Affiliation:
1. Institut für Mathematik , Universität Klagenfurt , Universitätsstraße 65–67, 9020 Klagenfurt , Austria
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.
Subject
Applied Mathematics,Numerical Analysis,Analysis
Reference52 articles.
1. R. A. Adams and J. J. F. Fournier,
Sobolev Spaces, 2nd ed.
Pure Appl. Math. (Amsterdam) 140,
Elsevier/Academic, Amsterdam, 2002.
2. R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ,
Measures of Noncompactness and Condensing Operators,
Oper. Theory Adv. Appl. 55,
Birkhäuser, Basel, 1992.
3. H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,
SIAM Rev. 18 (1976), no. 4, 620–709.
4. J. Appell,
Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator,
J. Math. Anal. Appl. 83 (1981), no. 1, 251–263.
5. J. Appell, N. Guanda, N. Merentes and J. L. Sanchez,
Boundedness and continuity properties of nonlinear composition operators: A survey,
Commun. Appl. Anal. 15 (2011), no. 2–4, 153–182.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献