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Eulogy
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Homework Statement
For the space of continuous functions C[0,T] suppose we have the metric ρ(x,y) =sup [itex]_{t\in [0,T]}[/itex]e[itex]^{-Lt}[/itex][itex]\left|x(t)-y(t)\right|[/itex] for T>0, L≥0.
Consider the IVP problem given by
x'(t) = f(t,x(t)) for t >0,
x(0) = x[itex]_{0}[/itex]
Where f: ℝ×ℝ→ℝ is continuous and globally Lipschitz continuous with
respect to x.
Find an integral operator such that the operator is a contraction on (C[0,T],ρ) and hence deduce the IVP has a unique solution on C[itex]^{1}[/itex][0,T]
The Attempt at a Solution
I was able to show that the metric space (C[0,T],ρ) is complete, but I'm having problems finding an integral operator that is a contraction on the space. I've tried the operator
(Tx)(t) = [itex]x_{0}[/itex] + [itex]\int^{t}_{0}f(s,(x(s))ds [/itex]but I was not able to get a contraction. Any help would be much appreciated!
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