uo(X) + CQ - o + (c SX - + (S Cs - sin sOx), 1 1 I (55) 00 00 1 1 L2i 17

Under the conditions of Criterion I we shall require that these functions are interior points of the set defined by relations (6) of Theorem I. 13. CRITERION I If the function f given in (46) for all | e< co satisfies all conditions of Theorems I, II, III with given constants T, N, N1, N2, L, M, M1, M2, M3, bo, bl, b2, and in addition f is Lipschitzian with respect to x and y in R, if C / 0, and the functions Xl(x), O2(y), U(x), V(y) defined in (50) and (55) are of class C1 with Lipschitzian first derivative, and lu(o)l< No < N, IU(xl)-U(x2)1< Njolx1-x2l, N1o < Ni, IV(yl)-V(y2)< N2o I -yl 21 N20 < N2, then there is some eo, 0 < Eo < co, such that Equation (46) for all leI< co, possesses at least one periodic solution ~(x,y) of period T in x and y, which is Lipschitzian in E2 together with Ax, (y,,xy: oxy - f(x,y,/,x,/y), /(x+T,y) - /(x,y) = /(x,y+T). The periodic functions Uo(x) = O(x,o) = O(x,T), vo(y) = 0(o,y) - 0(o,o) = z(T,y) - O(T,T), satisfy moreover relations (6) of Theorem I. 14. PROOF OF CRITERION I Let us denote by k4o, k5o the Lipschitzian constants of U(x) and V(y) respectively, and let k4, ks be arbitrary numbers k4 > k4o, k5 > k5o. Let us denote as usual by kl, k2 the Lipschitzian constants of f with respect to x and y respectively in R. Let S be the set of all pairs w = [uo(x),vo(y)] of functions Uo(x), vo(y) periodic of period T, of class C1, with derivatives Uo'(x), Vo'(y) Lipschitzian of constants k4, ks, and satisfying relations (6) of Theorem I, that is luo(o)l< N, |uo(xl)-Uo(x2)1< Nlx1-x21, Vo(O) = 0, 1vo(y1)-Vo(Y2)I< N21Y1-Y221 Then wb = [U,V]ES. We shall consider S imbedded in the Banach space of all pairs of periodic functions of class C1 with norm il||w = maxlu(x)| + maxlu'(x)| + maxlv(y)| + maxlv'(y)l. (56) For every w = [uo(x),vO(y)]eS we shall determine the solution W = [.,Em,En,qL]

of the modified problem relative to (46). Since this solution can be determined by the method of successive approximations of No. 10, we see that W can be written in the form ~(x,y,e) =,0(x,y) + ~!(x,y,E), m(y,e) = mo(y) + m(y,e), n(x,E) = no(x) + n(x,E), +(e) = o + (e), where, m, n, p. = 0(1) uniformly as e + 0, and o, mo, no, Pio are given by formulas (48). Using the functions u(x), v(y) as in No. 8, equations i = 0, m(y) - 0, n(x) 0 reduce to Cv(y) + gov'(y) = - c(y) - m(y,), Cu(x) + eou'(x) = - n2(x) - (xe), (57) 00 00 aoo + C(ao - s) - ) = 1 1 For eo = 0 we have u(x) = -C-l.2(x)+n(x,e); (58) for eo 0 we have rX u(x) = exp(-eO Cx)[K+e' / exp(eO CQ)(E2(()+ n(,e))d5, 0 (59) T x K = -e eoxp(eo Cx)dx exp(eo CO( T ))2 +n(S,E))d5, and hence u'(x) = - Cl(d/dx)( 72(x)+n(x,e)) if e = 0, (60) u'(x) = - eo (Cu(x)+ e2(x)+n(x,c)) if eo ~ 0. Analogous formulas hold for v(y). This determines u(x), v(y) and hence all coefficients an, Pn, Yn, )n, n = 1,2,... By Remark 1 we know that m, n are Lipschitzian functions, and 19

so are m, n as well as Xli, 2. Thus u(x), u(y) are periodic functions of mean value zero, of class C1, with Lipschitzian first derivative. Thus, the series Znq, ZYn are absolutely convergent, and (47), (57) yield 00 00 %IO - -(a oo as Olo = - Cl(aoo-p(e) + 7 s + 7 1-~~~ l.~~~~ ~(61) - C- (ao-(E)) - u(o) - v(o)), 00 00 uO(x) = u(x) + Oo - as, vo(y) = v(y) - s (62) 1 1 Note that these functions, when we take m = n - 0, reduce to U(x) and V(y) respectively, and thus the convergence of the series Zan, ZYn of No. 8 is proved above. Actually, for every w = [u (x),v (y)]eS, we can first determine m, n, t as in theorems I, II, III, and the method of successive approximations of No. 10) then we determine i, ', and finally the second members of formula, (58), (59), (60), (62) and analogous ones determine new functions, say = [o(x),7o(y) 1. Thus, we have a map', —. -;. -^w = w, weS, whose fixed elements w = 7w, if any, have the property that p = 0, m(y) - 0, n(x) - 0 We have already chosen the uniform topology of class C1 on w and w by means of (55). Let us choosethe uniform topology of class C~ on m, n as in Theorem III, as well as on m, n. We know already from Theorem III that m, n are continuous functions of w, and so are m, n. The second members of (58), (59), (60), (62) defines continuous functions of m, n. Thus ~C is a continuous function of w for weS in the topology defined by (55). By Theorems I, II, III we know that m(y,e), n(x,e) are Lipschitzian functions. The same property holds for m(y,e), '(x,e), but these functionsas well as their Lipschitzian constants-have a uniform bound of the form Me for some M > 0 and all |1e< e. Then, by choosing convenient constants k, kl, we have 20

l(yE)l, l (x,)]< k~, 1^11< kE, l (Yie)-m(y2,~)I< kelYil-Y2, I(xi,eC)-'(X2,E) l< kelxi-X21, |UO-U|< kle, IVo-VI< klE, liuO -U1 I< kil, 1vo'-V' 1< kle, |ITo(x)-U)xi)-uo(x2)+U(x2) I< kle, IVo(yI)-v(Yl)-V(Y2)+V(Y2) k1e, l0o' (xi)-U' (xI)-Uo' (X2)+U' (X2) I1 klc, vo' (Y1)-V' (yl)-v' (y2)+V' (Y2) 1< kle. If k4, k5 are the Lipschitzian constants of U', V', and Co = min[eo,k1 (N-No), kl (Ni-Nio), kl (N2-N2o) then for IeI< Co we have Iuo(o) 1< U(o) + kie < No + kle < N, |IU(X1)-Uo(x2) I< (N1o+ke) Ixi-x21 < Ni X-x2 1, I' (x)-u (x2) I< (k4+kie) Ix-x2j, vo(y1)-vO(y2) I< < (N20+kie)IY1-Y21< N2Y1Y2-y, IVo' (Y1)-v' (y2) I< (k5+klE) IY1-Y21 This shows that, for ei <, E, maps S into itself, '%: S + S, and S is a convex closed compact subset of a Banach space. By Schauder's fixed point theorem ~ possesses at least one fixed element w = oweS, w = [uo(x),vb(y)], with uo(x), vO(y) satisfying relations (6) of Theorem I. Criterion I is thereby proved. 15. EXAMPLE The equation Uxy = E(l+u) + 2g(x,y,u,ux,uy) where g is periodic of period 2ir in x and y, has a periodic solution,(x,y) of the same period O(x,y) = 1 + 0(e) The analogous equation uxy = e(sin x - cos y - sin y + cos y + u + ux - uy) + e2g(x,y,uuxuy) with g as above has a periodic solution = cos x - cos y + 0(e). 21

16. ANOTHER EQUATION CONTAINING A SMALL PARAMETER Let us consider the differential equation uxy = f(x,y,u,ux,uy), (65) f = 'V(x,y) + Cu + ui(y)ux + *r2(x)uy + eg(x,y,u,uxuy), where e is a small parameter, and \r, it, V2 are as in No. 12. We assume here that, for e = 0, E'quation (63) possesses a known periodic solution of period T in x and y, 0o(x,y) = u0(x) + v0(y), where no(x), vo(y) have Fourier series (47), and hence r(x,y) = - Cuo(x) - CVo(y) - l(y)uo (x)-2(x)vot'(y). Under the hypotheses below, we shall prove that for IelIO sufficiently small, (63) possesses a solution O(x,y) = 0o(x,y)+O(e) which is periodic of period T in x and y. 17. CRITERION II If the function f defined in (63) for all e < co satisfies all conditions of Theorems I, II, III with given constants TN,N,N,N2,LM,MMM2,M3, bobl,b2 and in addition f is Lipschitzian with respect to s and y in R, if C A O0 if (63)possesses for e = 0 a solution 0o(x,y) = U(x)+V(y) with U, V periodic of period T, if the functions U(x), V(y), i,(x), kJ2(y) are oi class C1 with Lipschitzian first derivative, and I u(o)l< No < N, IU(xl)-U(x2)I< NIolx-x2i, N10 < Ni, V(yl) -V(y2) < N2oly-yl 21 N20 < N2, then there is some eo 0 < o< Co, such that Equation (63) for all Ile< e, possesses at least one periodic solution O(x,y) of period T in x and y, which is Lipschitzian in E2 together with Ox, Oy,,xy xy = f(x,y,,Yx, y), O(x+T,y) = O(x,y) = O(x,y+T). The periodic functions Uo(x) = O(x,o) = O(x,T), Vo(y) = 0(o,y)-O(o,o) = 0(T,y)-0(T,T), satisfy moreover relations (6) of Theorem I. 22

18. PROOF OF CRITERION II As in No. 12 let us apply formally the method of successive approximations of No.10 to Equation (63) with arbitrary initial values uO(x), vo(y). Then the quadruple [~,m,n,p], solution of the modified problem for Equation (6), is given by jZ$= 0(x,y,~) + 0(xyy), m(y,) = mo(y) + m(y,e), n(x,E) = no(x) + n(x,), t(e) = [o + 1(e), where 0,m,n,. = 0(1) uniformly as e + 0, and yo,mo,no,yo are given by formulas (48). In addition we know that for Uo(x) U(x), Vo(y) - V(y), we have po = O, mo(y) 0, no(x) 0O. We can now repeat with obvious variants the argument of the proof of Criterion I. 19. EXAMPLES The equation Uxy = - 1+ + (y) + (X)y + (y)ux + g(x,y,u,Ux,Uy) for e = 0 has the obvious solution u = 1. Since C f 0 the same equation has a periodic solution i of period T in x and y for all 1el sufficiently small. Analogously, the equation uxy = - cos x + sin x + cos y - sin y + u + ux + uy + eg(x,y,u,Ux,uy) has, for e = 0, the obvious solution u = cos x - cos y. Since C A 0, the same equation has a periodic solution of period 2i in x and y for every Iei 0 O sufficiently small. 20. APPLICATION TO THE WAVE EQUATION Let us consider the differential equation utt - us = f(t,),u,utu), (64) where f is periodic in t and 5 of period T. Then the transformation t = x + y, x = x - y, x = 2-l(t+), y = 2-l(t-), (65) changes (64) into Uxy = F(x,yu,UxUy), 23

where F = f(x+y,x-y,u,2-ux+2 uy,2 ux-2 u) ( and F is periodic of period T in x and y. The theorems I, II, III and the criteria should now be applied to (66). Other transformations beside (65) can be used. As an example, let us consider the equation utt - u = E[X(t, )+Cou+Xl(t, )ut+X2(t,S)uS] + e g(t,,u,ut,uy), (67) (ty) = Ao+B1 cos 2T + C1 sin 2t + B2 cos 2+ + C2 sin 25 + D1 cos(t+5) + E1 sin(t+5) + D2 cos(t+S) +E2 sin(t-), (68) li(t,&) = A + B cos(t+~) + C sin(t+S) + D cos(T-4) + E sin(t-5), k2(t,) = A' - B cos(t+S) - C sin(t+S) + D cos(t-S) + E sin(t-0), where Ao, B..,.E are constants, Co f O0 and g is of period u in t and ~. By the transformation t = 2-'(x+y), t = 2-1(x-y), x = t+5, y = t-_, (69) Equation (67) is changed into uxy = e[C(x,y)+4 COu+l(y)ux+I2(x)uy] + e g(2- x+2 y,2 x-2 yu,ux+uy, (70) Ux-Uy) where the second member has period 2t in x and y, and with V(x,y) = aoo + aio cos x + col sin x + aol cos y + bol sin y + all cos x cos y + bll cos x sin y + cll sin x cos y + d1l sin x sin y, 1l(Y) = eo + el cos y + fl sin y, 2((x) = go + gi cos x + hi sin y, (71) 4aoo = Ao, 4alo = D1, 4coi = E, 4aol = D2, 4bol = E2, 4all = B1+B2, 4bll = C1-C2, 4cll C+C2, 4dll = -B1+B2, 4eo = A+A', 2ei = D, 2fl = E, 4go = A-A', 2g = B, 2h = C. 24

By Criterion I we conclude that, if Co f 0 and 1e1 sufficiently small then Equation (67) has at least a periodic solution $-(x,.y.,) of period 2a in x and y, and then Xquation (64) has at least a solution u(t,S,) = ~(t+S,t-S,), also of period at in t and ~. 21. ANOTHER EXAMPLE Let us consider the differential equation utt - u = (t,S)+ Cou + il(t,y)ut + X2(ty)ug + eg(t,S,u,ut,ut), (72) where X, i1, %2 are given by (68) and again Co A O. By the same transformation (69) Equation (72) is changed into Uxy = *(x,y) 4-f1oU + Vl(y)ux+V2(x)uy + eg(2-1x+2-ly,2-lx-2-ly,u,ux+uy,Ux-y), (73) where V, V1, *2 are given by formulas (71). It is immediately seen that (73) for e = 0 has a solution of the form u(x,y) = U(x) + V(y), U(x) = ao + a1 cos x + Pi sin x, V(y) = yi cos y + 61 sin y if and only if B1 = AlCo(-ED1+DEl)+A 1(A+A.' )(DDl+EFl)+A 2Co(CD2+BE2)+A 2(A-A' )(BD2-CE2), B2 = AlCO(ED1+DE) +A1(A+A. )(DDl-EEl)+A2CO (-CD2+BE2)+A2(A.-A. )(BD2+CE2), C1 = AlCo(-DDl-EE1)+t (A.+A' )(-EDl+DEl)42Co(-BD2+CE2)-A2(A-AT )(CD2+BE2), C2 = AlCo(-DD1+EEi)+A (A.+A.1 ) (EDi+nEl)+A2CO(BD2+CE2)+A2(A-A' )(CD2-BE2), A1 = (C2+(A+A )2)-1, A2 = (C2+(A-A. )2)-1. (74) In this situation, then a:1 = A1(-CoDl+(A+A')E1), 1 = Al(-CoEl-(A+A.' )D1), 71 = A2(-CoD2+(A-A' )E2), 6 = A2(-CoE2-(A-A')D2), Ob = - CoA1o. 25

By Criterion II we conclude that, if Co, 0, e] sufficiently small, and relations (74) hold, then Equation (73) has a periodic solution z(x,y) of period 2ri in x and y, and (72) has a solution u(t,,E) =,(t+,t-,~) also of period 2g in t and ~. For instance, for the equation utt - ut = D1 cos(t+S) + E1 sin(t+5) + D2 cos(t+t)+E2 sin(t-S) + u + ut + Eg(t,S,u,ut,u ) where DI, E1, D2, E2 are arbitrary constants, and g periodic of period iC in t and ~, we see that relations (74) are all satisfied with BI C1=C = B2 = C2 = t0 B = C = D = E = 0, Ao = 0, CO = 1, A = 1, A' = 0, A1 = A2 = 2-1. The corresponding Equation (73) is 4uxy = D1 cos x + El sin x + D2 cos y + E2 sin y + u + ux + uy + eg. For e = 0 this equation has the periodic solution PO(x,y) of period 2t in x, y given by -2~o(xy) = (D1-E1)cos x + (DI+E1) sin x + (D2-E2)cos y + (D2+E2) sin y and hence (75) for e = 0 has the periodic solution uo(t,j) = -2-'(D1-El)cos(t+S)-2-1(Dl+El)sin(t+ )-2-l(D2+E2)cos(t- ) -2-l(D2+E2)sin(t- ). Thus, for all |e sufficiently small Equation (75) has a periodic solution of period 27c in t and ~ of the form u(t,t) = Uo(t,y)+O(E). 26

BIBLIOGRAPHY 1. N. A. Artem'ev, Periodic solutions of a class of partial differential equations. Izv. Akad. Nauk, SSSR Ser. Mat. 1, 1937 (Russian). 2. L. Cesari, (a) Periodic solutions of hyperbolic partial differential equations, Intern. Symp. of Nonlinear Differential Equations and Nonlinear Mechanics (Colorado Springs, 1961), Academic Press, 1963; 33-57; Intern. Symp. on Nonlinear Oscillations (Kiev. 1961), Izdat. Akad. Nauk, SSSR, 2, 1963, 440-457. (b) A criterion for the existence in a strip of periodic solutions of hyperbolic partial differential equations. To appear. (c) Existence in the large of periodic solutions of hyperbolic partial differential equations, to appear. 3. B. A. Fleischman and F. A. Fiken, Initial value and time periodic solutions for a nonlinear wave equation. Comm. Pure Appl. Math. 10, 1957, 331-356. 4. V. N. Karp, (a) Application of the wave-region method to the problem of forced nonlinear periodic vibrations of a string. Izv. Vyss. Vc. Zaved. Mat. 6 (25), 1961, 51-59 (Russian). (b) On periodic solutions of nonlinear hyperbolic equations, Dokl. Akad. Nauk. Uzbek. SSR 5, 1953 (Russian). 5. A. P. Mitryakov, (a) On periodic solutions of the nonlinear hyperbolic euqation. Trudy Inst. Mat. Meh. Akad. Nauk Uzbek. SSR 7 1949, 137-149 (Russian). (b) On solutions of infinite systems of nonlinear integral and integro-differential equations. Trudy Uzbek. Gosudarstv. Univ. 37, 1948 (Russian). 6. C. Prodi, Soluzioni periodiche di equazioni alle derivate parrziali di tipo iperbolico nonlineari. Annali. Mat. Pura. Appl. 42, 1956, 25-49. 7. P. V. Solov'ev, Some remarks on periodic solutions of the nonlinear equation of hyperbolic type. Izv. Akad. Nauk. SSSR Ser. Mat. 2, 1939, 150-164 (Russian). 8. C. T. Sokolov, On periodic solutions of a class of partial differential equations. Dokl. Akad. Nauk. Uzbek. SSR 12, 1953, 3-7 (Russian). 9. 0. Vejvoda, Nonlinear boundary-value problems for differential equations, Proc. Conference Differential Equations and Their Applications (Prague 1962). Publishing House Czechosl. Acad. Sci. Prague 1963, 199-215. 10. M. E. Zabotinskii, On periodic solutions of nonlinear partial differential equations. Dokl. Akad. Nauk SSSR (N.S.) 56, 1947, 469-472 (Russian). 27

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