  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 parametrised 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 non-monotone 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-quasinilpotency 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 juvenile-adult and a structured consumer-resource population
model, as well as for a selection-mutation model with distributed recruitment
process.
