 Implicit midpoint Lipschitz estimate

For a uniformly Lipschitz continuous state field $f$ with constant $L$, an implicit midpoint rule step is uniquely solvable when $hL/2<1$, by the contraction mapping theorem. Write its step map as ${\Psi }_h(u)=u+h{\Phi }_h(u)$, where ${\Phi }_h(u)=f(t+h/2,u+h{\Phi }_h(u)/2)$. Subtracting the increment equations givesTherefore $\mathrm{Lip}({\Phi }_h)\le L/(1-hL/2)$, and the triangle inequality gives the bound for ${\Psi }_h$. Combining it with the implicit midpoint consistency bound yieldsfor $L>0$. This follows by summing the geometric error recurrence with forcing $K{h}^3/(1-hL/2)$. For $hL\le 1$ and $nh\le T$, $R_h^n\le e^{2LT}$, proving second-order global error with exact starting data. If $L=0$, the direct bound is $\Vert e_n\Vert \le \Vert e_0\Vert +KT{h}^2$.