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Keywords:
Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle
Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle
Summary:
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.
References:
[1] V. Barbu: Ecuatii Diferentiale. Ed. Junimea, Iasi, 1985.
[2] H. Brézis: Analyse Fonctionnelle. Théorie et applications. Masson, Paris, 1983. MR 0697382
[3] A. Halanay: Ecuatii Diferentiale. Ed. Did. si Ped., Bucuresti, 1972. MR 0355142 | Zbl 0258.34001
[4] Gh. Morosanu: Ecuatii Diferentiale. Aplicatii. Ed. Academiei, Bucuresti, 1989. MR 1031994
[5] L. Pontriaguine: Equations Différentielles Ordinaires. Mir, Moscow, 1969. MR 0261056 | Zbl 0185.15701
[6] S. Sburlan, L. Barbu and C. Mortici: Ecuatii Diferentiale, Integrale si Sisteme Dinamice. Ex Ponto, Constanta, 1999. MR 1734289
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