On the implicit function theorem for Lipschitz mappings

New estimations for Lipschitz constant of solutions in the Clarke's implicit function theorem are proved.
Let $U=B_{x_{0}}^{n}(r_{1})\subset \mathbf{\mathbf{R}}^{n},\;V=B_{y_{0}}^{m}(r_{2})\subset \mathbf{R}^{m}$ and $F:U\times V\rightarrow \mathbf{R}^{m}$ be a local Lipschitz mapping in some neighbourhood of point $(x_{0},y_{0})$. Let $\partial_{y}F(x_{0},y_{0})$ is of maximal rank. Then for every $\Delta ^{\ast },\;\Delta <\Delta ^{\ast },$ there exist $R,\;0<R\leq \min \{r_{1},r_{2}\}$, and a unique Lipschitz mapping $G:B_{x_{0}}^{n}(R)\rightarrow B_{y_{0}}^{m}(\Omega R)$ such that
\begin{equation*} G(x_{0})=y_{0},\quad F(x,G(x))=F(x_{0},y_{0}),\;x\in B_{x_{0}}^{n}(R), \end{equation*}
and
\begin{equation*} \left\vert G(x_{2})-G(x_{1})\right\vert \leq \Delta ^{\ast}|x_{2}-x_{1}|,\;x_{2}, x_{1}\in B_{x_{0}}^{n}(R). \end{equation*}

Moreover, we have $\lim\limits_{r\rightarrow 0+}Lip(G,B_{x_{0}}^{n}(r))\leq\Delta.$