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Consider the differential equation $\dot x = f(x)$. The standard proofs are

  1. The Picard iteration based proof of existence/uniqueness for Lipschitz $f$.

  2. The Peano existence theorem for continuous $f$.

  3. The Caratheodory existence theorem for measurable $f$.

My question is as follows. Assuming a Lipschitz $f(x)$, are there any other proofs out there for existence of solutions (in some reasonable sense) for ODEs?

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Why is another proof wanted? –  Deane Yang Aug 26 '10 at 17:49
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Because it's in the spirit of the game –  Pietro Majer Aug 26 '10 at 18:03
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Not only. Each method can be extended and generalized in different ways and points at different directions, so the more tools you have, the better. – Piero D'Ancona 0 secs ago –  Piero D'Ancona Aug 26 '10 at 19:03
    
I remember a proof in a Bourbaki volume which used a very nice approximation scheme with piecewise linear functions. The right hand side was a 'fonction reglee' and the method was called Euler's method if I remember correctly. Sorry for the vagueness, not able to check at the moment. –  Piero D'Ancona Aug 26 '10 at 19:59
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3 Answers

Numerical methods like Euler or Runge-Kutta, consisting in approximating a solution of an ODE with solutions of suitable discrete difference equations, in particular give proofs of existence.

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But is this really a new proof? It seems to me that the proof is always along the lines of rewriting the ODE as an integral equation, using a discrete approximation of the definite integral, and iterating. Isn't using Euler or Runge-Kutta just using a better discrete approximation for the integral. –  Deane Yang Aug 26 '10 at 20:09
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Well, a different proof of a theorem, to be really different, should have at least a different thesis, possibly opposite ;-) However I agree with you, at least partially. It has to be said that it happens often that proofs of a theorem, that are very different in spirit, rely on very similar computations in the core. –  Pietro Majer Aug 26 '10 at 22:32
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A very interesting kind of existence (though not uniqueness) proofs are proofs that use one of the various fixed point theorems and the tools from fixed point theory: The Schauder fixed point theorem can be used to prove Peano's existence theorem or simple existence theorems for boundary value problems. The theory of Brouwer-degree of certain mappings (not just between manifolds but also between banach spaces) can be used to prove several existence theorems, for example existence of periodic solutions for certain ODEs.

Granas / Dugundji - Fixed point theory is a very good (and very densely written) book about all kinds of fixed point theorems and one of the main application that frequently occurs throughout the book are existence theorems for differential equations.

Then there is the variational approach: Finding minima of functionals often is the same thing as finding solutions of PDEs (Euler-Lagrange-equations !). So very much of functional analysis can be applied to prove various existence theorems for PDEs. For example have a look at Guisti - Direct methods in the calculus of variations. Any book on the finite element method (Braess comes to mind, but I'm not sure at the moment if it is in english or in german... may be there are two versions?) will show you how Hilbert space methods can be applied to prove existence theorems.

The whole theory of distributions was invented for dealing with (linear) PDEs, their (weak) solutions and the regularity theory of these solutions. The Ehrenpreis-Malgrange-theorem is a very strong existence theorem that says that all linear PDEs with constant coefficient have distributional solutions. In Fact there exist (tempered) Green's functions for every such PDE.

And there is of course a more heavy machinery too: Morse theory and generalizations of it were used for (invented for?) the proof of the Arnol'd conjecture which also shows the existence of certain periodic solutions of differential equations.

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I was considering variational principle as an "alternative". But the, the local existence/uniqueness for solutions of variational principle again relies on local existence/uniqueness of ODEs in a coordinate chart. Right? –  Sujit_Nair Aug 27 '10 at 3:11
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No, not necessarily. As I said a solution to some PDE can often be obtained as a (the) minimum of a functional {Space of functions}$\to\mathbb{R}$. For example $u \mapsto \int_\Omega \|\nabla u\|_2^2$ has a unique minimum on $\lbrace u\in W^{1,2}(\Omega) | u_{|\partial\Omega}=g\rbrace$ if $g$ satisfies a certain condition. This minimum is exactly the solution of the Dirichlet equation $\Delta u=0, u_{|\partial\Omega}=g$. Existence and uniqueness of this minimum can be proved without any local considerations. –  Johannes Hahn Aug 27 '10 at 9:36
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The simplest proof (though 1 variable is probably not what you want) is in Peter Lax's Calculus book, Springer, 1976, page 442. It uses only the fundamental theorem of calculus. Existence, uniqueness, and asymptotics are proved for the case there.

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