ODELC1

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Show that {\begin{bmatrix}1+x_{1}\\x_{2}^{2}\\\end{bmatrix}} satisfies a Lipschitz condition when x lies in any bounded domain D (i.e. |x|<M where M is constant), but cannot satisfy a Lipschitz condition for all x.


First find f({\vec  x},t)-f({\vec  y},t)={\begin{bmatrix}x_{1}-y_{1}\\x_{2}^{2}-y_{2}^{2}\\\end{bmatrix}} and {\vec  x}-{\vec  y}={\begin{bmatrix}x_{1}-y_{1}\\x_{2}-y_{2}\\\end{bmatrix}}


{\frac  {\partial f}{\partial x}}={\begin{bmatrix}1&0\\0&2x_{2}\\\end{bmatrix}},\left|{\frac  {\partial f}{\partial x}}\right|=1+2x_{2}\,


If x is in a bounded domain |x|<M then f satisfies a Lipschitz condition with constant L=\max _{{x_{2}\in D}}(1+2x_{2})=1+2M\,. If the domain is not bounded then the max will not be bounded and so f will not be Lipschitz.


Main Page : Ordinary Differential Equations : Lipshitz Conditions