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Zbl 1007.47031
Mallet-Paret, John; Nussbaum, Roger D.
Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. (English)
[J] Discrete Contin. Dyn. Syst. 8, No.3, 519-562 (2002). ISSN 1078-0947; ISSN 1553-5231/e

This is an interesting survey on the nonlinear eigenvalue problem $f(x)=\lambda x$, 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 $$f(x)(s)= \max\{a(s,t) x(t): \alpha(s)\le t\le \beta(s)\}$$ in spaces of continuous functions $x: [0,\mu]\to \bbfR$; here $\alpha$ and $\beta$ are continuous on $[0,\mu]$ with $\alpha(s)\le \beta(s)$, while $a$ is continuous and nonnegative on $[0,\mu]\times [0,\mu]$. In particular, the authors compare the spectral radii $$r(f)= \sup_{x\in K} \limsup_{n\to\infty}\|f^n(x)\|^{1/n},\ \widetilde r(f)= \lim_{n\to\infty} \sup_{x\in K,\|x\|= 1}\|f^n(x)\|^{1/n},$$ $$\widehat r(f)= \sup\{|\lambda|: f(x)=\lambda x\text{ for some }x\in K\setminus\{0\}\}$$ and give conditions under which these numbers coincide. It seems that they are unaware of contributions by {\it M. Martelli} [Ann. Mat. Pura Appl., IV. Ser. 145, 1-32 (1986; Zbl 0618.47052)] and {\it G. Fournier} and {\it M. Martelli} [Topol. Methods Nonlinear Anal. 2, No. 2, 203-224 (1993; Zbl 0812.47059)] to this field.
[Jürgen Appell (Würzburg)]

MSC 2000:: *47J10 Nonlinear eigenvalue problems
47H07 Positive operators on ordered topological linear spaces
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: homogeneous cone maps; max-plus operators; nonlinear eigenvalue problem; condensing operators; measure of noncompactness; spectral radii

Citations: Zbl 0618.47052; Zbl 0812.47059

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