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- W2005224823 abstract "Introduction. In 1842 Cauchy proved an existence theorem for analytic nonlinear differential equations of the second order, which was extended in 1875 by S. Kowalewski to general analytic nonlinear systems of differential equations and became known as the Cauchy-Kowalewski theorem. In his proof [1; 6] Cauchy uses the method of majorants (which he calls Calcul de limites). This method consists of (a) inserting an analytic expansion for the solution into the differential equation and thus obtaining recursive formulas for the coefficients of the expansion, and (b) estimating these coefficients and thus proving that the formal solution is a genuine one. The essence of the method lies in the fact that a direct estimation can be avoided. Indeed, Cauchy constructs a simple majorant problem which can be solved explicitly and its solution establishes the desired estimates. In 1874 S. Kowalewski, apparently unaware of the work of Cauchy, extended (in her thesis) his results to general nonlinear analytic systems. Her work, which appeared in 1875 [18] contains the result of Darboux [7] (the same year) who has also extended the work of Cauchy. The proof as it is known today is a simplification of the proof of Kowalewski and it is due to Goursat [15] (see also [16; p. 360]). All the above proofs are based on the very same ideas of Cauchy, mentioned above. A method based on direct estimates of the coefficients was given by P. Lax [19]; see ?3. Some work has been done in the direction of extending the C-K (CauchyKowalewski) theorem to nonanalytic equations. For the system" @default.
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- W2005224823 date "1961-01-01" @default.
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- W2005224823 title "A new proof and generalizations of the Cauchy-Kowalewski theorem" @default.
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