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A generalization of the Nagumo uniqueness criterion


Author: Thomas C. Gard
Journal: Proc. Amer. Math. Soc. 70 (1978), 167-172
MSC: Primary 34A10
DOI: https://doi.org/10.1090/S0002-9939-1978-0470288-8
MathSciNet review: 0470288
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Abstract: The classical Nagumo criterion for uniqueness of solutions of the initial value problem $ x' = F(t,x),x({t_0}) = {x_0}$, assumes continuity of F as well as a modulus of continuity condition involving $ 1/t$. Recently, it has been shown that $ 1/t$ can be replaced by $ \alpha /t$, for $ \alpha > 1$, or by $ 1/{t^2}$ provided that the continuity condition on F is suitably modified; in each case it is required that $ F(t,x)$ at least approach a limit at a certain minimal rate as $ (t,x)$ tends to the initial value. In this note we extend both of these results, and in so doing, clarify the relationship between the two types of conditions needed.


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  • [1] S. R. Bernfeld, R. D. Driver and V. Lakshmikantham, Uniqueness for ordinary differential equations, Math. Systems Theory 9 (1976), 359-367. MR 0447673 (56:5983)
  • [2] J. M. Bownds and F. T. Metcalf, An extension of the Nagumo uniqueness theorem, Proc. Amer. Math. Soc. 27 (1971), 313-316. MR 0288331 (44:5529)
  • [3] J. B. Diaz, On existence, uniqueness, and numerical evaluation of solutions of ordinary and hyperbolic differential equations, Ann. Mat. Pura Appl. 52 (1960), 136-181. MR 0120452 (22:11206)
  • [4] J. B. Diaz and W. L. Walter, On uniqueness theorems for ordinary differential equations and for partial differential equations of hyperbolic type, Trans. Amer. Math. Soc. 96 (1960), 90-100. MR 0120451 (22:11205)
  • [5] E. Hille, Lectures on ordinary differential equations, Addison-Wesley, Reading, Mass., 1969. MR 0249698 (40:2939)
  • [6] V. Lakshmikantham and S. Leela, Differential and integral inequalities, Vol. 1, Academic Press, New York, 1969.
  • [7] M. Nagumo, Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung, Japan. J. Math. 3 (1926), 107-112.
  • [8] H. Okamura, Condition necéssaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano, Mem. Coll. Sci. Kyoto Imp. Univ. Ser. A 24 (1942), 21-28. MR 0015590 (7:442g)
  • [9] T. Rogers, On Nagumo's condition, Canad. Math. Bull. 15 (1972), 606-611. MR 0311965 (47:527)
  • [10] W. Walter, Differential and integral inequalities, Springer-Verlag, New York, 1970. MR 0271508 (42:6391)
  • [11] T. Yoshizawa, Stability theory by Liapunov's second method, Math. Soc. Japan, Tokyo, 1966.

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DOI: https://doi.org/10.1090/S0002-9939-1978-0470288-8
Article copyright: © Copyright 1978 American Mathematical Society

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