Conditions for unicity of system solutions of a non locally lipschitz IVPs 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.

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 $\Bbb R^n$.