THE UNI VERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Progress Report No. 10 THE IMPLICIT FUNCTION THEOREM IN FUNCTIONAL ANALYSIS Lamberto Cesari ORA Project 05304 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP-57 WASHINGTON, D.C, administered through: OFFICE OF RESEARCH ADMINISTRATION August 1964 ANN ARBOR

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THE IMPLICIT FUNCTION THEOREM IN FUNCTIONAL ANALYSIS* by Lamberto Cesari We consider a functional f(y,z), or f: yoxZo- F, where F is a linear space, Yo, Zo are subsets of linear spaces Y and Z of functions from an arbitrary space X into some Banach space E, and YCZ, YoCZo, (Y, Z Banach spaces, E finitely dimensional). The process usually associated with f may, or may not, involve a "loss of derivatives" in the terminology of J. Moser (Refo 6). Under various sets of hypotheses we prove that there exists at least one function EcYo0'Zo such that f(4,) = 0. (1.1) Under one of the sets of hypotheses taken into consideration, and for which the process usually associated with f involves no loss of derivatives, we prove a Theorem A (Section 2) one of whose corollaries (Section 5) includes a statement which we shall apply in a coming paper (Ref. 1) on hyperbolic partial differential equations. A slight restriction of the hypotheses guarantees (Theorem B) a local uniqueness of the function 4 above. A number of applications are made. The proof of Theorem A is based on the remark that, by the usual argument of the implicit function theorem in functional analysis (see, for instance Ref. 5, pp. 174-195), an element yzcYO can be proved to exist, for every zcZo, such that f(yz,z) = 0, and that the ensuing mapZC: Zo - Yo restricted to YO can be proved to satisfy Schauder's fixed point theorem. The usual argument is here given for locally convex topological vector spaces. In the case in which the process usually associated with f involves an actual loss of derivatives, the previous remark still applies and an analogous theorem can be proved (Theorem C) provided full use is made of the implicit function theorem that J. Moser (Ref. 6) has recently proved-and applied to a number of problems. Also, use is made of other considerations of the papers of J. Nash and J. Schwartz (Ref. 7 and 10) which were point of departure for the results of J. Moser (Ref. 6). Applications, other than those mentioned above, will follow. Considerations in the line of J. Leray's are also included (Section 1, No. 8). *Research partially supported by NSF Grant GP-57 at The University of Michigan.

SECTION 1. THE IMPLICIT FUNCTION THEOREM OF FUNCTIONAL ANALYSIS l THE EQUATION f(y) =. Let Y be a Hausdorff, complete, locally convex topological vector space, and f(y) a (not necessarily linear) functional f: Yo - F on a subset Yo of Y to a linear space F. We shall assume that an "approximate" solution yO to equation f(y) = 0 is known. Let (V) be a base for the neighborhood systems of 0 in Y, each element V of which is balanced, absorbant, and convex, and assume that Ve(V), X > 0 implies XVe(V). Let S be a balanced convex closed subset of Y with yo+SC Yo. We shall denote bye(B) the null space of an operator B. We shall need the hypothesis: (Gl) There are numbers o a5, 0 <0 o < 1, 0 <a_ < 1-o, and linear operators B: F -+ Y, A: Y - F, withat(B) = 0O such that yl, Y2CYo+S, Y1-y2eVE (V), implies B[f(yl)-f(y2)-A(yl-y2) ]Eo(SfnV), (1.2) Bf(yo)etdi, BA ' I. (1.5) Let T: S -+ Y be the map defined by Ty = y - Bf(y), yeS. (1o4) (i) Under hypothesis G1, there is one and only one element 4 in S with f(f) = 0, and 4 is a fixed point of T in S. Proof of (i). By (1.3) and (1.4) we have Tyo = Yo - Bf(yo), Bf(yo)S. (1.5) For any two elements yl, y2eS, by (1.2) and (1.4), and by taking V = Y, we have also Tyl - Ty2 = B[f(yl)-f(y2)-A(yl-y2)]eoS. (1.6) Finally, for every yeS, by (1.5) and (lo6), Ty - yo = [Ty-Tyo] + [Ty-yo] e (+?o)SCS. (17) 1

Thus T: S -+ S The remaining part of the proof is now rather standard. Let Yn+ = TYn, n = 0,1,.... Then YneS for all no Let Vc(V)o Since V is absorbant, there is some X > 0 such that yl-y2eXV, and, XVE(V) Let us prove that n Yn+l - YnEgXV, n = 0,1,... (.8) This relation is true for n = 0. Assume (1.8) is true for 0,1,..., n-l, and let us prove it for n. Indeed n-l Yn+l - Yn = B[f(n)-f(yn-A( Yn-yn-l)E ) o(eco V) = Thus (1.8) is proved for every n. We have Yn+p - Yn = (Yn+l-Yn) +''+ (Yn+p-Yn+p-l) (1.9) tea)Va +..... + i^n P lhVE^(l-) (1., where 9 < 1. Thus, given any V (V], there is some n such that n > n p > 1 implies (l- o)-' K < 1/2, and yn+p-ynE(l/2)Vc2l/2)VC V. Thus [Yn] is a Cauchy sequence in the topology of Y. Since Y is complete, = lim Yn as n -+ oo exists, fEY, *eS. Finally, (1.9) implies that r-yn belongs to the closure of ((l-Wo)-'ok)V, and since the numerical factor is < 1/2 for n > n, we conclude that f-yne(1/2)V for n > n. Thus, for n > n, we have - T = (i-Yn) + (Yn-T n-l) + (TYn-T*) (1.10) E (1/2)V + -o(l-o-l -1) lCl/2)V + (1/2)V = V where V is an arbitrary element of (V} Since Y is Hausdorff, we conclude that *-Ti = 0, or i is a fixed element of T. The uniqueness of i in S follows by standard argument. Indeed, if y, z are fixed points of T, then y = Tn z = Tnz for every n, and, if Ve(V}, and X is so chosen that y-zeXV, then y-z = Tn(y-z)eoXV, for every n. We can take n so large thatol x < 1, hence y-zeV for every Ve(V). Since Y is Hausdorff, we have y-z = 0. Note that, from y1-yoeXVj we have deduced (1.9), hence, yn+p-Yoe(l-4) -l( V). This can be reworded by saying that, if Tyo-yoeVe(V), then I-yoe(l"o)-lV. Now i = Ti implies, by (1.4), Bf(i) = 0, and since the null space 4(B) of B is zero, we conclude f(i) = 0. Conversely, if f(y) = 0^ yeS, then y = Ty, and, by uniqueness, y = i. 2

2o CONTINUOUS DEPENDENCE OF * ON PARAMETERS Let us assume that f(y,z) depends on y as in No. 1 and on a parameter z varying in a subset ZO of a locally convex topological vector space Z, hence f: YoxZo - F. Let (W) be a base for the neighborhood system of 0 in Z. Let 7 denote some closed subset of Z with ZCZoCZ. We shall assume that for a fixed subset S of Y as in No. 1, and for fixed numbers: *Q,:~ as in No. 1, hypothesis (G1) holds for every zcZ, and thus uniformly with respect to Z. In particular, both operators Az, Bz may depend on z. By (i), for every zeC there is a unique element rZES such that f(4z,z) = O, which is the fixed point of the map Tz: S - S, defined by Tzy = y-Bf(y,z), or *z = Tzfz, zVcS, zcW. Let C: Z + S be the map defined by 4z = Cz, zEc, rze-S. We shall need the hypothesis: (G2) Given Vc(V) there is some Wc[(W such that zl, Z2JC, zl-z2EW, yES imply BZlf(y, z) -Bz2f(y, z2)V (1.11) The analogous hypothesis that, given VEcV) and any compact subset C' of Z, there is some Wc[W) such that zi, z2eCnC', z1-z2EW, yeS imply (1.11), will be denoted by (G2)c. (ii) Under hypotheses (G1) (uniformly in zcZ), and (G2),m": Z + S is uniformly continuous on Z. Under (G2)c, T is uniformly continuous on every subset C'CZ, C' any compact subset of Zo Proof of (ii). Let Vc[V}, and Vo = 2-1(l-4)V. By (G2) there is an element Wc(W) such that z1, Z2Ce, z1-z2EW, yeS, imply Bzlf(yzil) - BZ2f(y,Z2)eV Let yi,n+l = TziYin, n = 0,1,..., i = 1,2, with yio = Y2o = yo. Then Yil = TziYo = - Bzif(yozi) i = 1,2, and hence yll - Y21 = Bzlf(yo,zi) = Bz2f(yO,Z2)EVo. Let us prove that yYiYn n n-l T Tzy (l++..+n 1 yln - Y2n z= T,1y Tz2yo E( o+.. n 1,, (1.1.) 5

This is true for n = 0. Let us assume that (1.12) is true for O,1,...,n, and let us prove it for nl+. Indeed Y1in+l - Y2)n+l B= - Blf(ylnZli) + Bz2f(Y2n,Z2) + Yln - Y2n - B [f(ln zl) -f( Y2n zl ) -A z ln-Yn n)] - Bzlf(Y2n,1) + BZ2f(Y2n, Z2), and then n-l n YJn+l - Y2,n+l o(i+ o o )V +V o =+ (1+ of+..+ o)VO. Thus (1,12) is proved for every n. As n - oo, we obtain that Jr -4r belongs to the closure of (1- o) lV, hence to the closure of (il)Vo and hence to V. We have proved that, given Vc(V) there is WctW) such that zl, z,2E, z1-z2EW, imply.zl-4z2EV. The first part of statement (ii) is thereby proved. The second part can be proved with obvious changes. 5, CONTINUITY OF l Under the hypotheses of No, 1, let us assume that Y is a space of functions y: X + E, where X is any metric space with distance function p(xlX2), that E is a locally compact Banach space (hence, finitely dimensional). If (ii,.,rm) is any base for E, then y = yjl+.lt y+ m Ym1, l,,ym real. Thus Y can be thought of as a space of vector valued functions Let ([C be a given collection of compact subsets C of X, with the property that any compact subset Co of X is contained in at least one element C (CJo We shall assume that Y is precisely a space of functions y: X -t E, with the topology of the uniform convergence on the compact subsets of X, Hence we can define the topology of Y by means of the seminorms jlylJCY = sUPly(x) lE, Ce C) (1153) xeC It is equivalent to use the seminorms IIYlliCY = sUPYi(X), Ce(C) i = l1. m where | denotes absolute value. It is possible that X be an open subset of a Euclidean space Ed, that each yeY is of class CN in X, that is, each yi is continuous in Y together with all its partial derivatives, say DSy, of 4

all orders s = 0,1,...,N, D~y = y. Then we shall take the topology of the uniform convergence of y and all DSy, 0 < s < N, on each compact subset C of X. This topology is defined by the seminorms IIYi!;Y - supIDsyi(x)I, CecC}, i = 1,..,m, s xEC = 0,1,...,N. or equivalently the seminorms YI ICY max s = O,1,....,N supllDS(x) lE, Ce(C). xeC (1.14) (G5) For every Ce(C) and 5 > O, there is a > 0 such that xl,x2CC, p(xl,X2)< a, yeS imply I (Bf(y))(xl)-(Bf(y))(x2)-y(xi)+y(x2)IIE <. (1.15) In the case yEC in X as above, we must replace (1.15) by lDS(Bf(y) )(xl)-D(Bf(y))(x2)-DXy(xl)+Dy(x^); < i, s = 0,1,...,N. (1.16) (iii) Under hypothesis (G1) and (G3), the element 4 of (i) is (uniformly) continuous on each compact subset of X and continuous on X. If yeC in X as above, all DS*, 0 < s < N, are (uniformly) continuous on each compact subset of X, and continuous on X. Proof of (iii). Let Ce(C} and 5 > 0 arbitrary. Let a > ber defined in (G3) in correspondence of C and the number. and vi = 4(xi), i = 1,2. Then * = Tr, and 0 be the numLet xl, x2eC, IV1-V211E = II(T )(xi)-(T T)(x2) IE = lI(X1)-(Bf() )(X1)-*(X2)+(bf() )(X2)IE <. This proves the uniform continuity of 4 on each Ce(C). Since each compact subset of X in contained in some Ce(C), the first part of (iii) is proved. The same if all yeY are of class CN as above. We shall only take vsi = DSr(xi), i = 1,2, s = 0,1,...,N. Since X is metric, hence Hausdorff, X satisfies the first axiom of countability and X is a k-space. Thus 4, being continuous on each compact subset of X, is continuous on X. We now could assume that f(y,z), or f: YoxZo -+ F depends on ycYo and on zcZo Z, Z parameter space, as in No. 2. 5

(iv) Under hypothesis (G1) (uniformly for zcW), (G2)c and (G3), the function 0(x,z), xcX, ze~, is uniformly continuous in any set CxC', C, C' compact subsets of X and Z, C'COW. Proof of (iv). In No. 2 we have proved that given 5 > 0 and C', there is a WE(WJ such that zl,z2EC'n Z, Z1-z2EW, imply [lrzl-rz21 _ 5, and hence ||1Z 1i(x)- Z2(X) ||E < 5. On the other hand, we have proved above that there is a' > 0 such that Xl, x2eC, p(xl,x2) < a' imply |IIZ2(X1)-nZ2(X2)lIE -< 5 Thus xi, x2EC, zl, z2EC'nZ, p(Xl,X2) < a, zl-z2EW, imply || (xl,z1)-0(X2,Z2)]|E <ll0(X1,z1)-0(x1,Z2)ilE + j11(X1,Z2)-0(X2,Z2)IIE =IIrZ1(X1)-*Z2(X1) _E + =IlZ2(X1)-4Z2(X2) IE 5< + = 25. This proves the uniform continuity of 0 on Cx(C'ln ), and hence the equicontinuity of the functions 4z: C + E (for each CE(C}). The same reasoning holds when all yeY are of Class CN, and O(x,z), rz(x) are replaced by DSO(x,z), DSfz(x), s = 0,1,...N. 6

SECTION 2. THE EQUATION f(y,y) = 0 1. HYPOTHESES Let X be any metric space with distance function p(xl,x2), let E be a locally compact Banach space (hence finite dimensional) with norm |lell| Let F be any linear space, let Y. Z be locally compact topological vector spaces of functions y: X - E and z: X - E, with YCZ, Y Hausdorff and complete. If (sl~,.,m) is a base for Y, then y = yijl+a. +ymm for every yeY, and hence ycY has a representation y = (yl,.,ym) as a real vector-valued function on X. The same holds for zcZ. We shall take in Y the topology of uniform convergence on compactao We shall denote by (C) a given collection of compact subsets of X covering X, such that every compact subset Co of X is contained in at least one element cc(C}. To simplify our considerations we shall consider two alternate situations. Either X is any metric space, Y is a space of functions X + E which are bounded on each compact subset of X, and we then define the topology of Y by means of the seminorms IlyllcY = supl(x)11E, Xe(C (2.1) xcC Alternatively, X is an open subset of an Euclidean space Ed, and Y is a space of functions y: X + E possessing partial derivatives, say DSy, of all orders 0 < s < N, for a given N. which are bounded in every compact subset of X. Then we define the topology of Y by means of the seminorms IIYIICy max sup||DSy(x) lE, CC(C} (2,2) O<s<N xcC Other analogous situations can be taken into consideration. Let yo be a particular element of Y. For each Ce(CJ let bC > 0 be a given number, and let S denote the subset of Y defined by S = Sb = [yeYillylljy < b C C}]CY (23) Thus, S is a balanced, convex, closed subset of Y. Let Z denote a fixed subset of Z with SCZ, 7

Let f: SxZ -* F be a (not necessarily linear) functional, or f = f(y,z) with yeS, zeZ, f(y,z)eF. In Theorem A we shall prove under the set of hypotheses of this number that there is an element 4eSCZ such that f(4,r) = 0. Concerning Z we need only to know that YCZ, and the topology of Z is defined by means of seminorms i||zlCZ with the following property: For every finite system Cs, s = 1,..., M, of elements Cse(C) and corresponding arbitrary numbers as > 0, there are numbers 5s > O, s = 1,..., M, such that yl,y2eS, llyl-y211CsY _< s, s = 1,..., M,implies 1IY1-Y211CsZ < a, s = 1,..., M. (2.4) Let (V}, (W] be bases for the system of neighborhoods of 0 in Y and of 0 in Z. We assume that each set V is of the form V = [YEY IYyIC sY < Vs, v > 0, C E(C], s = 1,...,M < + ] (2.5) and that the sets W have an analogous form with the seminorms of Z. We shall need the following hypotheses: Hi. There are numbers 6, 0, 0 < Ko < 1,2o +_< 1, and for zeE, linear operators Bz: F - Y, Az: Y -+ F with (B) = 0, such that yl, y2EYo+S, y1-y2VeV(V), implies Bz[f(yl, z)-f(Y2, z)-Az(yj-y2) ]( (SnV), (2.6) Bzf(yo,z)eS, BzAz = I. (2.7) H2. For every Ve[V) there is a We(Wj such that zl, z2eZ, yeS, zl-z2eW imply BZlf(y,zi) - Z2f(yz2)eV. (2.8) H3. Given 5 > 0 and Ce{C), there is a = a(y,C) > 0 such that xl,x2eC, p(xlx2) < a, yeS, zEZ imply l|(Bzf(y,z))(xl)-(Bzf(y,z)(x2)-y(xl)+y(x2) IIE < e (2-9) In the second alternative considered above we must replace (2.9 by }|D (Bzf(y,z))(xl)-D (Bzf(yz))(x2)-Dsy(xl)+Dsy(2)IE, s = 0,1,...,N. (2.10) 8

We shall use below the following forms of Ascoli's and Thychonoff's theorems. Ascoli's Theorem Let Y be the family of all continuous functions on Hausdorff k-space X into a Hausdorff uniform space E and let Y have the topology of uniform convergence on compacta Then a subset F of Y is compact if and only if (i) F is closed in Y; (ii) the closure of F[x] is compact for every xeX, and (iii) F is equicontinuous on each compact subset of X. (J. L. Kelley and A. P. Morse, see Ref. 2, pp. 234, No. 18). Tychonoff's fixed point theorem. If Y is a locally convex topological vector space, M is a convex closed subset of Y; and t: M + M a continuous map of M into itself, then t admits of at least one fixed point in M (Ref. 11, and also 8c, p. 848). 2. THE EXISTENCE OF r THEOREM A Under hypotheses (H123) there is a continuous function r: X + E, tCY, VeS, such that f(r,r) = 0. Proof of theorem A. Since X is metric, X is Hausdorff, satisfies the first axiom of countability, and is a k-space (Ref. 2, pp. 49-50, p. 120, No. 11, and p. 231, No. 13). The linear space E is Banach, hence Hausdorff, and uniform. Since YCZ, we denote by j: Y + Z the inclusion map, and by 7, 1 the images of an element yeY, or set ACZ under j, that is y = j(y),? = j(A). Hypothesis SCZ can now be written more precisely in the form j(S) = CCZ, and (2.4) becomes: given C and a > 0 there is b > 0 such that yeY, JIYJlCy < b implies I|Y|CZ < cr' By Section 1 we know that, for every zEZ there is exactly one element *z = 'CzeS such that f(rz,z) = 0, and C: Z +- S is a continuous map (in the topologies of Z and Y). By Section 1 we know that, for every C, Ce(C], and zeC', the functions 4z: X - E are equicontinuous on C. First note that 4z is continuous on X, since Az: X - E is continuous on each compact subset of X and X is Hausdorff and satisfies the first axiom of countability (Ref. 2, p. 225, No. 7). Secondly, if C = C' and zeC, the functions Az: C + E admit of a "modulus of continuity," that is, there is some real-valued continuous monotone function pC(p), 0 < p < + oo, with piC(o) = O, such that xl,x2eC, zeC, implies llz (X1)-Z(X2)IIE < CP(XC[p( X2)] - (2..11) 9

In the second alternative above we have to replace this by an analogous relation for each Dsr, 0 < s < N. Let us consider the family Woof all elements yeY, yES, satisfying the same relations (2.11), that is, such that xi, x2eC implies IIY(x1)-y(x2) IE < P[p(xl,x2)]. (2.12) Obviously ZcSCY, j(=Z) = cZcrz m: Qi- '' VG: 27~ v'C Now C: PZ. /is continuous in the topologies of Z and Y, precisely, we know, by Section 1, that given Ve[V), there is some We(W) such that zl, Z2c2I z1-z2EW implies zi -z ZeV. Actually, z1-Z2EW means llZl-Z2llcsZ < vSI, vS > 0, CsE(C) s = 1,.. M, for certain elements Cse(C) finite in number. By (2.4) there are new numbers Vs > O, s = 1...,M, such that Hyi-y2l1C y < vs. s = 1,.o.,M, implies IIY1i-Y2lCsZ < vs' s = 1...,M, and finallyt - eV. We shall denote by Vo the neighborhood of 0 in Y defined by IIY|ICsY < vs. We conclude that, given Ve(V}, we have determined VoE(VJ such that yl-y2eVo implies (LCj)(yi) -(C j)(y2)EV. Thus,< j: G/ +2u is a continuous map in the topology of Y. Obviously, 'Z^is a closed subset of Y because of the topology we have chosen in Y. For every xEX, the set ZtU[x] is a subset of E contained in the sphere Ile-yo(x)llE < bc, where C is any of the elements CE(C] with xeC, Since E is locally compact we conclude that 2[x] has a compact closure. Finally, /ais equicontinuous on each compact subset C of X. By Ascoli's Theorem, is compact. Finally,?/ is convex, and hence, by Tychonoff's fixed point theorem,-C has at least one fixed element in c/, say ' = L~C, and f(,A) = Oo Theorem A is thereby proved. 3. THE UNIQUENESS OF ' We need the further hypothesis (H4) There is a number, 0 <9 < 1^ such that yeS, zl, Z2eS, zi-z2eVe(V} imply 10

BZlf(y,zi) - BZ2f(y,Z2) e (l-(1'V (2.13) THEOREM B Under hypotheses (H1234) there is one and only one function t: X - E, teS, such that f(&,4) = 0. Proof of Theorem B. For every zeE, the element *z = " z is the fixed element of the mapping Tz corresponding to (1.4) of Section 1, or Tzy = y - Bzf(y,z). (2.14) Since SCZ, we take elements z belonging to S (properly, to the image of S in E under the inclusion map j: S ->.). For zl,z2CS, ri = *zi = ( j)zi, Yi = TiYi = $zi, i = 1,2, zl-z2eVe(V), we have '1 - *2 = ( rjj)zI - (Qtj)Z2 = TZ 11 - TL2.2 ~Zj~ Z = - BflBcVf i2Z1) + Bz2f (*2,z2) + 41 - 42 1= - B[f((Z1)l-f((2,Z1) -Az (i1-i-2)] - BZlf(42,yl) - BZ2f(a2,z2). By (H14)-we have I1 - f2^E-4V + (1l-to)tV = XV, (2.15) where X = <4+(l-ao)X' < 1. Now, for every Ce(C), let that V is a set of the form VC = 11l1-'211CY. Note V = [YEYlllyllcs < vs, vs > 0, Cse(C), s = 1,...,M < + co]CY. Hence, z1-z2EV implies also z1-z2eV' where V' is the same set V above where the numbers vs are replaced by numbers vs+es = vCs+eS, es>O arbitrary, s = 1,...,M. Then (2.15) implies I11-V2llCYy < ( llz-Z2llCSY+Es), s = 1...,M. The numbers es being arbitrary, we conclude that 11

ilIl-211CCY I< Al Z1-Z211CY, s = 1,..,M. This relation essentially says that 'Ij: S + S has a contraction type property, and this definitely excludes that Do can have more than one fixed point. Now assume, if possible, that f(y,y) = 0 has two solutions zlz2eS. Then, from (2.l4), we would deduce zi = Tzizi, i = 1,2. Since, for every zeS the equation y = Tzy has only one solution y = 4z in S, we conclude that Zi = (rj)Zi, zieS, i = 1,2, hence z1 = z2 by (2.15). Theorem B is thereby proved. 4. A LERAY'S TYPE THEOREM Let us assume that G(y,z,\) depends on y and z as in No. 1 and on a real parameter x varying in a finite interval J = [Koa<l]. We shall assume that it satisfies the properties H123 for every XeJ and uniformly with respect to N. Thus, for every XcJ there are sets Wk and Sbk, and a map '7: WxxJ - S%, or y = r (z,X) with XeJ, zeWx, yeSb,z and G( (z,X),z,x) = 0 for every keJ, zeW%. Thus the equation G(y,y,X) = 0 was actually reduced in No. 7 to the equation y- C(y,) = 0 as in Leray's Theory, and t is actually a compact map on W%, since it transforms (bounded) sequences in Wk into compact sequences of elements of SbX. In addition YCZ, and SbC Wk as in Leray's Theory. Nevertheless we do not assume that Y, Z are Banach spaces, we do not assume that bounded subsets of Y are compact subsets of Z, and wejdo not assume that G has a Frechet differential as a function of (y,z,i), but as a function of y only. Whenever Y and Z are Banach spaces (as in the sections below), then for every x the topological degree of the map y-/t(y,.): SbX -> Y can be defined at every point yEY which is not the image of points of the boundary aSb% of Sb.. Let us now assume with Leray that L1. y - (C(y,,) = 0 for every XeJ and yeSb%. L2. For X = Xo the topological degree of the map y -C (y,Xo) is known. and is a number m i 0. Then we have THEOREM C Under the hypotheses above, then (i) the topological degree of y-r(yX): Sb% -+ Sbr is the same for every XeJ; (ii) there is a continuum 12

subset C of USb, containing at least one element (y,X) with y- 0(y,X) = 0 for every XeJo 13

SECTION 5. MOSER'S THEOREM FOR THE EQUATION f(y) = 0 1. HYPOTHEESE Let X be an open subset of a Euclidean space Ed, let Y be a Banach space of functions y: X + E with partial derivatives DSy of all orders O < s < m. Let f(y) be a (nonlinear) functional defined in a subset Yo of Y, or f: Yo + E. The existence of an exact solution y of the equation f(y) = 0 in a neighborhood of an approximate solution yo was proved by J. Moser (Ref. 6) when the usual process involves a "loss of derivatives." We restate below, with few changes, hypotheses, statement, and proof of Moser's Theorem (i), both for the convenience of the reader and in order to emphasize the "uniform" character of the statement. This will enable us to complete (i) with statements (ii) and (iii) of continuous dependence of solutions upon parameter and uniform continuity on X, which are analogous to those of Section 1, and then to obtain (Section 4) existence theorems for solutions of equations of the form f(y,y) = 0 analogous to those of Section 2. Let Yu, u = 0,1,...,YO be linear spaces of functions y: X -+ E bounded in X and possessing bounded partial derivatives, say for the sake of simplicity DSy: X + E, of all orders 0 < s < u, and 0 < x < + oo respectively, and YoCYu+lC Yu. In each Yu, u = 0,1,.., we take the norm Illlu = max supiDSy(x)llE, YEYu (.l1) O<s<u xeX Let TN be a smoothing operator TN: Yu + Yo depending on the real parameter N > 1 such that, if RN = I-TN (I identity operator, RN error), we have v+6 (Koa) IITNYllu+v hN IlYlu, u,v > 0, yeYu T NyCYoCYu+ (Kob) ||RNYI|| < hN IlyIu+vu > 0, v > 56 y, RNYEY +v for given constants h > 0, 6 > 0, 6 integer which are independent of y,uv. Below, r, A,, y > 0 denote fixed integers, a+P<r, y<r, and we assume that Yr is a Banach space Let f(y) be a (not necessarily linear) functional defined in a subset Yro of Yr with values in YrC, or f: Yr' + Yr-C_, We shall also assume that f: Yr'f Ys + Ys-a for every s > r. Thus, f involves the "loss" of Ca derivatives. 14

As before, we shall denote by yoEYr' an approximate solution of the equation f(y) = 0. In (Klb) we shall suppose yo sufficiently smooth, that is, yoeYs for some s > r sufficiently large, and in (Klc) we shall suppose the error f(yo) sufficiently small. Also, we shall introduce in (Kla) operators L and A (L a right inverse of A as stated in (Kla)), each L and A involving the loss of P and a derivatives respectively. We shall denote by Sb the subset of Yr defined by IIy-yollr < b for some b > O. Let K be a given constant l<k<2, and take 2s = k(a+P+b), t = k(2s+P+y++), (3.2) and let us choose first p1 > 0, and then an integer X > O, such that 2s+(k-2). < 0, (t+k2 l)+(1-k) < - X (3.3) 2s+(l-k)h < O, X > oa + P + +, (Kla) The integers a, P, y, 6, r > 0 and the constant b > 0 are so chosen that SbCXr', and, for every ycSbnYr+%, there are linear operators L(y): Yr+*-aC + Yr+%-a-P t A(y): Ys + Ysy, s = r+y-a —, s = r+y-a, s = r, such that A(y)L(y)z = z, yesbnfYr+X, zeYr+X-, (3.4) Ilf(y+w)-f(y)-A(y)wlra < holwjl, y,y + wcSbnYr+, (35) IIL(y)zllrC-P < ho lzllr a, yESbnYr+%, zeYr+%-C, (3.6) IIA(y)zllr- < hollzllr+y-a, yeSb Y+, zeYr+y- (37) IIA(y)z|lry < hollzllr, yeSbnYr+, zCYr. (3.8) (Klb) yoeKr+%, and yeSb Yr+k, lIy-yollr+\ < M, M > 1 imply |L(y)f(y)||r+-acp < ho M. (3.9) 15

Let us take c = max [h2h3 hho2, h2ho2, h, ho, b]. (3.10) We choose an arbitrary number No > 1 such that N 2 - X(l-k) 2s+(1-k)% -1 -2s No > 4c, 2No < 1, 2No <1, 2hhoNo < 1, (2hho) No 5, (3.11) I L(yo)f(yo)llr+x-a-P hoNo -1 -2s -1 -2s(Klc) IIf(Yo)IIlr-c min [(2hho) No b, (2chho) No- N 2. MOSER' S THEOREM (Ref. 6) (i) Under hypotheses (KOab), (Klabc) there is at least one function 4eSb, 4EYr, such that f(V) = 0. There is a method of successive approximations yo, yi,..., starting at yo which converges in Yr toward a function 4 as above. Proof of Moser's Theorem. Take Nn+l = Nk, n = 0,1,..., and Yn+l = Yn - TNn+l L(yn)f(yn), n = 0,1,.... (3.12) By induction we prove that for every n we have YneYr+\. IYn-Yollr < b, lYn-Yollr+% < N, | hL(y,)f(yn)llNr+\-a-1 < h( * (3.13) We use the notations Ln = f(yn), An = A(yn), fn = f(Yn), vn = -Lnfn. The first three relations (3.13) are obvious for n = 0, and the fourth one is true because of (3.9). Assume y eYr+k, |Yv,-Yollr < b for r = 0,1,...,n. Then Yn - n f n+ll-Yr+Y+ fneYr+Q-a VnEYr+a.-cr-a RNnVneYr+-a-, Anvn = - AnLnfn = - fneYr+%-a' Then, by (Koa) with v = a+D, and u = r-a-P,

U+PB+E 2s llyn+l-Ynllr = ITNn+ivnilr < hNn+i lvnilr-a-P = hNn ilLnfnjIr-a-P, (3.114) llYn+i-Ynilr+X = IITNln+nllr+X < hNn+i ilvnllr+%-a-P = 2s hNn |fLnfnllr+\-O-P,, (35.15) where 2s is defined by (3.2). By (3.13) we have now IlYn+i-Ynllr < hhoNn llfnIlr-a. (3.16) On the other hand, Yn-Yni = VTN n = v - RNnnfn-1 + An-ln-1 = fn-1 fn-l = 0 n-i An- i n-i n-i - (3.17) (3.18) fn-1 + An-l(Yn-yn-) = fn-l + A.n-V = n-Rl Nnn-n A.n-lRNnn-i (3.19) By llfnilr-a (3.5), (3.17), (3.7), = [If(yn-i+(yn-yn-) ) llr-a lfn-i+A.n-(yn-yn-)ilr-O + hollYn-Yn-llr lAn-lRNnVn'n-llr- + hoilYn-yn-illr hollRNnvn-lllr-ar+y + holln-Yn_-illr By (Kob) with u = r-a+7y, v =- -P-y, 1fnI1r-a < hhoNn I n-illr+X-a-P + hojlYn-yn-illr By (Klc), (3.6), (3511), Ilyi-yollr < hhoNo llfollr-_ < min[b,(2c) N ], (3.20) (3 521) 17

yl-yo|r hhN fl — 2s < hh2+2s IIyY-yollr+N < hhoNo 11follr+-0- < hho+2s 2 N2s+(l-k)X X, = (hhON0 )N1 < Nj, (5.22) llv ll r+-0- < hoN1. (3.23) Thus, relations (3.13) are proved for are proved for 0,1,...,n. Then Yn+leYrt+ a (31.3) 2s 2s+< IIYn+l-Ynllr+2 < hNn llvnlilr+-a- _< hhoNn n = 0 and n = 1. Assume that they.s proved above, and by (3.15), 2s+(l-k)X < x = (hhoNo )Nn+i < Nn+i n n llyn+i-yollr+ _ < IlY+i-Yvll < NV v=o v=o < (l-k)X (l-k)X? < NnlNo [l-N ]< Nnl, llvn+ilL+-_-pI = lLn+lfn+ill r+X-a-P < hoNn+l~ Finally, (3.14), (3.6), (3.20) yield IIY +-Ynllr < hhoN [hhoNn +- hollYn- (3.24) (3.25) < C[N " Ily"_Y,_, 12+N- Nk-l _r n n-] where c and t are given by (3.2) and (3.12). If 6n denotes bn = Nnllyn-yn-illr then (3.25) becomes < c[N n 2s+(k-2)b 2 l(t+k2n)+(l-k), n+1 - n nin-1 where the exponents are < 0 because of (3.3). Hence n+l < c[6 n+Nn ], n = 1,2,..., where b1 = N||Iyi-yol|r < (2c)-1, N > 4c by (3.11) and (3.21). we prove that bn < (2c)-1 for all n, and hence By induction llY +-Y l < (2c)-' Nn+l n r n flyn+i-Yollr < I v=l lYv+l-Yvllr < (2c) n = 4-J - 1 Nn <c <b. L 18

Thus, all relations (3.13) are proved for n+l and, therefore, for every n. Also, n+1-l l1Yn+p-Ynllr < l yY-Yv-illr < c 1Nn v=n and, therefore, ' = lim Yn n->co is an element of Yr and rceSb. We have -1 - t lV-Ynllr < c Nn (3.26) Ilf(Yn)llr-a < hhN1 + hho ||yn-yn- 2s+(l-k)% -2 s( < cn-1 + Nn-1 On the other hand, for every n, 1lf(t)llo lf(Yn,+(V-yn))llo = lf(yn+(t-yn) - fn - An(t-yn) + fn + An(*-yn)llo < Illf(yn+(-n)-fn-An(-yn)llo + llfnllo+IIAn(*-yn)llo < Ilf(yn+(*-Yn)-fn-An(*-Yn)llr-_ + 1lfn11r-O + IIAn(*-yn)llr-y < holl*-ynllr + 11fnllr-a + holl*-ynllr c< c- 2+ cN2s+(l-k) + + N-1 + N rn 'n- n-1 n Hence, l|f(V)||o = 0, or f[4(x)] - 0 for every xeX, and finally f(r) = 0. Theorem (i) is thereby proved. 3. CONTINUITY OF r The continuity of the function \r of Moser's theorem can be assured under a very simple set of additional hypotheses: 19

(K2a) The functions Yo(x), Dsyo(x), s = 0,1,...,r, or X -+ E, are uniformly continuous in the topologies of X and E, that is, given ~ > 0, there is a > 0 such that x, x'eX, p(x,x') < a imply IIDSyo(X)-DSyo(X)IIE <, s = 0,l,...,r. (5.28) (K2b) For every yeSbn Yy+k, the functions L(y)f(y)(x), DSL(y)f(y)(x), s = 0,1,...,r-ca, or X + E are uniformly continuous, that is, given e > 0 and yeSbxYr+., there is a' > 0 (which may depend on y) such that xx'EX, p(x,x') < a' imply IIDSL(y)f(y)(x)-DSL(y)f(y)(x' )IE, = 0,.,...,r-c. (3.29) The same hypotheses when the uniform continuity is required only on every compact subset C of X, will be denoted by (K2a)c, (K2b)c. Also, we shall add to the properties (Koa, (Kob) the properties (Koa (Kob) of the smoothing process T the following one: (Koc) If l(p), 0 < p < oo, is a real-valued continuous monotone function with ou(o) = 0, and I|D y(x)-D y(xt )| < c[p(x,x' )],,X'EX, < s r-v, then lDsTNy(x) -DSTNy(x' )IE < hN [p(x,x')], x,x'X, 0 < s r. (5.30) (ii) Under hypotheses (Klabc) and (K2abc), the functions r(x), DSl(x), s = 0,1,...,r, are uniformly continuous on X. If (K2ab) is replaced by (K2ab)c, then 4(x), DSV(x), s = O,l,...,r, are uniformly continuous on each compact subset C of X. Let us prove first that the functions yl(x), DSyl(x), s = 0,1,...,r, are uniformly continuous function of x in X. Indeed by (K2b) there is a function u(p) as in (K2c) such that x,x'eX, s = 0,1,...,r-O-P, imply IIlDL(yo)f(yo)(X)-DSL(yo)f(yo)(X' ) IE < W[P(X,x' )], Then given i > 0, let a be a number as in (K2a), and a' a number such that Cw(a') _<. Then for x,x'eX, p(x,x') < a" = min[',a'], we have, for Lo = f(yo), fo = f(yo), and s = 0,1...,r, 20

lID yj(x)-Dsy1(x' )iE = IIDYO(X)-DS Y(X')+D5TN (LOfO)(X)-D TN (Lofo)(X' )IE < IDSyo(x)-DSyo(x' )IE + IDTNo[ ( Lofo)x-(Lofo)x' ]lIE < g + hNo u[p(x,x')] a++5, a+5+ < I + hNo w(a') < [l+hNo P ] This proves the uniform continuity of yi(x), Dyl(x), xeX, s = O,1,...,r, in X. By induction, we can prove that, for each n, the functions Yn(x), Dsyn(x), xeX, s = 0,l,...,r, are uniformly continuous on X (uniformly in X, not uniformly with respect to n). Since 00 \ = Yo + (yv+i-Yv), v=0 where l||y+l-Yvllr < vw, Ev numbers independent of n with ZEv < +4o, we conclude that the functions t(x), DSr(x), xeX, s = 0,1,...,r, are uniformly continuous on X. 4. CONTINUOUS DEPENDENCE OF r ON PARAMETERS Let us assume now that f depends also on a parameter z varying in a topological space Zo. To simplify we may assume that Zo is a subset of a metric space with distance function A. We assume that all hypotheses of No. 1, in particular (Klabc) holds for every zeZo and uniformly, that is, with the same constants ra,...,c,No. We may assume that yo(z) be either a constant, or vary with z in Zo. Then the element ecSb determined univocally by the method of successive approximations described above is a continuous function of z. To prove this we need only the simple assumption of uniform continuity: (K3) Given 5 > 0 there is a a > 0 such that z, z'tZo, y,y'ESb, IIz-Z"IZ < a, lly-y' 11 < a imply IlL(y, z)f(y - z)-L(y' z',)f(yz1 ),lr- < ay Ilyo(z)-yo (z)llr < a. The same hypothesis when z,z' are restricted to belong to the same compact subset C of n, (C arbitrary), will be denoted by (K2)c. (iii) Under hypotheses (Klabc) and (K3), the element 4 determined by the process of successive approximations (5.12) is a uniformly continuous function 21

of z in Zo (in the topologies of Yr and Z) o Under hypotheses (Klabc) and (K2)cj c is a uniformly continuous function of z on every compact subset C of Z. The proof is the same as above. Under hypotheses (Klabc), and (K2abc) uniformly in z, and (K3a) uniformly in x, we could now assume that the functions r(x,z), DS4(x,z), s = 0, 1,...,r, or XxZo + E, are uniformly continuous on XxZo. 22

SECTION 4. FURTHER RESULTS ON EQUATION f(y,y) = 0 1. HYPOTHESES Let the spaces Yu, u = 1,2,..., be the same normed linear spaces and YO a linear space, YOCYu++1CYuCYo, as in No. 1 of Section 3. Let TN, RN be the same smoothing and error operators as in No. 1 of Section 3, with constants h > 0, and integer b > 0. Let ra,P,y > 0 be integers as in No. 1 of Section 3, Ca+p < r, 7 < r, and assume that Yr be a Banach space. Let r',o' > 0 be new integers with O < r-r' < Oa-. Then O < r-a < r'-a', r' < r, and YrDYr. Also, if YEYr, then yeYr, and lylIr > IlyIrt (4.1 Let f(y,z) denote a (not necessarily linear) functional defined for all y of a subset Y' of Yr and all z of a subset Z' of Yrt with Y'CZ', and values in Yr-ca or f: Y'xZ' + Yr-c. We shall also assume f:(Y'O Yr+I)x(Z'T Yr+~) Yr+-aO for all I > O. Actually, we need this last statement only for all O < I < y, where y is the integer which is defined below. Thus f involves the "loss" of a derivatives on y. Let us denote by yo a given element of Y', hence yocY'CZ'. Let Sb the usual set Sb = [ycYrH Y-Yollr < b], b > 0, for which we shall assume SbCY'CYr. We shall consider a subset Z of Y', with Sb Z and ZCZC'Yr. We shall denote by r,C,7gyb, t.hhocNN o constants as in No. 1 of Section 3, and we assume (Llabc). The same as (Klabc) for every zEcriWr+~. (L2abc). The same as (K2abc) for every zZcWr+>k and uniformly in znw. n lWr+%* (L5). The same as (K3) for every zcnWr,. Theorem C. Under hypotheses (Llabc)(L2abc)(L3), there is at least one function teSb such that f(4,V) = 0. Proof of Theorem C. For every zEZ0fWrt+X there is a unique element *z = L-z which is the fixed point of the map Tz, on *z = Tz4z, 4zES, as in Section 3. The mapt': ZRWrT+ +- Sb so defined is continuous in the topologies of Yr and Yr'. Thus, we can extend Y to all of L in the topology of Yr and Yr', into a mapLC: L -+ S which is continuous in these topologies. 25

Since XCZ, we have also C: S -+ S, and relation (K1) assures that t: S + S is a continuous map also when we use the topology of Y for both S as a domain and range. The hypotheses (L) are worded so that the functions *z of the range of C are uniformly continuous in X together with all their partial derivatives DSy, 0 < s < N. In other words, the functions Vy with their partial derivatives D fy 0 < s < N. admit of a unique "modulus of continuity" I(p) in Xo Moreover X is a bounded open subset of Ed, hence X is compact, and by Ref. 12, the functions 4z: X -> E and their partial derivatives DS8z, 0 < s < N, admit of a continuous extension in X with the same modulus of continuity pi, and we shall use for them the same notation 4z, DSrz. Finally, we define the class?ri7of functions y: X - E, with yES, having the same modulus of continuity p. Then &Z1.: t/z-+ 2- and 2z/is a complete class in the topology of Y by the same Ascoji Theorem mentioned in No. 2 of Section 2 since X now is compact. The existence of * follows now from Tychonoff's Theorem of No. 2 of Section 2 (as well as Schauder's Theorem).

SECTION 5. SIMPLE APPLICATIONS TO THE EQUATION f(x,y(x),y) = 0 1. FIRST APPLICATION In the situation considered in Section 2, when N = 0, and f(y,z) depends only on the evaluation y(x) of y at x with ycY, and on the function zcX, it is convenient to deduce from (A) and (B) statements which are much easier to apply, though they are more restrictive. Let X be a compact matrix space with distance function p, let E, En be a Euclidean space, and Y = Z the space of all continuous (bounded) functions y: X + En, or y = (yi,..,yn), with norm Ily| = myall = max|y(x)||E 11Y[ = [Y-]u xX n Let yo = (yio,.*,Yno) be a point of En, S the sphere of center yo and radius b > 0, Z the set of all functions yeY with values y(x)cS. Let f(x,y,z), or f: XxSxZ - En be a given functional. In this situation, the Frechet differential of f with respect to y is an nxn matrix A(x,y,z) depending on x and z, so that, if we replace y by any given function ycY, we obtain a matrix A(x,y(x),z), depending on x and z only. Instead of the operators A, L of Sections 1 and 2 we may take matrices Axz, Axz. We shall denote by yo also the constant function yo(X) = Yo, xEX. Note that, with these conventions, the subset of functions yEY denoted in Section 2 by S, coincides with Z. (H1) There are constants M > 0, Y > 0, A_> 0, with My < 1,? < 1i-My, such that Ilf(x,ylz)-f(xy2,z)-Axz(Y1-y2)lIE < YIIY1-y2-lE ( 5l) |1f(x,yoz)||E < bCllM, |iAXzl < M, (502) for all xeX, zeZ, yl,ys2S. Hypothesis (5.2) can be further replaced by the hypotheses aX1: o, a2! o, 9~ +^ <X<G i-MY, 25

Ilf(x, yo)x),y, )iIE < bXl/M,, (5.5) llf(X,Yo(x),z)-f(x,yo(x),Yo)llE < b 2/M. (5.4) for all xeX, zeZ. Then (5.3) states a bound for the admissible error to which the "approximate" solution yo(x) = Yo satisfies the equation f(x,y(x),y) = 0, and (5.4) states a bound for the admissible deviation of f(x,yo(x),z) from f(x,yo(x),yo) when z describes E. (H2)' |lf(x,y,z)l|E < P, xeX, yeS, zeZ. (H3)' Given 5 > 0 there is a > 0 such that xl,x2EX, yeSb, Z1YZ2CZ, p(x1,X2)Ea&, |zl-z211y < a imply I|f(xi,y,zl)-f(x2,y,z2)lE < (5.5) -1 -1 lPz1-^Ax2z2ll < x. (5.6) As a corollary of Theorem A we have Theorem D. Under hypotheses (H123)' there is at least one continuous function 4: X + E, such that V(x)eS, f(x,y(x),y) = 0, xEX. (5.7) Also, we shall need the hypothesis (H4)' There is a number X', 0 <1 < 1, such that xEX, yeS, zl,z2EC, imply l|f(x,y, Z1)-f(x,y,Z2) IIE < ((l-My) -S/M)Z-Zl-z2llY (5.8) Then, as a corollary of Theorem B, we have Theorem E. Under hypotheses (H1234)' there is one and only one continuous function 4: X + E, such that (5.7) holds. Remark 1. Under hypotheses (H123)' of Theorem D, if we know that for some xocX we have f(xo,yoz) = 0 for all ze., then, among the functions 4 satisfying (5.7) there is at least one ' with r(xo) = Yo. 26

2. SECOND APPLICATION As an application of Theorem D we consider a functional f = (f,...,fn), or f(t,y;z(S), 0 < K < t) with values in En, depending on the real variable t, O < t < tl, the real vector ycEn in some set YoCEn, and the values in En taken by a function z: [O,ti] -+ En of some family YO in the interval 0 < e < t. As a corollary of Theorem D we shall prove under conditions the existence of at least one function V(t), O < t < a, a > 0 sufficiently small so that f(t,l(t); v(O), 0 < ~ < t) = O, tI = [O,a]. (5.9) For the sake of simplicity we write f(t,y,z), and then (5.9) takes the form f(t,1(t),r) = 0. Let i = (,l.. -.,n) be a point of En, and Yo the sphere in En of center i and radius bo for some bo > O. Let Zo be the family of all continuous functions z(5), 0 < a < a, with values in Yo and some ao, 0 < ao < ti. Suppose f: IxYoxZo + En. We shall assume that, for t = 0. f does not depend on z, and we may write f(O,y,-). As in the implicit function theorem we shall assume f(O,k.) = O. We shall take for the continuous functions z: I + En the norm Izl = max|z()|I for geI. Corollary 1. If f: IxYoxZo - En is bounded and continuous in IxYoxZo, and possesses first order partial derivatives aij(t,y,z) = afi/aYj, i,j = l,...,n, which also are bounded and continuous on IxYoxZo, if f(O,,*)=O, and det Ao=O, with Ao = [aij(O,i, )], then there is some a > O, b > 0, and a continuous function r(t), 0 < t < a, such that V(o) = tl, 4|(t)-l|l K b, and f(t,r(t),y)=O for all 0 < t < a. Proof. Obviously, aij(Oy,.) does not depend on the third argument. Let us take for yo the constant function yo(t) = Yo, 0 < t < a. We have fi(ty2 z)i(t(t,yl, z)-Aoji( (2-Y1) 1 = fo Zj[aij(t,Y1+T(y2-Y1),z)-aij(O,., -)](y2j-ylj)dT, where Aoi is the ith rate of Ao, and yj = (yi,...,yin), Y2 = (y21,...,Y n) Let M = IA-111 be the spectral norm of A-1, and take r = 1/2 M, so that My = 1/2. Then, take b and a, 0 < b < bo, 0 < a < ao, sufficiently small so that laij(t, Y+T(Y2-Y), z)-aij(O,,' *) | 1/2 n2 27

for all 0 < t < a, all Y1,Y2 withlyi-Cpl < b, i = 1,2, and all z with Iz(t)-yo(t)l Ib, 0 < t < a, Then, by Schwarz inequality, Ifi(t,(t,y2,z)-fi(t,yl,z)-Aoi(y-yl)I < (2n)-lly2-ll, i = 1,...,n, I f(t,y2, z)-f(t,yl, z)-Ao(Y2-Yl) 1I 2-11 Y2-yl Finally, we may restrict a further, if needed, so that |f(t,I,z)-f(O0,p, ') < b/2M, 0 < t < a. Then lf(t,p,z)| I b/2M = (l-My)b/M. All conditions of Theorem D are satisfied for the restricted interval X = I = [Oa]. We shall need the previous statement in Ref. 1. Concerning the uniqueness of the function \r we deduce, from Theorem B, the Corollary 2. Under the hypotheses of the previous corollary, if there are numbers a', b', 0 < a' < a, 0 < b' b, such that 0 < t < a, I y-II bt, Izi(t)-Pil b', IZ2(t)-p1 < b' for 0 < t < a', imply If(t,y,zl)-f(t,y,z2)I < (1/2M) max Izl(I)-Z2() I, (5o10) 0<5 <t the function r(t), 0 < t < a', of the previous corollary is uniqueO 3, A PARTICULAR CASE As a particular case of the previous corollaries we may consider functions f,g: IxEnxEn + En, and the functional equation in f: I + En, t f(t,4(t), Jo g(tT,$(T))dT) = 0. (5.11) Here we assume that f(tu,v) = (fi,..,fn) g(tuv) = (g,.oogn) are continuous in a set 0 < t < a0, lu-uol < bo, lv-uol < b, that Igl K N in this set and the partial derivatives afi/ayj exist and are continuous, and that det Ao f O, A0 = fu(o,uoo) 0. If M = IIAol1, f(oUoO) = 0, and 9, y are chosen so that My < 1 0 < = 1 - My, then in a set 0 < t < a, |u-uol < b, Iv-vol < b, and a, b sufficiently small the relations 28

lf(t~yul~v)-f(t,u,,v)-Ao(ul-u2)1 -< YlUl-U21, f(t,uov)|| I< b/M, |f(tUf tg(t,T,Z(T))dT) < *b, are satisfied for lui-uol, lu2-uol, Iv-uol < b, and all continuous functions z(T), 0 < T < a, with values |Z(T)-UoI < b. Then the existence of at least one solution to (5.11) follows from corollary 1. 4. A KNOWN PARTICULAR CASE For f = u-uo-v(5.11)reduces to the equation I(t) = uo + Jf g(t,Ty,(T))dT. (5.12) Then we can take A = I, M = 1, y = 0, a6 = 1 and a solution in te small certainly exists. The differential system with initial condition dy/dt = G(t,y), y(o) = uo reduces to the equation r(t) = uo + Jo G(T,4(T))dT of the type (5.12). The usual example dy/dt = yl/2 uo = 0, for which the solution is not unique, shows that the conditions of Theorems A and D do not assure uniqueness. The existence analysis of the problems of the present number in terms of fixed point theorems is Schauder's..ANOTHER PARTICULAR CASE Instead of (5.11), we may consider the integral equation f(x,f(x), fv g(x,Sy,x())di) = 0 (5.13) where x = (xl,...,m),eEm, 4(x)EEn, f(o,uo,o) = 0, and V is the variable sphere 29

V = [~eEmllt <_ |Ixl] Analogous results hold for the integral equation (5.13) where V = V(x,h) is the variable sphere V = V(x,h) = [SeEmjl I-xl _ h] and h > 0 is a fixed number sufficiently small. 6. ANOTHER APPLICATION As an application of Theorem D, let us consider the integral equation ti f(t,y(t), to g(tT,(T )dT))d = 0 (5.14) where I = [to < t < t1] is a fixed interval of the real axis, f(t,u,v) = (fi,...,fn), g(t,u,v) = (gi,...,gn) are continuous functions f,g: Ix EnxEn - En, and an "approximate" solution zo(t), teI, is known, with values in the sphere S = [ueEnl lu-uol < b] for some b > 0. We shall assume that f and g are continuous in the set H = IxSxS, and f has continuous first order partial derivatives. If fu is the nxn matrix of these derivatives, assume that set fu # 0 in H. Let Ao = fu(t,zo(t),v), and assume that If(t,ul,v)-f(t,u2,v)-Ao(ul-u2)l1 < Yl1-u2l, jg(t,T,u) < N, llf1u(t,u,v) ll < M, ti f(t, zo(t) ito g(t,T,Zo(T))dT| <bX/M, |f(t, zo(t),v)-f(t, Zo(t)J,fto g(t,T, Zo(T) )dT) < b 2/M for all t,TEI, u,veS, where 7, I1, 2 > O, My < 1, a+2 < 1-M. Then, by Theorem D, there is at least one solution r for equation (5.14). As particular cases of (5.14), we may consider the Uryshon equation tjl j(t) = h(t) + Ito g(t,T2,(T))dT, and the Harmerstein equation ti V(t) = h(t) + fto g(t,T)k(*(T))dT. 30

7. GENERALIZATION Instead of (5.14) we may consider the equation f(x,r(x),fG g(x,(,,()))d ) = 0, xeGo, where x = (xli...,xn)cGoCEm, f,g: GoxEnxEn + En, where Go is a compact subset of Em. The conditions for existence are analogous. 31

BIBLIOGRAPHY 1. L. Cesari, A criterion for the existence of periodic solutions of hyperbolic partial differential equationso Contributions to differential equations. To appear. 2. J. Lo Kelley, General Topology, Van Nostrand, New York 1955, xiv + 298ppo 35 Mo A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, MacMillan, New York 1964, xi+ 395pp 4o J. Leray and J Po Schauder, Topologie et equations fonctionnelles, Ann, Sci. Ecole Norm. Sup. (3) 51, 1934, 45-78. 5. Lo A, Liusternik and Vo Lo Sobolev, Elements of functional analysis, F Ungar, New York 1961, ix + 225 pp. 60 J. Moser, A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Scio 47, 1961, 1824-1831. 7. J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math, 63, 1956, 20-63. 8. Ho H. Schaefer, (a) Uber die Methode der a priori Schranken, Matho Annalen, 129, 1955, 415-416; (b) News Existenzsatze in der Theorie nichlinear Integralgleichungen. Sitzgsbero Sachs. Akad. Wiss. Math,nat. kl. 101, 1955; (c) On nonlinear positive operators, Paco J. Math, 9, 1959, 847-860. 9. Jo Po Schauder, (a) Zur Theory stetiger Abbildungen in Funktionalraumeno Math. Zo 26, 1927, 47-65, 417-431; (b) Der Fixpunktsatz in Funktionalraumeno Studia Math, 2, 1930, 171-180. 10o J. Schwartz, On Nash's implicit function theorem. Communications pure applied Math, Volo 13, 1960, 509-530. 11o A. Tychnoff, Ein Fixpunktsatz. Math. Annalen 111, 1935, 767-776, 12. Ho Whitney, Analytic extensions of differentiable functions defined in closed setso Trans. Amer. Math. Soc. 36, 1934, 63-89. 32

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