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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
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] Federico Cafiero, Sui teoremi di unicità relativi ad un’equazione differenziale ordinaria del primo ordine, Giorn. Mat. Battaglini (4) 2(78) (1948), 10–41 (Italian). MR 0028505
  • [2] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [3] L. C. Piccinini, G. Stampacchia, and G. Vidossich, Ordinary differential equations in 𝑅ⁿ, Applied Mathematical Sciences, vol. 39, Springer-Verlag, New York, 1984. Problems and methods; Translated from the Italian by A. LoBello. MR 740539
  • [4] Josef Kalas, Nonuniqueness for the solutions of ordinary differential equations, Czechoslovak Math. J. 29(104) (1979), no. 1, 105–112. MR 518144
  • [5] Walter Rudin, Nonuniqueness and growth in first-order differential equations, Amer. Math. Monthly 89 (1982), no. 4, 241–244. MR 650671, 10.2307/2320223
  • [6] Mansour Samimi, Nonuniqueness criteria for ordinary differential equations, Applicable Anal. 13 (1982), no. 4, 291–296. MR 670663, 10.1080/00036818208839400
  • [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: http://dx.doi.org/10.1090/S0002-9939-1991-1028290-X
Keywords: Ordinary differential equations, initial value problem, uniqueness
Article copyright: © Copyright 1991 American Mathematical Society