Smooth Dependence of Solutions of Differential Equations on Initial Data: A Simple Proof*
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1 Smooth Dependence of Solutions of Differential Equations on Initial Data: A Simple Proof* J. SOTOMAYOR ABSTRACT. A simple proof of smooth depehdence of solutions of ordinary differential equations. with respect to ini.tialcon,j[iions, is given. The proof uses the Fibre Contraction Theorem. A proof of the following theorem can be found in [1, 2]. 1. FIBRE CONTRACTION THEOREM. Let (X, d) and (X', d') be complete metric spaces and let I~: X x X' ~ X X' be a map of the form Assume that f(x, x') = (F(x), F'(x, ~')). a) F: X ~ X has an attracting fi point p, that is: F(p) = p, and lim F"(x) = p, for ecerv _x 9 X. b) The map x.-, F'(x,x') is continous in X, for every x' ~ X'. c) For every x e X the map F'~ X' ~X' defined by Fi,(x') = F'(x,x') is a ),-contraction, with 2 < 1. This means that for all xex and x',.v'sx'. d'(f.i~(x'), Fi,(v')) < 2d'(x', y') Then if p' denotes the unique attracting fixed point of F'p, the point/~ = (p, p') ex x X' is an attracting fixed point of ft. *Recebido pela SBM em 30 de agosto de
2 REMARKS I) Condition a) above is satisfied if, for instance, F is a,;.-contraction with < 1. This is the well known shrinking Lemma. The proof of Theorem 1 is essentially elementary, although slightly technical. 2) In [1,2] Theorem l was used to prove the smoothness of the invariant' manifolds associated to hyperbolic fixed points. 2. THEOREM ON SMOOTH DEPENDENCE ON. INITIAL CONDITIONS. Let.f he a map of class C t (continuous with continuous first partial dericatives) in an open set A ~ R", with values in R ~. For each point x o ~ A there are associated positive numhers :~, fl and a unique map ~ of class C l in with values in A, such that l, B e = {(t,x); It <,Ix-; ol < ~, (0) go (t, x) = f(r x)), 4(0, x) = x & for all (t,x) e l~ x B e. PROOF. Let b >0 be such that ~ ={x; Ix-xo[ <bl c A and let m = sup l f(x)l, l= sup ll Df(x)l[, for xeb--bb. Take ct and fl such that arm + fl < b and 2 = let < 1.'Denote by X the space of bounded continuous maps of I, x B a, endowed with the metric Denote by s d(~5,,,9 = sup{i oh(t, x) - 't'(t, x) l; (t, x) s I= x B,l. the space of linear endomorphisms of [R", endowed with the norm Ilgll = {suplgl; [xl = l}. Let X'be the vector space of bounded continuous maps of I= x B e in s endowed with the metric v') = supl I1 - v'(t, x)ii; (t, x) e I= x B,,',. Define F: X 56, X by F((,5) (t, x) = x + f' d 0.f(c~(s, x)) ds~
3 and I."".\ x,\" --,.\" by t"'(,h, ~//)U,.\)= t:, ~- t" [ l)f (,h(s,.\)1 ~fi' ls. \')ds, where E is the identiy element of '_/'. The map /g = (F, F') satisfies the hypofllcsis of Theorem I. In fact" a) F is a 2-contraction, since, by the mean value Theorem: d(f(~), F(~) = sup t i do I f(~5(s, x))-.f(w(s, x))l ds <_ sup, 11 ~5(s, x) - hv(s, x)[ ds < odd(o, ~) = 2 d((l~, '~). Hence F has a (unique) attracting fixed point ~, e X. b) Is immediate since Df is uniformly continuous in /~. t c) d'(f~,(0' ) - F~(~")) = sup [I f)f(o(s, x)) I gs'(s, x) - tp'(s, x) l dsll <_ ),d'((ff, ~'). i do The (unique) attracting fixed point of f is ot: the form (fi = (0, (~'), where F(qS) = 4) and F'(qS')= 95'. Relation (0) is obtained differentiating with respect to t both membrs of F(qS) = qs. Continuity of #5 is I n x B~ is immediate since q5 ~ X. To prove that q~ is of class C 1 it is enough to verify that D2q5 is equal to ~5' which is continuous in I~ x B~, since Dig5 = fo q5 is continuous in I~ x B~. The sequence (0,, gs',) = F"(4%, 0;), where qs0(t, x) = x and 4/ -= E, satisfies 4,,, 4) and q~',, qs', uniformly in I~ x B~ ; moreover, every qs, is of class C 1 and, for every n, DzqS, = ~b',. This follows by induction. Therefore, since 4), = D2q5. is continuous because it is an element of X', it follows, by the theorem on the exchange of the order of taking uniform limits and differentiating, that Dzq~ exists and is eq.ual to 4)'. This ends the proof. 57
4 REMARKS 1) The same.arguments in the proof above lead to the somoothness of solutions of "'non autonomous" differential equations x' = f(t, x), X(to) = x, where f is continuous with D z f continuous in an open set of the (t, x)- space. 2) Other classical theorems of Analysis, like the Inverse Function Theorem, can be proved using Theorem I. We outline such proof in what follows. The interested reader can fill in details. If g is of class C 1 in an open set of [t~ with L = Dg(xo) non singular, it can be assumed, by translating Xo and 9(Xo) to 0 ~ [W and composing with L-~, that 9 is of the form v=9(x)=x +A(x), with A(0)=0 and DA(0) =0. Let B6 be a ball centered at 0 with radius 6, where {I DA]I < 89 Call X the space of continuous maps ), of B6/2 with values in B6/2, and let X' the space of bounded continuous maps ~/' from B6/2 tos the space of linear endomorphisms of R". Apply Theorem 1 to the map f = (F, F'), where F: X ~ X is given by F(7)(.v) = -A(y + 7(.v)), and F': X, X' is defined by F(,,,, y'x.v) = -DA(y +,/(.v))o [E +?,'(.v)], and obtain the existence of the inverse of 9, 9-1, in the form x = v + ),(v), with y ~ X, and its differentiability. Actually, going backwards, this form of 9-1 motivates the definition of F, since it follows, by substitution in 9, that v +,/(v) + A(y + ;~(v)). = v, ar~tt,/(y) =- A(.v + ~.,(.v)). Also, the definition of F' is obtained heuristically differentiating this last equation. 58
5 REFERENCES [l] HIRSH, PUGH, Stable manifolds.for hyperbolic sets, Proc. Syrup. in Pure Math., vol. XIV, AMS, [2] I. C. DE Ot.,vEm^, Variedades Invariantes de Pontos Fixos Hiperb6- licos, S~io Paulo, Inslituto de Matemiltica Pura e Aplicada Rio de Janeiro - BRASIL 59
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