On the implicit function theorem

I. V. Zhuravlev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2016), pp. 66-74.

Résumé

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

Détail

Keywords: the implicit function theorem, the inverse function theorem, Clarke derivative, Lipschitz mappings, Lipschitz constant.

Bibliographie

[1] F. Klark, Optimization and Nonsmooth Analysis, Nauka Publ., M., 1988, 280 pp.

[2] A.\;N. Kolmogorov, S.\;V. Fomin, Elements of the Function Theory and Functional Analysis, Nauka Publ., M., 1976, 543 pp.

[3] L.\;A. Lyusternik, V.\;N. Sobolev, Elements of Functional Analysis, Nauka Publ., M., 1965, 520 pp.

[4] F. Clarke, “On the inverse function theorem”, Pac. J. Math., 64:1 (1976), 97–102 | DOI | MR | Zbl