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A Nagumo-like uniqueness result for a second order ODE

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Abstract

In this note, we extend to second order nonlinear ordinary differential equations (ODEs) a recent generalization of Nagumo’s uniqueness theorem.

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References

  1. Athanassov Z.S.: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jpn. 35, 351–367 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Bertozzi A., Majda A.: Vorticity and incompressible flow. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  3. Chemin J.Y., Lerner N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Equ. 121, 314–328 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantin A.: On the unicity of solutions for high-order differential equations. Istituto Lombardo (Rend. Sci.) A 130, 171–181 (1996)

    MathSciNet  Google Scholar 

  5. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin A.: On Nagumo’s theorem. Proc. Jpn. Acad. 86(A), 41–44 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Constantin A.: A dynamical systems approach towards isolated vorticity regions for tsunami background states. Arch. Rat. Mech. Anal. 200, 239–253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kaper H.G., Kwong M.K.: Uniqueness for a class of nonlinear initial value problems. J. Math. Anal. Appl. 130, 467–473 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mejstrik, T.: Some remarks on Nagumo’s theorem. Czechoslov. Math. J. (to appear)

  10. Mustafa, O.G.: On the uniqueness of flow in a recent tsunami model, Applicable Analysis, accepted. http://arxiv.org/abs/1103.2470

  11. Nagumo, M.: Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung. Jpn. J. Math. 3, 107–112 (1926) Reprinted In: Mitio Nagumo Collected Papers, Eds. M. Yamaguti, L. Nirenberg, S. Mizohata, Y. Sibuya, Springer, Tokyo, 1993

  12. Nagumo, M.: Eine hinreichende Bedingung für die Unität der Lösung von gewöhnlichen Differentialgleichungen n-ter Ordnung, Japan J. Math. 4, 307–309 (1927) Reprinted In: Mitio Nagumo Collected Papers, Eds. M. Yamaguti, L. Nirenberg, S. Mizohata, Y. Sibuya, Springer, Tokyo, 1993

  13. Strauss, W.: Nonlinear wave equations, CBMS Regional Conference Series in Mathematics. Am. Math. Soc. Providence (1989)

  14. Ni W.M., Nussbaum R.: Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u, r) = 0. Commun. Pure Appl. Math. 38, 67–108 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peletier L.A., Serrin J.: Uniqueness of positive solutions of semilinear equations in \({\mathbb{R}^{n}}\) . Arch. Rat. Mech. Anal. 81, 181–197 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wahlén E.: Uniqueness for autonomous planar differential equations and the Lagrangian formulation of water flows with vorticity. J. Nonlinear Math. Phys. 11, 549–555 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wintner A.: On the local uniqueness of the initial value problem of the differential equation \({\frac{d^{n} x}{d^{n}t} = f(t,x)}\) . Boll. Un. Mat. Ital. 11, 496–498 (1956)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Octavian G. Mustafa.

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Communicated by Adrian Constantin.

Octavian G. Mustafa is on leave from the University of Craiova, Faculty of Mathematics and Computer Science, Al. I. Cuza 13, Craiova, Romania.

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Mustafa, O.G. A Nagumo-like uniqueness result for a second order ODE. Monatsh Math 168, 273–277 (2012). https://doi.org/10.1007/s00605-011-0324-2

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