Entrainment of frequency in evolution equations

AbstractThe existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ϵf(t, u) when the linear problem is totally degenerate (e2πA = I) and the period of f is entrained with ϵ (T = 2π(1 + ϵμ)). The approach is to solve the periodicity equation u(T,p,ϵ) = p for an element p(ϵ) in D, the domain of A, as a perturbation from an approximate solution p0. p0 is a solution of the nonlinear boundary value problem 2πμAp + ∝02π e−Asf(s, eAsp) ds = 0 obtained from the periodicity equation by dividing by ϵ, applying the entrainment assumption, and letting ϵ → 0. Once p0 is known, the conventional inverse function theorem is applied in a slightly unconventional manner. Two particular cases where results are obtained are ut = ux + ϵ{g(u) − h(t, x)} with g strongly monotone and ddtvw = 0ddxddx0vw + ϵv3h(t,x), where in both cases D is a certain class of 2π-periodic functions of x