Journal of Inequalities and Applications
Volume 2005 (2005), Issue 3, Pages 221-234

On the domain of the implicit function and applications

Marco Papi1,2

1Istituto per le Applicazioni del Calcolo “Mauro Picone”, viale del Policlinico 137, Roma 00161, Italy
2Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, viadella Ricerca Scientifica, Roma 00133, Italy

Received 31 July 2003; Revised 22 December 2004

Copyright © 2005 Marco Papi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequalities. Also in this case, we state an estimate for the size of the domain. An application to the local Lipschitz behavior of solution maps is discussed.