On restricted uniqueness for systems of ordinary differential equations

Bownds, J. M.; Díaz, J. B.

doi:10.1090/S0002-9939-1973-0304739-1

On restricted uniqueness for systems of ordinary differential equations
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by J. M. Bownds and J. B. Díaz

Proc. Amer. Math. Soc. 37 (1973), 100-104

DOI: https://doi.org/10.1090/S0002-9939-1973-0304739-1

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Abstract:

A uniqueness theorem is proved, for not necessarily Lipschitzian systems of ordinary differential equations, $y’ = f$. This theorem compares with one of Okamura and Murakami, in that, here, at the expense of assuming a certain additional smoothness for f on open sets, no assumption is made regarding the existence of an auxiliary positive definite (Lyapunov) function. An example compares the relative applicability of the two theorems.

References

Hirosi Okamura, Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano, Mem. Coll. Sci. Kyoto Imp. Univ. Ser. A. 24 (1942), 21–28 (French). MR 15590
Haruo Murakami, On non-linear ordinary and evolution equations, Funkcial. Ekvac. 9 (1966), 151–162. MR 209617

Differential and integral inequalities

Fred Brauer and Shlomo Sternberg, Local uniqueness, existence in the large, and the convergence of successive aproximations, Amer. J. Math. 80 (1958), 421–430. MR 95303, DOI 10.2307/2372792
Felix E. Browder, Non-linear equations of evolution, Ann. of Math. (2) 80 (1964), 485–523. MR 173960, DOI 10.2307/1970660
Tosio Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan 5 (1953), 208–234. MR 58861, DOI 10.2969/jmsj/00520208
John M. Bownds, A uniqueness theorem for $y^{\prime } =f(x,\,y)$ using a certain factorization of $f$, J. Differential Equations 7 (1970), 227–231. MR 254305, DOI 10.1016/0022-0396(70)90107-5
A. W. Hales and N. J. Fine, Problems and Solutions: Solutions of Elementary Problems: E1784, Amer. Math. Monthly 73 (1966), no. 6, 672. MR 1533869
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 171038