Abstract
In this paper 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 theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.
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Mortici, C. Approximation Methods for Solving the Cauchy Problem. Czech Math J 55, 709–718 (2005). https://doi.org/10.1007/s10587-005-0058-1
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DOI: https://doi.org/10.1007/s10587-005-0058-1