Urysohn and Hammerstein operators on Hölder spaces

Abstract

Abstract We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of Hölder 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 (Nyström methods). Under suitable Carathéodory 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 Hölder continuous functions defined on metric spaces.