Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case. (English) Zbl 1215.47052

Cavazos-Cadena, Rolando; Hernández-Hernández, Daniel

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 7-8, 3303-3313 (2010).

Let \(f\) be a monotone and homogeneous self-map of the \(n\)-dimensional positive cone. Let \(\mathcal{F}\) be a family of functions constructed by multiplying the components of \(f\) by positive numbers. Let \(M^f\) be a communication matrix, i.e., the matrix whose \((i,j)\) entry indicates whether or not the \(i\)-th component of \(f\) is an unbounded function of the \(j\)-th coordinate. Assuming that the function \(f\) satisfies a weak form of convexity, the authors obtain two results concerning the eigenvalues for family \(\mathcal{F}\). First, the structure of matrix \(M^f\) which is necessary and sufficient for the existence of an eigenvalue of each function in the family \(\mathcal{F}\) is determined. Second, a necessary and sufficient criterion is given in terms of the recession function of the function \(f\).

Reviewer: Leszek Gasiński (Kraków)

MSC:

47J10	Nonlinear spectral theory, nonlinear eigenvalue problems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Keywords:

eigenvalue problem; generalized Perron-Frobenius theorem; Collatz-Wielandt relations; minimal closed set; communication matrix

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