Abstract and Applied Analysis
Volume 2005 (2005), Issue 6, Pages 581-597

An extension of the topological degree in Hilbert space

J. Berkovits1 and C. Fabry2

1Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
2Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, Chemin du Cyclotron 2, Louvain-la-Neuve 1348, Belgium

Received 16 June 2004

Copyright © 2005 J. Berkovits and C. Fabry. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We define classes of mappings of monotone type with respect to a given direct sum decomposition of the underlying Hilbert space H. The new classes are extensions of classes of mappings of monotone type familiar in the study of partial differential equations, for example, the class (S+) and the class of pseudomonotone mappings. We then construct an extension of the Leray-Schauder degree for mappings involving the above classes. As shown by (semi-abstract) examples, this extension of the degree should be useful in the study of semilinear equations, when the linear part has an infinite-dimensional kernel.