# MR3269340 Reviewed O'Regan, Donal Lefschetz type theorems for a class of noncompact mappings. J. Nonlinear Sci. Appl. 7 (2014), no. 5, 288–295. (Reviewer: Calogero Vetro) 47H10

Risultato della ricerca: Review article

### Abstract

Lefschetz fixed-point theorem furnishes a way for counting the fixed points of a suitable mapping. In particular, the Lefschetz fixed-point theorem states that if Lefschetz number is not zero, then the involved mapping has at least one fixed point, that is, there exists a point that does not change upon application of mapping.\newlineLet $f=\{f_q\}:E \to E$ be an endomorphism of degree zero of graded vector space $E=\{E_q\}$. Let $\tilde{E}=E \setminus \{x \in E : f^n(x)=0, \mbox{ for some }n \in \mathbb{N}\}$. Define the generalized Lefschetz number $\Lambda(f)$ by $$\Lambda(f)=\sum_{q \geq 0}(-1)^q\mbox{Tr}(f_q),$$ where $\mbox{Tr}(f)=\mbox{tr}(\tilde{f})$ is the generalized trace of $f$, tr'' is the ordinary trace and $f:\tilde{E} \to \tilde{E}$.\newlineLet $X$ be a Hausdorff topological space and $\phi:X \to X$ be such that, for each selected pair $(p,q)$ of $\phi$ with $\phi(x)=q(p^{-1}(x))$ for $x \in X$, the linear map $q_\star p_\star^{-1}:H(X) \to H(X)$ is a Leray endomorphism, where $H$ is the \u{C}ech homology functor with compact carries and coefficients in the field of rational numbers $K$ from the category of Hausdorff topological spaces and continuous mappings to the category of graded vector spaces and linear mappings of degree zero. Define the Lefschetz set by $$\mathbf{\Lambda}(\phi)=\{\Lambda(q_\star p^{-1}_\star): \phi=q(p^{-1})\}.$$ The author gives four Lefschetz type fixed-point theorems for mappings which are general or general approximative (compact) absorbing contractions and for extension spaces of certain types. Precisely, Lefschetz type fixed-point theorems of O'Regan have the following form:\begin{theorem*}Let $X$ be a Hausdorff topological space and $F: X \to X$ be a general or general approximative (compact) absorbing contraction. Then $\mathbf{\Lambda}(F)$ is well defined and if $\mathbf{\Lambda}(F) \neq \{0\}$ then $F$ has a fixed point.\end{theorem*}
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