On the Lefschetz fixed point theorem. (English) Zbl 1006.55002

The abstract version of the Lefschetz fixed point theorem is proved which states the following: If \(X\) is a metric space and if \(X_0\subset X\) is such that \(X_0\) absorbs compact sets, \(f\:X\to X\) is a continuous map, \(f(X_0) \subset X_0\) and \(f/X_0\) is a Lefschetz map, then \(f\) is also (on \(X\)) a Lefschetz map. Here a continuous map \(f\:X\to X\) is called a Lefschetz map if the generalized Lefschetz number \(\Lambda (f)\) of \(f\) is well defined and \(\Lambda (f) \neq 0\) implies that the set Fix\((f)\neq 0\). From that theorem three relative versions of the Lefschetz fixed point theorem are deduced which are connected with condensing and \(k\)-set contraction mappings.


55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47H11 Degree theory for nonlinear operators
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