Poisson equations associated with a homogeneous and monotone function: Necessary and sufficient conditions for a solution in a weakly convex case

Given a monotone and homogeneous self-mapping ff of the nn-dimensional positive cone, a communication matrix MfMf is introduced and a family FF of functions is obtained by multiplying each component of ff by arbitrary positive numbers. Assuming that ff satisfies a weak form of convexity, a necessary and sufficient criterion on the structure of MfMf is given so that each function in FF has a (nonlinear) eigenvalue. An alternative necessary and sufficient condition in terms of the recession function of ff is also provided.