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I'm having some trouble with the Existence and Uniqueness (E&U) thms. for ODEs.

Consider the IVP:

[tex]x'(t)=f(t,x(t)), x(\xi)=\eta[/tex]

In order to prove Existence, we need f to be Lipschitz or at least locally Lipschitz. For the Uniqueness we use the Banach's fixed point thm. for the operator:

[tex](Tx)(t):=\eta+\int_{\xi}^t f(s,x(s))ds[/tex] which is equavalent to the initial ODE.

The problem lies in proving the operator T has only one fixed point: To do this, we need f to be bounded on a certain domain (which also corresponds to the Lipschitz condition), so we actually need the domain of f.

But defining the domain of f is in special cases what causes the problem.

Consider the IVP:

[tex]x'=\frac{t}{1-x}, f(0)=2[/tex]

From the ODE we obtain the domain of f:

f: R x R\{1} -> R

(t,x) -> f(t,x)

So f does not appear to be bounded at all ?! We could conclude that f is continuously differentiable in x, so f is also locally Lipschitz continuous and therefore must be bounded by some const. at least in some interval of R, but we cannot dfind neither this interval, nor the constant from the initial data. (can we?)

So the boundness is what we seek but cannot find.

I hope someone to bring more light into this topic for me :)

Regards,

Marin