Eigenvalues for a class of homogeneous cone maps arising from max-plus operators
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 \[ f(x)(s)= \max\{a(s,t) x(t): \alpha(s)\leq t\leq \beta(s)\} \] in spaces of continuous functions \(x: [0,\mu]\to \mathbb{R}\); here \(\alpha\) and \(\beta\) are continuous on \([0,\mu]\) with \(\alpha(s)\leq \beta(s)\), 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 \textit{M. Martelli} [Ann. Mat. Pura Appl., IV. Ser. 145, 1-32 (1986; Zbl 0618.47052)] and \textit{G. Fournier} and \textit{M. Martelli} [Topol. Methods Nonlinear Anal. 2, No. 2, 203-224 (1993; Zbl 0812.47059)] to this field.
scientific article; zbMATH DE number 1751461