Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

Abstract

We show that if u is a bounded solution on R + of u″( t) ϵ Au( t) + f( t), where A is a maximal monotone operator on a real Hilbert space H and f∈ L loc 2( R +; H) is periodic, then there exists a periodic solution ω of the differential equation such that u( t) − ω( t) 0 and u′( t) − ω′( t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in D(A) and that a smoothing effect takes place.