Existence and uniqueness of solutions for semilinear equations involving anti-selfadjoint operators

Abstract

We consider the problem of the existence and uniqueness of solutions to a semilinear equation in a Hilbert space, of the type $Lu=Nu$, where the linear operator $L$ is assumed to be anti-selfadjoint, and the nonlinear part N is controlled by two bounded selfadjoint operators $A$ and $B$. As an example of application, we study the existence and uniqueness of periodic solutions for a system of transport equations. Precisely, we look for solutions which are periodic in each of their variables, the periods being determined by the forcing term.