Reduction theorems for a class of semilinear equations at resonance

Abstract:

In solving equations of the form $Lu - Nu = p$ in a Hilbert space, where $L$ is linear and $N$ is nonlinear, the alternative method can sometimes be used to reduce the problem to one in a subspace. In this note previous reduction results are extended and at the same time the proofs are simplified. The approach is to use simple fixed point theorems in place of the traditional variational methods which are often quite delicate.