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Zbl 1081.34009
Mortici, Cristinel
Approximation methods for solving the Cauchy problem. (English)
[J] Czech. Math. J. 55, No. 3, 709-718 (2005). ISSN 0011-4642; ISSN 1572-9141/e

Summary: We give some new results concerning solvability of the 1-dimensional differential equation $y' = f(x,y)$ 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's theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.

MSC 2000:: *34A12 Initial value problems for ODE
34A34 Nonlinear ODE and systems, general
34A45 Theoretical approximation of solutions of ODE

Keywords: Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle

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