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2 | Existance -> Existence; formatting, grammar | ||
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Picard Existance Existence Theorem: Consider $z'(t)=f(t,z)$. Define $\mathbf{F}y:=y_0+\int_{0}^tf(x,y(t))dx$. \mathbf{F}y:=y_0+\int_0^t f(x,y(t))dx$. It sure would be nice if $\mathbf{F}$ has had a fixed point, so use Banach fixed point theorem to show that it has a fixed point. That would require $f$ to be a contraction, so sprinkle a hint of Lipschitz on $f$. |
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1 | [made Community Wiki] | ||
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Picard Existance Theorem: Consider $z'(t)=f(t,z)$. Define $\mathbf{F}y:=y_0+\int_{0}^tf(x,y(t))dx$. It sure would be nice if $\mathbf{F}$ has a fixed point, so use Banach fixed point theorem to show that it has a fixed point. That would require $f$ to be a contraction so sprinkle a hint of Lipschitz on $f$. |
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