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A uniqueness theorem for $ y'=f(x,y),\;y(x\sb 0)=y\sb 0$


Author: Armando Majorana
Journal: Proc. Amer. Math. Soc. 111 (1991), 215-220
MSC: Primary 34A12
DOI: https://doi.org/10.1090/S0002-9939-1991-1028290-X
MathSciNet review: 1028290
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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the initial value problem for a first-order differential equation

$\displaystyle y' = f(x,y),\quad y({x_0}) = {y_0}.$

In this paper a new uniqueness criterion is proved. This criterion is related to the numeric equation

$\displaystyle u = {y_0} + (t - {x_0})f(t,u).$

It is also shown that some well-known uniqueness theorems are consequences of our result.

References [Enhancements On Off] (What's this?)

  • [1] F. Cafiero, Sui teoremi d'unicita relativi ad un'equazione differenziale ordinaria del primo ordine, Giorn. Mat. Battaglini 78 (1948), 10-41, 193-215. MR 0028505 (10:457a)
  • [2] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [3] L. Piccinini, G. Stampacchia, and G. Vidossich, Ordinary differential equations in $ {{\mathbf{R}}^n}$: problems and methods, Springer-Verlag, New York, 1984. MR 740539 (85b:34002)
  • [4] J. Kalas, Nonuniqueness for the solutions of ordinary differential equations, Czechoslovak Math. J. 29 (1979), 105-112. MR 518144 (80c:34004)
  • [5] W. Rudin, Nonuniqueness and growth in the first-order differential equations, Amer. Math. Monthly 89 (1982), 241-244. MR 650671 (83f:34009)
  • [6] M. Samimi, Nonuniqueness criteria for ordinary differential equations, Appl. Anal. 13 (1982), 291-296. MR 670663 (84d:34003)
  • [7] V. Lakshmikantham and S. Leela, Differential and integral inequalities : theory and application, Academic Press, New York, 1969.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1028290-X
Keywords: Ordinary differential equations, initial value problem, uniqueness
Article copyright: © Copyright 1991 American Mathematical Society

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