Entrainment of frequency in evolution equations

The existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ?f(t, u) when the linear problem is totally degenerate (e^2πA = I) and the period of f is entrained with ? (T = 2π(1 + ?μ)). The approach is to solve the periodicity equation u(T,p,?) = p for an element p(?) 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 + ∝₀^2πe^?Asf(s, e^Asp) ds = 0 obtained from the periodicity equation by dividing by ?, applying the entrainment assumption, and letting ? → 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 u_t = u_x + ?{g(u) ? h(t, x)} with g strongly monotone and

$\text{d}\text{dt}\text{v}\text{w}\text{=}\text{0}\text{d}\text{dx}\text{d}\text{dx}\text{0}\text{v}\text{w}\text{+ ?}\text{v}{}^3\text{h(t,x)}$

, where in both cases D is a certain class of 2π-periodic functions of x.