An extension of the topological degree in Hilbert space

Abstract

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 $\left(S_{+}\right)$ 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.