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I am studying unicity of the solutions of IVP in ordinary differential systems.

In this context, we have the following result for differential equations:

If $f(x,t):R\times R\to R$ is continuous and decreasing in $x$ for every $t$ then the IVPs associated to the equation $x'=f(x,t)$ have unique solutions.

Is there an analog of this result for higher dimensional systems?

Something like:

If $f(X,t):R^n\times R\to R^n$ is continuous and decreasing in norm with respect to each $X_i$ for every $t$ then the IVPs associated to the system $X'=f(X,t)$ have unique solutions.

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Yes, there is an analog: $$ \langle X-Y,F(X,t)-F(Y,t)\rangle\le0\quad\forall X,Y\in\Bbb R^n,\quad\forall t\ge t_0. $$ Here $\langle,\rangle$ is the dot product in $\Bbb R^n$.

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  • $\begingroup$ Sweet! Can you point me to a proof or an intuition of why it is true? $\endgroup$ – Jsevillamol Dec 18 '17 at 16:08
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    $\begingroup$ Same proof as the one dimensional case. If $X(t)$ and $Y(t)$ are two solutions with the same initial value at $t_0$, differentiate $\|X(t)-Y(t)\|^2$ with respect to $t$. $\endgroup$ – Julián Aguirre Dec 18 '17 at 16:25

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