A uniqueness theorem for 𝑦’=𝑓(𝑥,𝑦),𝑦(𝑥₀)=𝑦₀

Majorana, Armando

doi:10.1090/S0002-9939-1991-1028290-X

A uniqueness theorem for $y’=f(x,y),\;y(x_ 0)=y_ 0$
HTML articles powered by AMS MathViewer

by Armando Majorana PDF

Proc. Amer. Math. Soc. 111 (1991), 215-220 Request permission

Abstract:

Consider the initial value problem for a first-order differential equation \[ y’ = f(x,y),\quad y({x_0}) = {y_0}.\] In this paper a new uniqueness criterion is proved. This criterion is related to the numeric equation \[ u = {y_0} + (t - {x_0})f(t,u).\] It is also shown that some well-known uniqueness theorems are consequences of our result.

References

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 28505
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
L. C. Piccinini, G. Stampacchia, and G. Vidossich, Ordinary differential equations in $\textbf {R}^{n}$, Applied Mathematical Sciences, vol. 39, Springer-Verlag, New York, 1984. Problems and methods; Translated from the Italian by A. LoBello. MR 740539, DOI 10.1007/978-1-4612-5188-0
Josef Kalas, Nonuniqueness for the solutions of ordinary differential equations, Czechoslovak Math. J. 29(104) (1979), no. 1, 105–112. MR 518144, DOI 10.21136/CMJ.1979.101582
Walter Rudin, Nonuniqueness and growth in first-order differential equations, Amer. Math. Monthly 89 (1982), no. 4, 241–244. MR 650671, DOI 10.2307/2320223
Mansour Samimi, Nonuniqueness criteria for ordinary differential equations, Applicable Anal. 13 (1982), no. 4, 291–296. MR 670663, DOI 10.1080/00036818208839400

Differential and integral inequalities : theory and application