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Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. (English) Zbl 1007.47031

This is an interesting survey on the nonlinear eigenvalue problem $f\left(x\right)=\lambda x$, with $f$ being a 1-homogeneous continuous operator which leaves a cone $K$ in a Banach space invariant. Particular emphasis is put on compact or, more generally, condensing operators (w.r.t. a suitable measure of noncompactness). Applications are given to operators of the form

$f\left(x\right)\left(s\right)=max\left\{a\left(s,t\right)x\left(t\right):\alpha \left(s\right)\le t\le \beta \left(s\right)\right\}$

in spaces of continuous functions $x:\left[0,\mu \right]\to ℝ$; here $\alpha$ and $\beta$ are continuous on $\left[0,\mu \right]$ with $\alpha \left(s\right)\le \beta \left(s\right)$, while $a$ is continuous and nonnegative on $\left[0,\mu \right]×\left[0,\mu \right]$. In particular, the authors compare the spectral radii

$r\left(f\right)=\underset{x\in K}{sup}\underset{n\to \infty }{lim sup}\parallel {f}^{n}{\left(x\right)\parallel }^{1/n},\phantom{\rule{4pt}{0ex}}\stackrel{˜}{r}\left(f\right)=\underset{n\to \infty }{lim}\underset{x\in K,\parallel x\parallel =1}{sup}{\parallel {f}^{n}\left(x\right)\parallel }^{1/n},$
$\stackrel{^}{r}\left(f\right)=sup\left\{|\lambda |:f\left(x\right)=\lambda x\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}x\in K\setminus \left\{0\right\}\right\}$

and give conditions under which these numbers coincide. It seems that they are unaware of contributions by M. Martelli [Ann. Mat. Pura Appl., IV. Ser. 145, 1-32 (1986; Zbl 0618.47052)] and G. Fournier and M. Martelli [Topol. Methods Nonlinear Anal. 2, No. 2, 203-224 (1993; Zbl 0812.47059)] to this field.

##### MSC:
 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 47H07 Monotone and positive operators on ordered topological linear spaces 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces