The Lefschetz fixed point theory for morphisms in topological vector spaces. (English) Zbl 1031.55002

Górniewicz, Lech; Rozpłoch-Nowakowska, Danuta

Topol. Methods Nonlinear Anal. 20, No. 2, 315-333 (2002).

Topological fixed point theory is one of the most useful and powerful techniques in nonlinear analysis. Motivated by the Lefschetz fixed point theory for compact mappings on metric ANRs, the authors extend the theory to a larger class of mappings for more general spaces that are not necessarily metric. A topological vector space \(E\) is admissible in the sense of Klee if for any compact subset \(K\subset E\) and for any open covering \(\alpha\) of \(K\) (in \(E\)), there exists a map \(\pi_{\alpha}:K\to E\) such that (i) \(\pi_{\alpha}(K) \subset E_K\) for some vector subspace \(E_K\) with \(\dim E_K <\infty\); and (ii) for any \(x\in K\), there exists \(U_x \in \alpha\) such that \(x\in U_x\) and \(\pi_{\alpha}(x)\in U_x\). Using the general notion of morphisms as introduced by L. Gorniewicz and A. Granas in [J. Math. Pures Appl., IX. Ser. 60, 361-373 (1981; Zbl 0482.55002)] (every such morphism determines a multivalued mapping), the authors define homological invariants for certain morphisms defined on admissible spaces. In particular, a Lefschetz type trace \(\Lambda (\varphi)\) is defined and a fixed point theorem is proven for a certain class of morphisms (compact absorbing contractions) \(\varphi\). Moreover, an existence theorem for periodic points and its relative version are given for these general settings, extending the corresponding results obtained by J. Andres, L. Gorniewicz and J. Jezierski in [Topology Appl. 127, 447-472 (2003; Zbl 1034.34014)]. A relative Lefschetz type fixed point theorem, similar to that of C. Bowszyc [Bull. Acad. Pol. Sci. 16, 845-850 (1968; Zbl 0177.51702)] is also presented.

Reviewer: Peter N.-S.Wong (Lewiston)

MSC:

55M20	Fixed points and coincidences in algebraic topology
54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

Lefschetz number; fixed points; periodic points