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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
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 $$f$$.

##### 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
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