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

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. ewline Let $f={f_q}:E o E$ be an endomorphism of degree zero of graded vector space $E={E_q}$. Let $ ilde{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 $$Lambda(f)=sum_{q geq 0}(-1)^qmbox{Tr}(f_q),$$ where $mbox{Tr}(f)=mbox{tr}( ilde{f})$ is the generalized trace of $f$, ``tr'' is the ordinary trace and $f: ilde{E} o ilde{E}$. ewline Let $X$ be a Hausdorff topological space and $phi:X o X$ 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 $q_star p_star^{-1}:H(X) o H(X)$ 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: egin{theorem*} Let $X$ be a Hausdorff topological space and $F: X o X$ be a general or general approximative (compact) absorbing contraction. Then $mathbf{Lambda}(F)$ is well defined and if $mathbf{Lambda}(F) eq {0}$ then $F$ has a fixed point. end{theorem*}