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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(x)=λx, 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


in spaces of continuous functions x:[0,μ]; here α and β are continuous on [0,μ] with α(s)β(s), while a is continuous and nonnegative on [0,μ]×[0,μ]. In particular, the authors compare the spectral radii

r(f)=sup xK lim sup n f n (x) 1/n ,r ˜(f)=lim n sup xK,x=1 f n (x) 1/n ,
r ^(f)=sup{|λ|:f(x)=λxforsomexK{0}}

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.

47J10Nonlinear spectral theory, nonlinear eigenvalue problems
47H07Monotone and positive operators on ordered topological linear spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces