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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 2, 1996, pp. 397–418
DOI: 10.4171/ZAA/707

Published online: 1996-06-30

On the Application of the Newton–Kantorovich Method to Nonlinear Partial Integral Equations

Jürgen Appell[1], Espedito De Pascale[2], A.S. Kalitvin[3] and P. P. Zabrejko[4]

(1) Universität Würzburg, Germany
(2) Università della Calabria, Arcavacata di Rende, Italy
(3) Pedagogical Institute, Lipetsk, Russian Federation
(4) The Academy of Sciences of Belarus, Minsk, Belarus

We discuss the applicability of the Newton–Kantorovich method to a nonlinear equation which contains partial integrals with Uryson type kernels. A basic ingredient of this method consists in verifying a local Lipschitz condition for the Fréchet derivatives of the nonlinear partial integral operators generated by such kernels. The abstract results are illustrated in the space $C$ of continuous functions and the Lebesgue space $L_p$ for $1 ≤ p ≤ \infty$. In particular, it is shown that a local Lipschitz condition for the derivative in the space $L_p$ for $p < \inftly$ leads to a degeneracy of the corresponding kernels. For ordinary integral operators, such a degeneracy occurs for $p ≤ 2$ only.

Keywords: Newton–Kantorovich method, nonlinear Uryson equations, partial integral operators, Chebyshev spaces, Lebesgue spaces, ideal spaces

Appell Jürgen, De Pascale Espedito, Kalitvin A.S., Zabrejko P. P.: On the Application of the Newton–Kantorovich Method to Nonlinear Partial Integral Equations. Z. Anal. Anwend. 15 (1996), 397-418. doi: 10.4171/ZAA/707