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 (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.
Highlights
We will introduce classes of mappings of monotone type with respect to a given projection P : H → E, where E is a closed linear subspace of a real Hilbert space H
We will give the main properties of the new classes of mappings of monotone type: (S+)T, T-pseudomonotone, T-quasimonotone and (M)T
It turns out that in certain cases we can apply the degree theory, provided the nonlinearity is of class (S+)T with respect to a suitable operator T
Summary
We will introduce classes of mappings of monotone type with respect to a given projection P : H → E, where E is a closed linear subspace of a real Hilbert space H. The main result of this note is the construction of a topological degree for mappings of the form. We will give the main properties of the new classes of mappings of monotone type: (S+)T , T-pseudomonotone, T-quasimonotone and (M)T. We show their relations to each other and to the traditional classes as well. It turns out that in certain cases we can apply the degree theory, provided the nonlinearity is of class (S+)T with respect to a suitable operator T
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