Geodesic
Approximation methods for solving the Cauchy problem
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 709-718.

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In this paper we give some new results concerning solvability of the 1-dimensional differential equation $y^{\prime } = f(x,y)$ with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if $f$ is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.
Classification : 34A12, 34A34, 34A45, 47H10, 47N20
Keywords: Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle 
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     author = {Mortici, Cristinel},
     title = {Approximation methods for solving the {Cauchy} problem},
     journal = {Czechoslovak Mathematical Journal},
     pages = {709--718},
     publisher = {mathdoc},
     volume = {55},
     number = {3},
     year = {2005},
     mrnumber = {2153095},
     zbl = {1081.34009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a12/}
}
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Mortici, Cristinel. Approximation methods for solving the Cauchy problem. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 709-718. http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a12/