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

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 $C$ of continuous functions and the Lebesgue space $L_p$ for $1\le p\le \infty$. In particular, it is shown that a local Lipschitz condition for the derivative in the space $L_p$ for $p<\infty$ leads to a degeneracy of the corresponding kernels. For ordinary integral operators, such a degeneracy occurs for $p\le 2$ only.