We define classes of mappings of monotone type with respect to a given direct sum decomposition of the underlying Hilbert spaceH. 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

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... d be useful in the study of semilinear equations, when the linear part has an infinite-dimensional kernel.