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$\boldsymbol{C}^{r}$ convergence of Picard's successive approximations

Author(s): Alexander J. Izzo
Journal: Proc. Amer. Math. Soc. 127 (1999), 2059-2063.
MSC (1991): Primary 34A12, 34A45
Posted: February 26, 1999
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Abstract | References | Similar articles | Additional information

Abstract: A simple, elementary proof of the existence, uniqueness, and
smoothness of solutions to ordinary differential equations is given. In fact, it is shown that for a differential equation of class $C^{r}$ , the successive approximations of Picard converge in the $C^{r}$ -sense.

References:

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[D]: J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 50:1782
[H-S]: M. W. Hirsh and S. Smale, Differential equations, dynamical systems, and linear algebra, Academic Press, New York, 1974. MR 58:6484
[L1]: S. Lang, Analysis II, Addison-Wesley, Menlo Park, California, 1969.
[L2]: -, Differentiable manifolds, Springer-Verlag, New York, 1985.
[S]: M. Spivak, A comprehensive introduction to differential geometry, 2nd ed., vol. 1, Publish or Perish, Wilmington, Delaware, 1979. MR 42:2369


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