THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report Noo 12 ON MULTIDIMENSIONAL INTEGRAL EQUATIONS OF VOLTERRA TYYPE. B. Suryanarayana ORA- Project 02416 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINiA administered. through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1970

ON MULTIDIMENSIONAL INTEGRAL EQUATIONS OF VOLTERRA TYPE* M. B. Suryanarayana I. INTRODUCTION Multidimensional integral equations of Volterra type have been studied extensively in the past [see N. V. Kasatkina [5], W. Walter [10], Arthur Wouk [11], for partial surveys], and were used to solve boundary value problems-a typical example of such an application is the classical Darboux problem: z (x,y) = f(x,y,z,z,z ),(xy)e[o,h]x[o,k], xy x y z(xo) = p(x); z(o,y) = *(y); (p(o) = (o), which corresponds to the integral equation z(x,y) = cp(x) + k(y) - cp(o) + f (.f(1)z Yzz )d (1.1) 0 0 x y Several forms have been proposed for multidimensional integral equations of Volterra type. One such form has been studied by W. Walter [10] namely, u(x) = gV(x) + H (X)kV(x'.u ()))(d ) (1.2) v Pv u = (uu2... un), v = 1,2... n, xeB c E *This research was partially supported by research project U. S. AFOSR-69-1662 at The University of Michigan. The author is greatly indebted to Professor Cesari for his valuable guidance and constant encouragement during the writing of this paper. 1

where H (x) c B(x) = (eB|S. < x., i = 1,2,...n}. Precisely, H (x) is assumed V 1 V by Walter to be contained in a p -dimensional hyperplane, 1 < p < m, parallel to the coordinate axes-i. e., in a translate of the linear manifold generated by p of the basis vectors (1,0,0...0), (0,1,0...0),... (0,0,...0,1) in E. W. Walter gave theorems for the existence of continuous solutions for such systems. One is now directed naturally to equations where the set H need not be in a p -dimensional hyperplane (with nonzero measure). In other words, one may consider equations where H may consist of several sets, each of which belongs v to a hyperplane of different dimension. An example of such an equation would be cp(x,y) = f(x,y) + JfA(x,y,s)p(s,y)ds + + YB(x,y,t)Zp(x,t)dt + C(x,y,s,t)p(st)dsdt (1. 3) which is not of the form (1.2) [see T. H. Gronwall [5]]. N. V. Kasatkina [5] has studied the following more general integral equation for local uniqueness of continuous solutions: t t x(t) = a. Ki i(ti ik 1 < i <..< i k < n i.. k 1 _ k < n l<k<n x(t,t...t. si' 1 t Si.....t )ds... ds. + f (t) (1. 4) 1 2 i i i k-+1 k n i k 1 1 1k k k where t = (t,t...t ), i... = (s,s...s ). His method involves dif1 2 n i i 1 i 1 k 1 2 k ferential inequalities and the results hold locally. 2

It is the purpose of this paper to study multidimensional nonlinear integral equations of Volterra type of the same general form as (1. 4), but with unknowns in spaces L, 1 < p < o, and not necessarily in C, as is usual. Besides, we require less demanding hypotheses on the integrands, as motivated by relevant applications which will be mentioned below. In this situation we establish global existence and uniqueness theorems as well as continuous dependence on initial values, for such integral equations. The canonical form, emphasized in the present discussion, seems to be the most general since it is found that many equations considered in the past can be put into this form. The theorems we obtain are global-and not local. Further, an analogy with the usual theory of differential equations is maintained as far as possible. The proofs in the present paper are based on fixed point theorems —precisely, on an extension of Banach's contraction theorem. Indeed, the equations under consideration-being of Volterra type-give rise to an operator T which is not necessarily a contraction by itself; but suitable powers of T are so. It is seen that still, there exists a fixed point for T, by a remark of F. F. Bonsall [1]. This fact allows us to relax the hypotheses. Essentially, the same argument applies to the space C of continuous functh tions, as well as to the spaces L of the p summable functions, for 1 < p < o, p and in each of all these cases, we assume a different set of hypotheses. If the equations are assumed to be linear, with analytic coefficients, then existence of analytic solutions is obtained by applying the argument to a space of analytic functions. The results of this paper are used in-and the present paper has been motivated by-problems of optimal control monitored by nonlinear system (1.1) (and corresponding Darboux data), particularly since the controls are known to be only measurable, and hence the integrands-which contain the controlsmay be assumed, at best, to be in some L -space, 1 < p < o. Besides,the p corresponding Pontryagin-type multipliers are known to satisfy linear Volterratype integral equations of the same form (1. 4), but again with integrands in an L -space. P 5

In ~2 we develop suitable notations, in ~3 we summarize basic statements, in particular Bonsall's remark, in ~4 we discuss the problem under consideration in class C, in ~5 we show that analogous results and essentially same proofs using Bonsall's remark hold in L, 1 < p < co; in ~6 we consider the linear case and show the existence of analytic solutions, in harmony with classical results. Applications to control problems will be given elsewhere. The special case of the classical Darboux problem (1.1) is discussed in the appendix. 2. NOTATIONS Let E denote the n-dimensional Euclidean space. Let G = (tEnt = (t,t,...t ), a. < t < a + h., i = 1,2...n}. Let a denote the multiindex 1 2 n 1 - i 1 1 (a,,...an)with arbitrary nonnegative integers i., i = 1,2...n. As usual, |c - a + +..a and a' = a!a!...a. Given any a, let P. denote the index of i 2 n 1 2 n J th the j nonzero element of the sequence (a,5... ). Let us consider the multiindex 1 2 n = (p,,...k ); k being some integer with 1 < k < n, determined by a. We shall i 2 k say that P corresponds to a. Thus, for example, (2,5) corresponds to (0,1,0,0,2); n (1,3,6,7) corresponds to (1,0,1,0,0,1,1,), etc. For tcE, let to denote 1 2 k 1 k k In particular let ti = (t,t...tti t....t n). Let Go = [seEkls (s s,...s, < s,. a +h i =,...P. Let it (e) = 7t(,e...E ) 1 2i'' k) i - a*-h.; 1 2 k 1.2 n 1 2 k = Z a 0 denote a polynomial of degree N in 0 with no constant term. 1 < |a l N a a a 5a e 2 n Here 0 denotes 0 9...0. 1 2 n 4

Let C (G) denote as usual the space of functions continuous on G, with supnorm and let L (G) be the space of the p-th summable functions for 1 < p < o and P L (G) be the space of all essentially bounded measurable functions on G. For m > 1 and X = C(G), L (G) or L (G), let X = XxXx...:X (m times) and for m CP= (,p...p) in Xm let![p = Z ill. We shall denote by i|p11c,lcplp and I||p| the norms in [C(G)]m, [L (G)]m and [L (G)], respectively. For i = 1,2...n, let Ti be an operator defined on Xm (with X = C(G), L (G) or L (G)] as follows: t. T.p(t) = 1cp(tis)ds, cpCX a. i' We define the product T.T. as composition, so that Tr T.(T )r 1 and T p = p;X. By using Holder's inequality, it is seen that IT i p < |lq|p h (r.pr1)-1/p, 1 < p < 00 (2.1) and TI ll < lal1 hi ( (2.2) Also, IT I < _||pTI hi (r:')1 (2.3) For any multiindex Ua = (Q,... c ) as in the beginning of this section, 2 2 n we shall denote by T the operator T 1... n; and given any polynomial t(e) = 1 n ZX 0 of degree N, we shall denote by i(T) the operator ZXCT a a0 a (a,a..a ) 1V 2 T 1 2 n 5

In this paper we shall consider the following (canonic) form of multidimensional integral equations of Volterra type: x(t) = f(t) + (jr(T)~F)(x)(t) f(t) + z T F(tsx(tBs5)) (2. 4) where f(t)cXm and for each a, with 1 < |a[ < N, the function Fa(t,ss,x) (FF2.. FM ) is defined on GxG xE and here D is the multi-index corresponding to a, as described earlier. Specific assumptions on f and F will be made later. 3. PRELIMINARIES In the sequel, we shall need a few preliminary statements. They are given below: If F:X - Y is a mapping of a metric space (x,p) into another metric space (Y,a) and there is a constant c > o such that o(Fx,Fx ) < c p(x,x ) for all 1 2 2 (i) (An extension of Banach's contraction mapping theorem) (see F. F. Bonsall [1]): Let F:X -~ X be a mapping of a complete metric space (X,P) into itself. Let F' = F and Fn = F(Fn-l) for n > 1. Let us assume that v (Fn) < + oo for every n and that v(F ) < o. Then F has a unique fixed point xoeX. nl= (3.ii) (F. F. Bonsall, [1]): If F:X + X is any continuous map of a comN plete metric space (X,o) into itself such that for some integer N > 1, F" is a contraction on X, then F has a unique fixed point xo in X. 6

(3. iii) Let G = [ttE nlt =(t,t...t ),a < t < a.+hi, i = 1,2... n as in 2 2 n i i - i1 ~2 and let G. = seE:a. < s < a +h.} i = 1,2... n. Let cp = cp(t,s) be a real valued function defined on GxG. such that (a) I.Cp(t,s)l < cm(s) for all (t,s)EGxG. where c is a constant and m(s) is integrable on G.. (b) cp is continuous in t for each fixed s~G.. (c) p is measurable in s for each fixed tcG. t Then the function T.cp(t) = J ip(t,s)ds is continuous on G. 1 a. 1 Following the notation of ~2, if Ti, i = 1,2...n are the operators defined a a there, if a = (a,c... ) is any multiindex, T = T...T and 5 is the multi1 2 n 1 n index corresponding to a, then T (p(t, s)e[C(G)], provided (a) Icp(t,s) I < cm(s ) for all (t,s )eGxG. (b) cp is continuous in t for each fixed s cG5. (c) CD is measurable in so for each fixed teG. (3.iv) Let a = (a,a... ) be any multiindex and let i = (P,5...P ) cor1 2 n 1 2 k respond to x as in ~2. Let cp(t,s ) be any real valued function on GxG which is continuous in tB for almost all (tsg)eC and belongs to L (G) for each fixed t eGs. Let there exist a constant B > C and a function m(tt,s) in L (G) such that for (t,ss) eGxG5 we have I|p(t,ss)I < Bm(t,sg). Then the function O(t) defined by 0(t) = T cp(t,s), is measurable on G. Let B be a measurable subset of E and let gi(x,y),i = 1,2...m be real1 i r valued measurable functions on B xE. Let p and p be two real numbers with i 1 2 1 < p,p < a. Let us consider the following condition: 1 2 (H) There exist m functions a.(x),i = 1,2...m, in L (B ) and a constant 2 b > C such that for each i = 1,2... m, we have 7

p/ p gi(x,y) < ai(x) + blyl 2 (5.1) n (3.v) (see M. A. Krasnoselskii, [6]) Let B be a measurable subset of E 1 and let gi(x,y),i = 1,2...m be real valued functions on B xE such that for 1 1 each i, g.(x,y) is continuous on E with respectto y for almost all x in B, r and measurable in x for each fixed y in E. Let Jz =(J z,J z,... J z) with 1 2 m J.z(x) = g.(x,z(x)). Then Jz is measurable whenever z is measurable. Further1 1 more, the operator J maps [L (B )]r into [L (B )]m if and only if condition 1 2 (H) holds. Following the terminology of R. C. Gunning and H. Rossi [4], a complex valued function f defined on an open subset B c C (the n-dimensional complex vector space) is called "holomorphic" in B if each point weB has an open neighborhood U,weU cB, such that the function f has a power series expansion V V f(z) = Z a (z - w )..(z - w ) which converges V v...V 0 V...V 1 1 n n 1 2 n 12 n absolutely for all zcU. A function f is said to be holomorphic on a closed set D c Cn, if f is holomorphic on an open set containing D. For functions of several real variables, the same definition above holds, with the word "analytic" being used instead of "holomorphic." The set of all functions holomorphic on D will be denoted by O(D). If D denotes the rectangle ({eCC li < Hi,i = 1,2... n} c C, then we can define a norm on O(D) as: (|x = Z a H where x = Z a C O(D) ]i o |= Ia = o (3.vi) The set 0(D) with the above norm is a Banach space. 8

4. CONTINUOUS SOLUTIONS In this section we shall discuss the existence and uniqueness of continuous solutions (as well as the dependence of solutions on the "initial" values) of the canonical system: x(t) = f(t) + Z iT F (t s,x(t',s )) (2.4) 1 aJJ<N Theorem 1 below shows the existence of continuous solutions of (2.4) under the assumption that the F are merely continuous in x, and not necessarily linear. In this situation, the modulus of continuity is required to be "small" and suitable bounds are given for the k. If all F are known to be Lipschitzian in x, the condition of "small modulus of continuity" can be removed due to the fact that the equation (2.4) is of Volterra type, and the a are then arbitrary. Besides, the solutions are unique in this case. Precise formulations are found in theorem 2. If Fa are linear in x, with analytic coefficients A, then solutions can be found which are analyticnot merely continuous. This is in harmony with classical results. This case is studied in ~6. In order to state a theorem on the existence of continuous solutions of the integral equation (2.4), we shall need the following set of hypotheses: (H ): Let f(t) be a given element of [C(G)]m, and let M > 0 be such that 1 1 if(t)| < M for teG. Let M > M and S denote ( cEm, < M. f1 t) 12 1 M 2 2 9

(H ): Let F (tsx) = (F1,F... FmQ) be functions defined on GxG xS where 2 cx Cc cM 2 D = (5lP2,...d ) corresponds to a, as described in ~2. Let (a) F (t,sa,x) be measurable is so for fixed (t,x); (b) there exist functions kl (s ) integrable on G such that on G x Gx x SM 2 we have |F (t,s,x)| < kl( s). (H ): There exist monotone nondecreasing continuous functions w.i with wi (O) = 0, 3 1 i = 1,2 and functions kC,(s) integrable on G such that for (ttx ),(t2,x ) in GxSM and s~EGB, we have 2 I|Fs(tl,sB x) - F(t2,s,)x )I < k (s [wda(t t1_|) + W2 a (Ix1-xlJ)] (4.1) It is to be noted that if we take H,(t,s,x) = (to - s) ((D-l)') c 2 da F (ts,x), then we have TH = T.T..T (H) = T F. Furthermore, as a consequence of the condition (H5) above, there exist functions k2a(s5) integrable in G- such that IH (t'l, s,x) - Hc(t2,S,X2)1 < < k a(sP)[wl (tl-t21) + w2 a(Jx -x )] (4.2) for (tx ),(t2,x ) in GxS and s cF. This is what we shall need in the sequel. ~~~~~~1 2 Let Di denote the product of intervals [akas + hx] for j = 1,2.s..i-l(i+)1...d where d is the number of nonzero elements in c = (a,F... F ). Let us consider the functions K. (tP ) defined on [ab,a + h ] as follows: a i 2i 1 1

K. (t ) = (tg-sg) (-)) k (s)s (4. 2) I i the functions found in (H2). Since kl (s ) are L-integrable in Gg, it follows that the functions Ki are continuous in [a,a + h ] Also, by (H1), the a i i function f is continuous on G. Thus, there exist monotone nondecreasing functions v(. ),a (.) such that v(O) = OCo (0) = 0, and ia' i If(t1) - f(t2)1 < v(ttl-t21) and IK t ) )I < (It - t 1). (4 3) 1 C 1 C6 1 1 Letk kf ( ki (sdsii = 1,2; 1 < 1a1 < N. (H4): Let there exist real numbers k such that M + Z |IX|k h 1 ((P-3l) < M 1 6 (6 - 2 (H5): Let there exist monotone nondecreasing, continuous functions r(. ) vanishing at zero such that i(09) > (e) + X |.lk w (n()) where.(e) - v(0) + 1 [Eec( 10 ) + k w (2)] and a = (0,.... ) with.i[O, hi], i = 1,2... n (4. 4) 1 2 n 1 Theorem 1: Let the above hypotheses H -H hold. Let K denote (xE[C(G)] I|x(t)I M and Ix(tl) - x(t2) < r( Itl1t2 ) for tl,t2EG}. Let r be an operator defined 2 on0 [C(G)]m by TX(t) = f(t) + E,T FC (t sx(,' Then, there exists at least one xcK with x = rx. 11

Proof: The set K defined above is a nonempty compact convex subset of the normed space [C(G)]m. Let us prove that T maps K into K and that T is continuous. For xcK, t, t2eG with t1 < t2 i = 1,2..n we have x(tl) - x(t2) < f(tl) - f(t2) + + X T F (t x(t s)) - T F(t sx(t s))] < v(It'-t21) + X [TH (t'ls x(t ls)) - T H (t2 sa xX(t2 s))] daO < V( tl -t21) + 1 iE ((-1):(t-S )) k a(S )s fv ^ r^^1-12')^^iP - x(t~,~)l) i v(Itlt21) +|lc1 [ZC (t1 2-t2 I) +k (w (Itlt2j) + w on(it1t21))] < ( ttl-t2 ) by (H ). (Here, as in (H ), E denotes [t,t ] x D ) This shows in particu5 3 i th 1 1 lar that TX is c(ln.tinuous on Ge Also, for xcK, 12

|Tx(f( < I + (t)l + T tsx(t S)) < M + &iC1 I TH (t,s,x(t,so)) M. + A.% 1 h ('k < M I c6 J- 2 Thus, T maps K into K. Now, let x and x be any two elements of K. 1 2 Then, ITX (t)-Tx (t) | < lX[T Fa(t,s,x) - -TF (t,s,x )]i< l\Clk w ( lx -x II) where IIx -x I| = sup (Ix (t)-x (t):tEG}. 1 2 1 2 H[ence for x,x EK,|l rx -Tx I| < l k w (|Ix -x I) (4.5) 12 2 2 2 2 This shows T maps K continuously into it itself. Now, by Schauder's fixed point theorem there exists at least one xeK with TX = x. This concludes the proof of theorem 1. [Note: Details of above calculations for n = 2, may be found in [8]]. Remark: Concerning the hypothesis (H ) of above theorem, let us consider 5 the case where F are Lipschitzian with constants A in x. Then w (v) = A v (see (4.1)). If ZIA aolh | < 1 13

then n(e) = C(e) + &1xiAah l n() yields (0e) = (1-liA.h i)-'(e) ) a au and hence by choosing this function as - in (H ), it follows by theorem 1 that 5 there exists at least one xcK satisfying (2.4). Further, in this case, i.e., if & A X h [ < 1 then the solutions of (2.4) are unique. Indeed, if x and x are solua<JC a\j(/ \J1 2 tions then x TX and x = TX Now with w (v) = Av, the inequality 1 1 2 2 2C (4.5) will reduce to IIx -x II ITX -TX I < IA x h I ||x -x 2 1 2 - 1 2 < IIx -x II 1 2 which is impossible if x x. 1 2 This proves uniqueness. On the other hand, if Fa are known to be Lipschitzian, then by a completely different argument one can prove the existence and uniqueness of continuous solutions of (2.4) —without the further condition &IA % h ~ < L Precise formulations follow. We shall omit the proof here, since it is the same as for L -solutions P which will be discussed in the next section. Theorem 2: Hypotheses: For each a, 1 < |aI < N, let F (t,s,x) = (F F2... F be defined on GxG xEm where P corresponds to a. Let F be continuous in (t.,t ) and be measurable in s5. Let there exist constants M > 0 and functions A (s ) in L (G ) such that for all (t,s)EGxG5 and x,x in E, we have IP P ~~1 1 2 14

IF (t,s,xl)-F (t,s,x )j <M Ix -x |a ai $)'2 1 2 and IF (ts,o) < A (t,s ). Given a sequence of real numbers fX 1 <,la < Nj and a positive integer r, let F dernote Z hC (U) where summation extends over r r < caj < rN and 4i is the coefficient of a in the binomial expansion of (/ i i OC ~r 00 ( ~ I[XIM 9 ) It is seen that 6 Z 6 < o. Let R > 1 be such that 1 < N | <o Nr 6 < 1. Let M > 0 be any real number such that M > (1-6 )Ilfii 1 C < A Nl h (al ) 1) (4, 6) RE(-)ltl 1 < lai <N a a where f is a given element of [C(G) m and [|j|| refers to the supremum rnormo Conclusio.n: Given fc[C(G)], \c real, 1 < | <a | N, and M> 0 satisfying (4.6), there exists a unique xc[C(G)]m with |Ix|l < M such that Zx:= x where, as i,n Theorem 1, Tx(t) f(t) + < T FA(tsx(t s&)) (4.7) 1 <! a a Further, if f and f are any two elements of [C(G)m a.nd if x, x are the cor1 2 1 2 responrding solutions of Tx = x, then |[ix -x 1I < (1-SR)'-f -f 1| (4.8) 1 2 *2 12 Thus, the solutions depend continu3ously on the "initial" values. Remark: The inequality (40 8) is readily obtained by repeated application of the fol,-wing 15

Ix (t)-x (t) = If (t)-f (t) + EX T [Ffts,x ) 1 2 1 2 C - Fc(t, s,x) < 2f (t)-f (t)l + Zl|I T M |x - Thus, for each r I|x -x | < 6|f -f i| + i |x-x II 1 2 2 r 1 2 so that IIx -x 11 < (1- >-'llf -f I 1 2 -- 1 2 where R is such that 6R < 1. 5. L -SOLUTIONS (1 < p < oo) P It is of interest to observe that theorem 2 with slight changes yields unique L -solution for the equation (2.4). The changes needed are made clear p by the following. Theorem 3: Hypotheses: Let F be as before defined on GxG xE; where P corresponds to a. Let F be continuous in t5 and be measurable (to,ss). Let there exist constants M > 0 and functions A (t',s) in [L (G)], 1 < p < o, such that a- Ca P p - - for all (t,s)eGxG and x,x in E 1 2 F (t,s,x ) - F (t,s,x )I < Mx - x (5.1) OC 1 2 - 1 2 and IF(ts,o)l < A (t,s) (5.2) Given a sequence of real numbers [( X1 < czl < N) and a positive integer r, let aO a -l/p 5 denote Z 4 h (a.p ) (with p = 1 in case of L ) where 4 is the r _ [ < rN coefficient of a_ in the binomial expansion of ( I k IM o ). It is 1 < a < N seen that 6 = Z 5 < a. Let R > 1 be such that 6 < 1. Let M > 0 be any real r=l r number such that 16

M> (1-5)- Olf + 2z I1 A I C (a.PC) ) (5.3) 1 <1-a11 < Na where f is a given element of [L (G)]. (In the case of L (G), p is taken as 1 p in (5.3)). Conclusion: Given f in [L (G)]m, X real, 1 Il < N and M > 0 satisfying (5-3), there exists a unique xe[L (G)] with I||x| < M such that — x = x, where p P T is defined as in theorem 2, by T X(t) - f(t) + T F(t, s,x(t,sk)). 1< <jaN. < Further, if f and f are any two elements of [L (G)], and x, x are the cor-' 1 2 P 2 responding solutions of TX = x, then i|x -x | < (1-R) )51|f -f i| "2i P- R 2 P Proof: Let us observe from (5.1) and (5.2) that IF (t,s,x) x < |F(t,sx)-Fs(t,s,o) + |IF(t,s,o) < M ixi + Ac(t,s) (5.4) As a consequence of the assumption on F and the inequalities (5.4), it follows (see 3.v in ~3) that F (t,sg,x(t,sf)) for fixed tD is in [L (G)] m Hence acq maps [L(G) into itself. T F (t,s,x(t',s)) is in [L (G)]m and consequently maps L (G)] into itself. a p p9 p p p Let us now show that for x,x in [L (G)]m and any integer r > 1, we have 1 2 p 1ne. (5.5) Indeed, ITx (t)-Tx (t)l = 1 2 l7

a a = | T Fa (t, s, x(t, s))-.-AT F~(t, s,, x 2(t,s)) < l| XT M| x -x1 (t,s) and Tx (t)- rx (t)T < zlX lT MaI x -T x | 1 2 1 2 It follows now by induction that i 2'I-p2 Now, it follows by using the inequalities (2. 1) that l ~rx- T X <6 | x X 2P r 1 2 P This concludes the proof of the inequality (5.5). As a consequence, it is seen that T = T is continuous on [L (G) m. Further, since 5, - 0 as r -> oo, there is p r an R > 1 such that 6R < 1. The corresponding operator T is a contraction on [L (G)]m. p As a further consequence of (5.5), we have IT+ -TxH < |6ilTx-xll,xc[L (G)]l i = 1,2... Hence, for r > 1, ITr xl< ||x|| + ri J xT' x 1x11 + x-T x I lrx|i + 1|TX-Xi|L| i For x = o, this inequality yields IIT (0)1 (? )11T(o)l1 = sIT(o0)I (5.6) Hence, by (5.5) II x1i< I||xll + 611iT(0)l. (5.7) 18

But, since |T(O) (t) < If(t)| + z|L T A(t?,s )I it follows by the inequality (2.1) that a a -1/p I1T(0) | < ifll + IA h (atp /P 1 < 1.1-< N Thus, by (5.7), 1 rx|| < 6 IX|| + 6(|I f| +Zj| X A c|ihCl(UIP )l/p (5. 8) Let us consider the set XM = {x[L (G)] Ilxl <IM) M pp This set is mapped into itself by T. Indeed it follows from (5.8) that for XEX M RR m [|Tx|| < MbR + M(1-BR) xVI Now, since T is a contraction on [L (G)], it is so on XM. Further, since X p M is a closed subspace of the Banach space [L (G)], and thus itself is a Banach P r space, it follows that there is a unique xcX with T x = x. But since T is a continuous operator on [L (G)]m it follows (see 3. ii in ~3) thatTx = x. This p concludes the proof of existence and uniqueness. Let f and f be any two elements of [L (G)]m and let x and x be the cor1 2 p 1 2 responding solutions of (2.4). Let f = If -f | and x = Ix -x i. It is seen 1 2 1 2 that x(t) < f(t) + (|kIXT MC)x(t). Applying the inequality again, we get 19

x(t) < f(t) + ZJ TC M(f(t) +(Z|l ITa M )x(t)) = (1 + Z|X|T M )f(t) + (zl I T M )2x(t) In general for r > 1, x(t) < [1 + (zlxl T M )+..+(zIX TM) Mr- ]f(t) + (Z I X TM )r.x(t). But then IXl< flf||p(l +6 +6F...+6,) ~6.Jx|| ||xP< lf 1 +| f | | + I2 rx r p If R is such that R < 1, then Xp(1 -) | if(p( + 5 +..+ |) <r-1 5 Pfl thus, i. e., |-x 1 x1< 68(1-86R)l|f -f | This concludes the proof of the theorem. 60 GENERAL REMARKS A. The arguments of the previous sections, when applied to a space of analytic functions yield a unique analytic solution of (2.4) provided the function F are assumed linear in x with analytic coefficients. Precise formulations follow: 20

Theorem 4: Let G = (tEn Iti < hi,i = 1,2...n) and let f(t), A (t), 1 < ta|; < N i 1. It be functions analytic on an open rectangle R = tcE E ti H, i - 1,2... n containing G. Then there exists a unique function x(t) analytic on G and satisfying x(t) = f(t) + T V e(A x)(t',sQ (6.1) 1 < |fa < N Proof: Let R = (tcE |til < Hi,i=l,2...n} be such that - -c:R P Ton D = c 1t. il < H.,i=l,2..n c C, where C denotes as usual, the set of complex numbers. Let O(D) denote the set of all functions holomorphic in D. Then O(D) is a Banach space with the norm given by 00 oc ~~6o 1Ilxl = IX l H for x - Ia =o (see 3.vi) Let f(s) and A (G) be natural holomorphic extensions in D, of f(t) and A (t) respectively. Letr be an operator defined on O(D) by Tx( ) = ) + x a)( ) where T T 1... T is analogous to T; for example,'-*v 1 n \j1 Tx(clt) fS x(q:, )d.,i = 1,2.. n. It is to be noted that for xeO(D), the integral defining T,, does not depend on the path. It is clear that maps O(D) into itself. Further, for any positive integer r and any x,x EO(D), we have 1 2 21

11TrxT -x II < Z (.) -x1 - x 1 2 r < |a| < rN where p1 is the coefficient of 0 in the binomial expansion of (1 <C < N | | A co )r-here, A = Sup(|A (|)I:cD). [To obtain the above inequality, we observe that if Z ~v x -X2 I==0av 1 2 |V|jOV: then IT (x - x )I < a vvl((a + v).) CY 1 2 VV < Ela IH v ((a + v)')<!la |HV H (at)- ] V V It follows that IC x -TX I < 6 I|x -x II where 6 = E 4H (Qa) 1 2 12 r r < |aI < rN Since this is true for any x,x eO(D) and 6 6 < oo-(it is majorised by an 1 2 r=l r exponential function)-and since O(D) is a Banach space, it follows by statement (3.i) that T has a unique fixed point in O(D). If x( ) denotes this fixed point, and if x(t) denotes the restriction of x(I) for: real, it is clear that x(t) is the unique analytic solution of (6.1). B. The canonical form suggested in this paper is very similar to the form studied by Kasatkina [5]. It is to be noted, however, that the notation proposed here simplifies the exposition. Further, it taKer care of repeated itegrals also, in a natural way. Of course, a repeated integral can be transfcrmed into a single integral and the author found that it made no difference in the estimates obtained here. The arguments remain the same too. 22

C. A function veL (G) will be said to be the generalized partial derivatives p of order a = (...a ) of a function ucL (G), or v =D u, in the usual sense (C. B. 1 n p Morrey [7], L. S. Sobolev [9]). We mention here that generalized partial derivatives of order one, have a simple characterization. A function xt L (G) is the i p generalized partial derivative of xcL (G) with respect to ti if and only if for p i almost all closed rectangles R c G, R = [tici < ti < d,i=l,2...n), we have dv xt dt J, x(d. s)-x(cis)]ds R i i where c' and d' are defined as usual, by c (c,,c!) and d = (d.,di) Here the expression "for almost all rectangles R = [c,d]" means that the set of all (c,d) forms a set of measure zero in GxG. It is not difficult to verify that if xcL (G) then T.x possesses generalp 1 ized partial derivative with respect to ti,i=1,2...n. Consequently, if x is an L -solution of an equation of the form P x(t) - f(t) + TZX cTP F(t,s,x(t,s)) for a given i, i = 1,2...n, and if f possesses generalized partial derivative with respect to ti, then x also possesses generalized partial derivative with respect to t,. An example of such a situation is the equivalent of Darboux problem. z(x,y) =- fSXF(a,,Pz(a,P))deidP 00 Any L -solution of this equation possesses generalized partials with respect to both x and y. 23

APPENDIX We shall discuss now the application of our existence theorems to the special case of the classical Darboux problem in a rectangle G = [a,a + h] x [b,b + k] c E2 z F (x,,z, z y), (xy)G, xy i. x y z (x,b) p (x); z (a,y) - r (y); cp(a) = (b), Z = Z (I 2,...2 z); i l,2o.. n (A 1) The precise results which will be stated below as corollaries of our theorem 3 of 95 will be applied in the optimal control problem mentioned in the introduction. We shall need the following hypothesis on (A. 1) to be able to apply theorem 3, ~5 and obtain solutions of (A. 1) belonging to a Sobolev class (see [2]). (H): The fuictions cp(x) - (cp, c2,...), and i(y) (12. ) are defined and absolutely continuous on> [a,a + h] and [b,b + k] respectively. The derivatives cp an.d y which exist almost everywhere, belong to L ([a,a + h]) x y p and L ([b,b + k]) respectively; here 1 < p < oo Further cp(a) = *(b). (H): F = F(x,yz,r,t) = (F F,.F ) is defined for all (x,y)eG 212 n 2 and (zrt)cE3. For each i, Fi is measurable in (x,y) for fixed (z,r,t)E3n. (H ): There exists a constant K > 0 such that for (x,y)eG and (z r,t ), 3 1 1 (z,r,t )EE'", we have 2 22?!

IF(x,y,z,r,t ) - F(x,y,z,r,t )I K( z -z1 + r -r + It -t I). I l 222 1 2 1 2 1 2 (H ): Let F(x,y,O,O,O)c[L (G)]. 4 P Remark: One may assume, instad of (H ) that, there exist constants K j, 3 1j K, K. > 0 with j = 1,2,...,n, such that, 2j 3j IF.(x,y,z,r,t ) - F.(x,y,z,r,t )| < 1 1 1 1 1 2 2 2 n < Z [K IzJ-zJl + K Jr-rJl + K ttJ-tJl]. - =1 =l 1 2 2j 1 2 3 2 But with no loss of generality we may set K = K = K = K > 0 and 1j 2j 33 K = nK so that the above inequality reduces to (H ). 3 Let W1(G) denote the Sobolev space of all z~L (G) with first order generP P alized partial derivatives (see ~6) z,z belonging to L (G). Let ||z1 = x y p I||z|| + + I z + I|z Iy denote the norm in [Wp(G)] n. " p xp yp p Theorem 5: Let the hypotheses H -H hold. Then, there exists a unique 1 4 zc[W (G)], 1 < p < oo (same as in (H )), such that (i) the generalized partial p 1 derivative z (x,y) exists and equals F.(x,y,z,z,z ) a.e. in G and (ii) z(x,b) = C(x); z(a,y) = (Y). Further, IzIl < (1-s )- [kl/P(||xp + 2 p'II ) + + hl/p(lllp + 2-'1ip ) + (h+k)lls(x,y)ll (A.2) = =p p = where s(x,y) = F(x,y,OOO); 6 = Z; 8= [n/2] p with p = (2K +)x (p /p) (hP + kP)/P if 1 < p < o and p = (k + 2 ) (h + k) if p = 1 or p = o. 25

The number k here is same as in (H ) and the number R in (Ao2) is that positive 3 integer for which 6R < 1. If (cp, ) and (cp, ) are any two pairs of functions satisfying (H ) and 1 1 2 2 1 if z,z are the corresponding solutions of (A. ) then 1 2 I llz ( -1 [kl/ + 2-1cp) + (llyp + 2 ip)] (A5)1 where z z z;,= cp - p; 4 = i - 1 2 1 2 1 2 t's are as aboveo Proof: Let us consider the integral equation z(x,y) = cp(x) + V(y) - cp(a) yF(aZ(,) z ( ) (a,))dad (A. ) where z and z are understood as generalized partials of z. Clearly, any x y solution of (A.4) (which is necessarily continuous on G and hence in [L (G)] ) p has, indeed, generalized partials z, z which satisfy the following x y w (x,y) =' (cp(x) +.(y)) + 2 Jx (a,y)da + 2.f (x, )dB i':0 2'b q (x,y) = x () + Jx O. a f+ f F(x,P,,(x,P))d3 e x a b (x,y) = (y)'+ / F(aCy, (ca,y))dc + fb O. dB (A 5) 3 ya b Where = z; = Z;' z; W= (w,w' ) (A.6) 1 2 3 y 1 2 3 Further, z exists and equals F(x,y,z,z,z ) ao e in G. Thus, every [W'(G) ] xy xy x p solution of (A 4) (and herce of (A. 1) corresponds in a unique manner (as in (Ao6)) to a [L (G)]3- solution W = (w,w,i ) of (Ao5)o Now, the system of P L 2 23 26

equations (A. 5) is exactly in the canonical form (2.4). Since the hypotheses H - H guarantee the assumptions in theorem 3, ~5, existence of a unique solu1 4 tion wc[L (G)]3n and hence of the corresponding ze[W (G)] is concluded. p P The norm bound (A. 2) for the solution z follows from the inequality (5. 3) with the observation that in the present case, ||w|| - ilZ |; N = 1 P A(xy) = F(x,y,O,0,0); a 1; f = (2('1( ),CP,); a - (1,0) or (0,1). It is to be noted that |. in (5.3) denotes the norm in L (G) while I||p| in (A. 2) denotes the norm of p(x) in L ([a,a + h]), and so on. The same observation leads us from the conclusion of theorem 3, ~5 to the inequality (A. ). This concludes the proof of Theorem 5. Special cases: (i) If F of (H ) does not depend on r and t, then (A.4) 2 is itself in the canonical form (2. 4). In this case, the norm bound for the solution z is given by ll 6(1 - sR) -1C2 (kl/pi + hl/P I) + p-l/P( ihkls + kl/P2 lh|lp 1l + h1 /2kP2-1k ll )], x y where 5 = (r) 2( Khk) if p = 1 or ap = o and r 5 = (IPr)- P(khk)r if 1 < p < oo and 00 Z E as before.!n=-l n' (ii) Let U c E and let F be any set of measurable functions \.G W j. Let f = f(x,y,z,r,t,u) be defi.ed on GxE3' xU and let f be measurable in (x,y), contlinuous in u, and Lipschitzian (as in (H )) in (z,r,t). Let for each 27

vcr, the function f(x,y,0,0o0,u(x,y) belong to [L ( )]. For a given uer, define F(x,y,z,r,t) = f(x,y,z,r,t,u(x,y)) (A.7) Then F satisfies H, H, H. By applying theorem 5, to this F we obtain a 2 3s 4 unique solution z of (A. 1) corresponding to the data ar, and - (which defines F). The inequality (A.2) gives the norm-bound on z, as before. If / and 2 are any two elements of F' (,l a, ) and (cp, ) satisfy (H ) and if zi is the 1 1 2 2 1 solution of (A. 1) correspondi.ng to the data (oi,'i: i), ill,2 then z = z -z 1 2 satisfies the inequality (A.2) with p = cp -:; 4 = -4r anc 1 2 1 2 s(x,y) - f(x,y,z,z,z,v ) - f(uyz,z,z, ) (A. 8) l(Xyy) f~xyyyz 1 Y ixy ly 1 l ix ay 2 Poin.,twise estimates: Since any solution z of (A.1) satisfies the integral equation (A. 4), it is absolutely continuous in the sense of Tonelli. Hence, there is a set E - G, with meas E = O such that for (x,y )eG-E, we have in view of (H ), 3;x(Xo Y) l < |Px(x,)I + b [s k(|Z + Iz [ + |Z yi)](x,A)dp (A. 9) where s(x,y) denotes F(x,y,0,0,0). Similar il:equality holds for z (x,y )o y.Yo Further z(x,y) = (y) t+ S z (a,y)da cp(x) + fYz (x, )do Using these a x b y facts along with repeated application of Gronwall's lemma one finds that | z(x,y)j < 11 HC/p| + 1lli* l + eff () s(a,P)dacdB + A (Ah + Ak + e(h+k)] z| (x,y)| < e (x) + AA and Iz (x,y)| e 9 (y) + AA (A. 10) 1 28 y 2 28

where (x) = e [lp P(x) + Kklp(x) I + lb s(x,) )d I1~~X b e (y) = eK [ly (Y) + Khlt(y) + a s(a,)da] 2 Ya A5 i.ph + llk/yl + + (I + h )Khk + ffs(a, )dadP ~5 x P V c c ~G and- A is a constant depending only on K,h and k, In the above 1o ]| denotes the supremum norm. Dependence on data: Let zi denote, as before, the solution of (A. ), correspo:ding to the data (Cpii vi). i = 1,2. (F being given by(A.7) ) It is seen then that z = z - z and z = z - z satisfy the inequalities (A. 9) and x ix 2X y ly Y hence pointwise estimates for z = z - z and its derivatives are also given 1 2 by (Ao 10) where z, cp, Jr and s are understood as follows: z z -z; cp = c - p; fr = -; 1 2 1 2 1 2 s(x,y) =- f(x,y,z,z, ) - f(x,y,z,z,z,v ix ly I ix y29 2 29

BIBLIOGRAPHY 1. F. F. Bonsall, Lectures on some fixed point theorems of functional analysis, notes by K. B. Vedak, Tata Institute of Fundamental Research, Bombay, India 1962. 2. L. Cesari, Sobolev spaces and multidimensional Lagrange problems of optimization, Annali Scoula Normale Sup Pisa, Vol. 22 (1968) pp. 193-227. 3. T. H. Gronwall, An integral equation of the Volterra type, Annals of Mathematics, ser. 2, Vol.16 (1915) No. 3, PP. 119-122. 4. R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice Hall Inc., New Jersey, 1965. 5. N. V. Kasatkina, Uniqueness theorems for a system of multidimensional Volterra integral equations, Differentialnye Uravneniya, Volo 3, No. 2 (1967) pp. 273-277. 6. M. A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, Engl. tr. ed. by Jo Burlak, Pergammon Press, New York 1964. 7o Co B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer Verlag, New York 1966. 8. M. B. Suryanarayana, Optimization problems with hyperbolic partial differential equations, Ph.D. thesis, University of Michigan, Ann Arbor 1969. 9. So L. Sobolev, Applications of functional analysis in mathematical physics, Izd 1950, Amer. Math. Soc. Transl., Vol. 7, Providence, R.I. 1963. 10. W. Walter, Differential-uund Integral-Ungleichungen und ihre anwendung bei Abschatzungs-und Eindeutigkeitsproblemen, Berlin, Springer 1964. 50

11. Arthur Wouk, Direct iteration, existence and uniqueness. Nonlinear integral equations (Proc. Advanced Seminar conducted by Math. Research Center, U. So Army, University of Wisconsin, Wisconsin 1963) pp. 3-33, University of Wisconsin Press, Madison, Wisconsin 1964. 3'1