On a criterion of properness of multimaps
Dmitry Vorotnikov
TL;DR
This paper establishes a general criterion for the properness of multimaps, linking it to closedness, with implications for topological degree theories and extending properties known for Fredholm maps.
Contribution
It introduces a new criterion for properness of multimaps, providing both necessary and sufficient conditions, inspired by properties of Fredholm maps.
Findings
Provides a general criterion for properness of multimaps.
Establishes direct and converse theorems relating properness and closedness.
Enhances understanding of topological degree theories for multimaps.
Abstract
The close relation between properness and closedness of maps is well-known. For instance, for Fredholm mappings of infinite dimensional Banach manifolds, these properties are equivalent. On the other hand, properness of maps plays an important role for construction of various topological degree theories. In this work we give a general criterion of properies of multimaps, congenial to the just mentioned property of Fredholm maps. The criterion is formulated in the form of two theorems: direct and converse.
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Taxonomy
TopicsNonlinear Differential Equations Analysis · Fixed Point Theorems Analysis · Optimization and Variational Analysis