JournalszaaVol. 15, No. 2pp. 397–418

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

  • Jürgen Appell

    Universität Würzburg, Germany
  • Espedito De Pascale

    Università della Calabria, Arcavacata di Rende, Italy
  • A.S. Kalitvin

    Pedagogical Institute, Lipetsk, Russian Federation
  • P. P. Zabrejko

    The Academy of Sciences of Belarus, Minsk, Belarus
On the Application of the Newton–Kantorovich Method to Nonlinear Partial Integral Equations cover

Abstract

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 CC of continuous functions and the Lebesgue space LpL_p for 1p1 ≤ p ≤ \infty. In particular, it is shown that a local Lipschitz condition for the derivative in the space LpL_p for p<\inftlyp < \inftly leads to a degeneracy of the corresponding kernels. For ordinary integral operators, such a degeneracy occurs for p2p ≤ 2 only.

Cite this article

Jürgen Appell, Espedito De Pascale, A.S. Kalitvin, P. P. Zabrejko, On the Application of the Newton–Kantorovich Method to Nonlinear Partial Integral Equations. Z. Anal. Anwend. 15 (1996), no. 2, pp. 397–418

DOI 10.4171/ZAA/707