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Appendix a Explicit Estimates in the Implicit Function Theorem

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Appendix a Explicit Estimates in the Implicit Function Theorem Appendix A Explicit Estimates in the Implicit Function Theorem In this appendix, we carefully examine the implicit function theorem. We extend this standard theorem to classes of functions with additional properties such as bound- edness and uniform and Hölder continuity. The crucial ingredient is the explicit formula (A.2) for the derivative of the implicit function, which allows us to transfer regularity conditions onto the implicit function. As an application of the implicit function theorem in Banach spaces, we will establish existence, uniqueness and smooth dependence on parameters for the flow of a system of ordinary differential equations. Essentially, these are standard results from differential calculus, see e.g. Zeidler [Zei86, p. 150,165] or [Irw72, Rob68]. We consider a general setting of ODEs in Banach spaces and show smooth dependence, both on the initial data, as well as on the vector field itself. Moreover, our extension of the implicit function theorem yields boundedness and uniform continuity results. We start with some results on inversion of linear maps. Lemma A.1 (Invertibility of linear maps) Let X be a Banach space and let A ∈ L(X) be a continuous linear operator with continuous inverse. Let B ∈ L(X) be another linear operator such that B < 1 . Then A + B is also a continuous A−1 linear operator with continuous inverse, given by the absolutely convergent series − − n − − − n A + B 1 = − A 1 B A 1 = A 1 − BA 1 . (A.1) n≥0 n≥0 −1 Proof First of all, note that there exists an M ≥ 1 such that A, A ≤ M.The base of the geometric series can be estimated in operator norm as −A−1 B < 1, so the series is absolutely convergent and the limit is a well-defined continuous linear operator, whose operator norm can be estimated as − n A 1 (A + B)−1 ≤ A−1 −A−1 B ≤ < ∞. 1 − A−1 B n≥0 J. Eldering, Normally Hyperbolic Invariant Manifolds, Atlantis Series 151 in Dynamical Systems 2, DOI: 10.2991/978-94-6239-003-4, © Atlantis Press and the author 2013 152 Appendix A: Explicit Estimates in the Implicit Function Theorem That the limit is again a well-defined linear operator follow from the fact that L(X) is a Banach space. Applying A + B to the left-hand side of (A.1), we see that the candidate is a right inverse: − − n − n − n+1 (A + B) A 1 − BA 1 = − BA 1 − − BA 1 = 1. n≥0 n≥0 n≥0 Similarly the candidate can be shown to be a left inverse of A + B.Nowwehave that the candidate is continuous and a full inverse and furthermore, A + B itself is clearly a continuous operator as the sum of two continuous operators, so the proof is completed. Corollary A.2 (Linear inversion is analytic) Let I : A → A−1 be the inversion map defined on continuous, linear mappings A ∈ L(X) with continuous inverse, where X is a Banach space. The map I is analytic with radius of convergence ρ(A) ≥ 1/A−1. When X is finite-dimensional, I is a fortiori a rational map. Proof Extending well-known results on analytic functions to Banach spaces (see e.g. [Muj86]), we read off from (A.1) that the inversion map I can be given around A by an absolutely convergent power series with ρ(A) ≥ 1/A−1 and is thus analytic. When X is finite-dimensional, det(A) = 0 implies that A−1 is a rational expression in the matrix coefficients of A according to Cramer’s rule. The inversion map I is locally Lipschitz, like every C1 mapping: − 2 n A 1 (A + B)−1 − A−1 ≤ −A−1 B A−1 ≤ B. 1 − A−1 B n≥1 However, when we restrict to a domain bounded away from non-invertible operators A, that is, when A−1 ≤ M, then the Lipschitz constant is bounded for small B. This implies that when A = A(x) depends on a parameter via a certain continuity modulus, then A(x)−1 will have the same continuity modulus up to the Lipschitz constant, at least in small enough neighborhoods. The standard implicit function theorem on Banach spaces can be stated as Theorem A.3 (Implicit function theorem) Let X be a Banach space, Y a normed k≥1 linear space, and let f ∈ C (X × Y ; X). Let (x0, y0) ∈ X × Y and assume −1 that f (x0, y0) = 0 and that D1 f (x0, y0) ∈ L(X) exists as a continuous, linear operator. Then there exist neighborhoods U ⊂ Xofx0 and V ⊂ Yofy0, and a unique function g : V → U such that f (g(y), y) = 0. Furthermore, the map g is Ck and the derivative of g is given by the formula −1 Dg(y) =−D1 f (g(y), y) · D2 f (g(y), y). (A.2) Appendix A: Explicit Estimates in the Implicit Function Theorem 153 See [Zei86, pp. 150–155] for a proof. Note that we do not need to assume that Y is a complete space, as the contraction theorem is only applied on X. Recall that we use notation where D denotes a total derivative, while Di with index i ∈ N denotes a partial derivative with respect to the i-th argument. Formula (A.2) for the derivative of the implicit function g will be crucial for the extension of the implicit function theorem to many classes of regularity, extending Ck smoothness. We use the Lipschitz estimate for the inversion map and require that the regularity conditions are preserved under composition, addition, multiplication and localization of functions. By Proposition C.3, the derivatives of g are expressed −1 in terms of D1 f (g(y), y) acting on a polynomial expression of same or lower order derivatives of f and strictly lower order derivatives of g. k,α As an example, let us take Cb functions. Using Lemma 1.19 and induction over k, this function class is preserved under products. For composition, we check Hölder continuity, ( ( )) − ( ( )) ≤ − α α ≤ ( α) − α f g x2 f g x1 C f Cg x2 x1 C f Cg x2 x1 for 0 <α≤ 1, when x2 − x1 ≤ 1. In case x2 − x1 > 1 however, we can directly use the boundedness of f : ( ( )) − ( ( )) ≤ ( ( )) + ( ( )) ≤ − α. f g x2 f g x1 f g x2 f g x1 2 f 0 x2 x1 Thus, Hölder continuity is preserved with some new Hölder constant, while bound- ∈ k,α ( )−1 edness is trivially preserved as well. We conclude that if f Cb , and D1 f is ∈ k,α globally bounded, then we can read off from formula (A.2) that g Cb .Thesame k results hold for the class of Cb,u functions, or any other class of functions whose properties are preserved when inserted into (A.2). Together, interpreting α = 0as an empty condition, these lead to ∈ k,α ≥ Corollary A.4 Let in the Implicit Function Theorem A.3, f Cb,u with k 1 and −1 0 ≤ α ≤ 1. Assume moreover that D1 f (x, y) ≤ M is bounded on U × V < ∞ ∈ k,α for some constant M . Then g Cb,u , and the boundedness and continuity estimates depend in an explicit way on those of f . Remark A.5 Formula (A.2) only provides control on the derivatives of the implicit function, but the size of g itself can be controlled by choice of the neighborhood U. In our applications, this will match up with choosing coordinate charts around the origin in Rn. ♦ Let us now consider an ordinary differential equation x˙ = f (t, x), x(t0) = x0, (A.3) ∈ k,α(R × ; ) ≥ where x takes values in a Banach space B and f Cb,u B B with k 1, 0 ≤ α ≤ 1. We consider solutions x ∈ X = C0(I ; B) equipped with the 154 Appendix A: Explicit Estimates in the Implicit Function Theorem supremum norm, which turns X into a Banach space.1 We choose I to be a closed interval I = [a, b] ⊂ R. The Picard integral operator t T : X → X : x(t) → F(x)(t) = x0 + f (τ, x(τ)) dτ (A.4) t0 has exactly the solution curves of (A.3) as fixed points. It also implicitly depends on ∈ k,α(R × ; ) ( , ) ∈ × f Cb,u B B and t0 x0 I B. From now on we denote by Dx a partial derivative with respect to the argument that is typically described by the variable x. This T is a contraction for |I | = b − a small enough: t T (x1) − T (x2) = sup f (τ, x1(τ)) − f (τ, x2(τ)) dτ ∈ t I t0 t ≤ sup Dx f (τ, ξ(τ))x1(τ) − x2(τ) dτ t∈I t0 ≤ sup |t − t0|Dx f x1 − x2 t∈I ≤ |I |Dx f x1 − x2. We restrict T to a bounded subset of argument functions f , F ⊂ k,α(R × ; ), ≤ . Cb,u B B sup f k,α R f ∈F | | ≤ 1 = 1 Thus, choosing I 2R turns T into a q 2 contraction, which shows that there is a unique x ∈ X satisfying T (x) = x and therefore (A.3). Next, we consider small perturbations of both (t0, x0) and f . To apply the implicit function theorem, we define F(x) = x − T (x). This function has a unique zero and DF(x) is invertible, as F(x + δx)(t) − F(x)(t) t = δx(t) − Dx f τ,ξ(τ) · δx(τ) dτ t0 t = δx(t) − Dx f τ,x(τ) · δx(τ) + O ξ(τ) − x(τ) δx(τ) dτ t0 = DF(x) · δx (t) + o δx (A.5) The neglected terms are o δx since Dx f is uniformly continuous on I ,soDF(x) exists.
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