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Some remarks on Nagumo’s theorem

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Abstract

We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x′ = f(t, x), x(0) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.

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  1. University of Vienna, Vienna, Austria

    Thomas Mejstrik

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  1. Thomas Mejstrik
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Correspondence to Thomas Mejstrik.

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Cordially dedicated to Prof.Adrian Constantin

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Mejstrik, T. Some remarks on Nagumo’s theorem. Czech Math J 62, 235–242 (2012). https://doi.org/10.1007/s10587-012-0008-7

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  • DOI: https://doi.org/10.1007/s10587-012-0008-7

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