Existence and uniqueness of solutions for semilinear equations involving anti-selfadjoint operators (Q2259321)

Let \(L\) be a linear operator and \(N\) be a nonlinear operator, both defined on a Hilbert space. It is further assumed that \(L\) is anti-selfadjoint and \(N\) is controlled by two bounded selfadjoint operators. This paper establishes a sufficient condition for the existence of a unique solution of the semilinear equation \(Lu=Nu\). The abstract result is illustrated with the problem of finding periodic solutions for a class of first order systems, the periods being determined by the forcing term. The proofs combine monotonicity and iterative arguments.