Smooth dependence of ODEs on initial conditions

Question

The following is a theorem known to many, and is essential in elementary differential geometry. However, I have never seen its proof in Spivak or various other differential geometry books.

Let $t_0$ be real, and $x_0 \in \mathbb{R}^n$ and $a,b>0$. Let $f:[t_0-a,t_0 + a] \times \overline{B(x_0,b)}\rightarrow \mathbb{R}^n$ be $C^k$ for $k\ge 1$.

Then $f$ is Lipschitz continuous, with which it is easy, using the contraction mapping theorem of complete metric spaces, to prove that the ODE: $\dfrac{d}{dt}\alpha(t,x)=f(t,\alpha(t,x)),\quad \alpha(t_0,x)=x$

has a continuous solution in an open neighbourhood of $(t_0,x_0)$. In other words, the ODE $x'(t)=f(t,x(t));x(t_0)=x_0$ has a family of solutions which depends continuously on the initial condition $x_0$.

The theorem that I'd like to prove is that, in fact, if $f$ is $C^k$, then $\alpha$ is $C^k$, for any $k\ge 1$.

I'd like an "elementary" proof that needs no calculus on Banach spaces or any terribly hard theory such as that, but hopefully something elementary, such as the contraction mapping theorem. I currently have an attempt of a proof that looks at perturbations of linear ODEs, but it is incorrect (I think). The proof can be found on page 6 of http://people.maths.ox.ac.uk/hitchin/hitchinnotes/Differentiable_manifolds/Appendix.pdf. I believe that there is a typo in the claim:

"Apply the previous lemma and we get

$\mathrm{sup}_{\left| t\right|\leq \epsilon}\left\|\lambda(t,x)y-\{\alpha(t,x+y)+\alpha(x)\}\right\|=o(\left\|y\right\|).$"

but more importantly, what it should be replaced by is incorrect. What is needed is that $\|A-B_y\|=o(\|y\|)$ but I do not see why this is.

Thank you for your time and effort.

Answer 1

If you differentiate the ODE one or more times with respect to time or the initial conditions you will find an ODE for the derivatives of the solution. Use standard existence theory to prove the new ODEs have a solution, then verify that these really are the derivatives of the original equation.

(Sorry to be so brief and sketchy – I am a bit pressed for time this morning.)

Answer 2

An elementary 'coordinate' proof is given in Ordinary Differential Equations by Philip Hartman. It doesn't even use the contraction mapping argument. The main effort is spent to show the $C^1$-regularity of solutions with respect to the initial data (as well as time and other possible parameters in the nonlinear term).

Basically, the proof goes as follows.

• Lemma. If function $f=f(t,x)$ belongs to $C^1(\Omega, \mathbb{R}^n)$, where $\Omega=(a,b)\times K$ and $K$ is an open convex set in $\mathbb{R}^n$, then for any $(t,x_1,x_2)\in (a,b)\times K\times K$, $$ f(t,x_2)-f(t,x_1)=\sum\limits_{k=1}^{n}f_k(t,x_1,x_2)(x_2^k-x_1^k)$$ with the $\mathbb{R}^n$-valued functions $f_k\in C((a,b)\times K\times K)$ given by $$f_k(t,x_1,x_2)=\int_{0}^{1}\frac{\partial f(t,sx_2+(1-s)x_1)}{\partial x^k}ds.$$
• Let $h$ be a scalar, $e_k$ be a normalized vector in $\mathbb{R}^n$, and let $x_0(t)=\eta(t,x_0)$ stand for a solution to the problem $\dot{x}=f(t,x)$, $x(t_0)=x_0$. Then $x_h(t)=\eta(t,x_0+he_k)\to x_0(t)$ uniformly on $t\in[a,b]$. This is just a corollary of continuous dependence of solutions on the initial data.
• Thanks to the Lemma
  $$[x_h(t)-x_0(t)]'=\sum\limits_{k=1}^{n}f_k(t,x_0(t),x_h(t))(x_h^k(t)-x_1^0(t)).$$
• Introduce the abbreviation $$y_h(t)=\frac{x_h(t)-x_0(t)}{h},\qquad h\neq 0.$$ We need to show the existence of $\lim y_h(t)$ as $h\to 0$. Since $x_h(t_0)=x_0+he_k$, $y_h(t_0)=e_k$. Therefore, $y_h(t)$ is the solution to the initial value problem $$\dot y=J(t,h)y,\qquad y(t_0)=e_k,\qquad\qquad\qquad(1)$$ where $J(t,h)$ is a $n\times n$ matrix in which the $k$th column is the vector $f_k(t,x_0(t),x_h(t))$.
• Thanks to the Lemma above, it follows that $J(t,h)\to J(t,0)$ as $h\to 0$ uniformly on $[a,b]$, where $$J(t,0)=\left.\left(\frac{\partial f}{\partial x}\right)\right|_{x=\eta(t,x_0)}.$$

• Consider (1) to be a family of initial value problems depending on a parameter $h$, where the right side $J(t,h)x$ of the ODE is continuous on the open set $a< t< b$, $|h|$ small, $x$ arbitrary. Since the solution of (1) is unique, the theorem on continuous dependence of solutions on initial data implies that the general solution is a continuous function of $h$ (for fixed $t$, $t_0$.) In particular, $y(t)=\lim y_h(t)$, $h\to 0$ exists and is the solution to the problem $$\dot{y}=J(t,0)y,\qquad y(t_0)=e_k. \qquad\qquad\qquad(2)$$ for $t\in(a,b)$. Hence $\partial\eta(t,x_0)/\partial x_0^k$ exists.

• Finally, to verify that this partial derivative is continuous with respect to $x_0$, note that (2) is a family of initial value problems depending on parameter $x_0$. Since $J(t,0)$ is a continuous function of $(t,x_0)$ and initial value problems associated with linear differential equations have unique solutions, we conclude that $y=\partial\eta(t,x_0)/\partial x_0^k$ is a continuous function of its arguments (again, the theorem on continous dependence of the solution on time and initial data is used).

The $C^k$-case is a direct corollary of the $C^1$-result (use mathematical induction or iteration in $k$).

Answer 3

There is also a proof in appendix B of the Springer book "Lie Groups" by Duistermaat and Kolk. It does use the contraction mapping and implicit function theorems on Banach spaces, though. This is a bit abstract, but if you are willing to step over that, then with no extra work you also obtain $C^k$ dependence on the vector field $f$.

The essential idea is to consider the contraction mapping

$F: \mathcal{B} \times \mathbb{R}^n \times C^k(\mathbb{R} \times \mathbb{R}^n;\mathbb{R}^n) \to \mathcal{B} : x(t),x_0,f \mapsto x_0 + \int_0^t \; f(\tau,x(\tau)) \; {\rm d}\tau$

on the space $\mathcal{B} = C^0(I;\mathbb{R}^n)$ of solution curves for a small time interval $I$.

Answer 4

There are proofs in textbooks such as Lang's Real and Functional Analysis and Conlon's Differentiable Manifolds but they do use some Banach space theory (not an awful lot). I have also seen in one well-known text, a completely botched attempt at proving it.

Answer 5

The books you really want to look at for this result are advanced ODE texts,Max-it's only tangently related to the study of differential manifolds.Still,there are VERY good discussions in John M.Lee's INTRODUCTION TO SMOOTH MANIFOLDS and Noel J.Hicks' NOTES ON DIFFERENTIAL GEOMETRY.For discussions related to differential equations,the BEST source is probably Vladimir Arnold's texts-both the third edition of ORDINARY DIFFERENTIAL EQUATIONS and GEOMETRICAL METHODS OF ORDINARY DIFFERENTIAL EQUATIONS discuss this matter at some length and with a great deal of geometric insight.

Answer 6

Beside other good texts cited if you can find it you can read Pontryagin's "ordinary differential equations". He uses an elementary method to demonstrate continuity and differentiability properties of solutions both with respect to parameters and initial conditions. On the web you can read the first edition, in the second edition something is changed but I think it is the best introduction in absolute to elementary geometrical theory of differential equations.