Quote:"... a map $h$ of an open set $\Theta$ of a Bnach space $B_1$ into a Banach space $B_2$ is Lipschitz in $\Theta$ if there exists $k\in \mathbb{R}$ such that $||h(a)-h(b)||\leq k||a-b||$ for $(a,b)\in\Theta\times \Theta$".
"... a map $h$ of an open set $\Theta$ of a Bnach space $B_1$ into a Banach space $B_2$ is Lipschitz in $\Theta$ if there exists $k\in \mathbb{R}$ such that $||h(a)-h(b)||\leq k||a-b||$ for $(a,b)\in\Theta\times \Theta$".
Quote:"0.33 The Cauchy Theorem. The differential equation $$(*)~~~~~~~~~~~~ x'=f(t,x),x(t_0)=x_0$$ has a unique continuous solution $x(t)$ if $f(t,x)$ is Lipschit in $x$ on $I\times \Omega$."
"0.33 The Cauchy Theorem. The differential equation $$(*)~~~~~~~~~~~~ x'=f(t,x),x(t_0)=x_0$$ has a unique continuous solution $x(t)$ if $f(t,x)$ is Lipschit in $x$ on $I\times \Omega$."
Quote:"0.34 Remark. If $f(t,x)$ is not Lipschitz, we cannot say anything in general. But if Banach space is $\mathbb{R}^n$, then there is a solution of (*), but it may not be unique."
"0.34 Remark. If $f(t,x)$ is not Lipschitz, we cannot say anything in general. But if Banach space is $\mathbb{R}^n$, then there is a solution of (*), but it may not be unique."
This made me thought about GR, with black hole, the curvature went to infinity, thus the function could not be Lipschitz. Does this imply that the dynamics around black hole was not unique?
