Urysohn and Hammerstein operators on H"older spaces

TL;DR

This paper develops a unified, application-oriented framework for analyzing Urysohn and Hammerstein integral operators on H"older spaces, establishing their properties and differentiability under broad conditions.

Contribution

It introduces new differentiability results for Urysohn operators and provides a comprehensive analysis of Nemytskii operators within a flexible measure-theoretic setting.

Findings

Established well-definedness and boundedness of operators

Proved continuous differentiability under Carathéodory conditions

Unified treatment of Urysohn and Hammerstein operators

Abstract

We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of H"older continuous functions over compact metric spaces. These nonlinear mappings are formulated by means of an abstract measure theoretical integral involving a finite measure. This flexible setting creates a common framework to tackle both such operators based on the Lebesgue integral like frequently met in applications, as well as e.g.\ their spatial discretization using stable quadrature/cubature rules (Nystr"om methods). Under suitable Carath{\'e}odory conditions on the kernel functions, properties like well-definedness, boundedness, (complete) continuity and continuous differentiability are established. Furthermore, the special case of Hammerstein operators is understood as composition of Fredholm and Nemytskii operators. While our differentiability results for Urysohn operators appear to be new, the section on Nemytskii operators has a survey character. Finally, an appendix provides a rather comprehensive account summarizing the required preliminaries for H\"older continuous functions defined on metric spaces.