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Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the first argument). By Picard-Lindelof, there is a unique function $x : [0,+\infty) \to B$ such that $x(0) = \mathbf{0}$ and for all $t\in [0,+\infty)$, $x'(t) = f(t,x(t))$. Define the family of functions $Euler_h : [0,+\infty) \to B$ to be the result of linear interpolation between the points obtained by the Euler method on $f$ with initial value $\mathbf{0}$ and step size $h$. Does it follow that $Euler_h$ converges compactly to the unique solution $x$ as $h$ goes to 0 from the right? If yes, is an explicit rate known?

The results I have found are only for the points provided by the Euler method, and not for the approximating functions obtained from them.

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Hubbard's TAM textbook "Differential Equations: A Dynamical Systems Approach. Part II: Higher Dimensional Systems" covers convergence of Euler's method for piecewise-linear approximating functions as you call them. He's dealing with finite-dimensional systems but the proof generalizes, of course. I imagine there are some more modern numerical analysis texts out there that would be a more comprehensive reference but that's off the top of my head. –  Ryan Budney Oct 1 '10 at 3:58
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I believe it's more or less a straight-forward exercise to deduce these types of results from the Gronwall inequality (stated in suitable generality) -- thinking of the Euler method as an exact solution to an approximation to the differential equation. So that might be a better thing to look for. –  Ryan Budney Oct 1 '10 at 4:01

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The Picard-Lindelof Theorem is not quite correctly stated in your question. Recall that it is usually referred to as the LOCAL existence and uniqueness theorem, and it only guarantees a solution on a certain maximal interval [0,T), and for as simple a system as $x'=x^2$ the maximal existence time T is finite. That said, it is correct (with caveats) that the Euler method approximating solutions will converge to this "true" solution at all points of this maximal interval. The detailed story, with all the error estimates, is too complicated to state here, but you can find a careful discussion of this not only for Euler's Method, but also for a number of other methods, in Chapter 5 (Numerical Methods) of the book Differential Equations, Mechanics, and Computation (which I wrote together with my son Bob). There is a website for the book at http://ode-math.com where you can download for free more than half the book. In particular clicking here:

http://ode-math.com/PDF_Files/ChapterFirstPages/First38PagesOFChapter5.pdf

will download the first 38 pages of chapter 5, where starting on page 144 you will find a careful discussion of the rate of convergence and stability properties etc, for Euler's method, starting from scratch.

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You may also want to look at the literature on backward error analysis for ODE methods.

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