Fixed points and positive eigenvalues for nonlinear operators (Q2581475)

The authors present some results related to fixed points and eigenvalues. They prove, in particular, a new fixed point theorem for nonexpansive mappings in Banach spaces, which is inspired by the result proved recently in \textit{J.--P.\ Penot} [Proc.\ Am.\ Math.\ Soc.\ 131, 2371--2377 (2003; Zbl 1035.47043)] and by the notion of the scalar derivative. The results related to eigenvalues are based on this new fixed point theorem and on a fixed point theorem given by the authors in [J.\ Math.\ Anal.\ Appl.\ 290, No.~2, 452--468 (2004; Zbl 1066.47056)] as a variant of Altman's fixed point theorem. This paper can be considered as a complementary part of the cited paper.