We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrized family of unbounded linear operators, which are generators of strongly continuous semigroups, and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with nonmonotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension n. In addition, we establish a non-quasi-nilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenileadult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process. © 2014 Society for Industrial and Applied Mathematics.
- Fixed points of multivalued maps
- Nonlinear evolution equations
- Positive steady states
- Semigroups of operators
- Spectral theory of positive operators
- Structured populations