Abstract
The 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 ( e 2 πA = I) and the period of f is entrained with ϵ ( T = 2 π(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 p 0. p 0 is a solution of the nonlinear boundary value problem 2 πμAp + ∝ 0 2 π e − As f( s, e As p) ds = 0 obtained from the periodicity equation by dividing by ϵ, applying the entrainment assumption, and letting ϵ → 0. Once p 0 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 + ϵ{ g( u) − h( t, x)} with g strongly monotone and d dt v w = 0 d dx d dx 0 v w + ϵ v 3 h(t,x) , where in both cases D is a certain class of 2π-periodic functions of x.
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