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 of continuous functions and the Lebesgue space for . In particular, it is shown that a local Lipschitz condition for the derivative in the space for leads to a degeneracy of the corresponding kernels. For ordinary integral operators, such a degeneracy occurs for only.
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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