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Article Dans Une Revue Journal of Mathematical Analysis and Applications Année : 2018

The Poincaré-Miranda theorem and Viability condition

Résumé

The aim of this note is to discuss the relation between the assumptions of the Poincaré-Miranda theorem and the viability condition, first used by Nagumo to prove existence of a solution to ODEs under state constraints (viable solutions). An interesting consequence of this observation is an extension of the Poincaré-Miranda theorem to arbitrary convex compact sets in locally convex Hausdorff vector spaces (instead of a parallelotope in an Euclidean space). We also recall a very short proof of this extension based on a Ky Fan inequality. This proof is not new, but seems to have passed unnoticed in the literature devoted to the Poincaré-Miranda theorem. In fact, recent variations of this theorem in 2 follow then in a simple straightforward way. The above extension also implies a generalization of the Lax intermediate value theorem to infinite dimensional spaces.
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Dates et versions

hal-02126114 , version 1 (10-05-2019)

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Hélène Frankowska. The Poincaré-Miranda theorem and Viability condition. Journal of Mathematical Analysis and Applications, 2018, 463 (2), pp.832-837. ⟨10.1016/j.jmaa.2018.03.047⟩. ⟨hal-02126114⟩
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