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Approximation methods for solving the Cauchy problem

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 Pica…