Abstract
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.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Papi, M. On the domain of the implicit function and applications. J Inequal Appl 2005, 373250 (2005). https://doi.org/10.1155/JIA.2005.221
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DOI: https://doi.org/10.1155/JIA.2005.221