Some first‐order nonlinear equations on a torus

1. The Main Result We consider here real functions of x =(xi, * a , x,) in W" which are periodic of period 21r in each variable, i.e., functions defined on the torus R, and on fl we consider the constant coefficient operator Let g : R X W-+ W be a C" function periodic in x such that (1) gu(x, u)>O for all x , u. Our purpose is to find a real C" periodic function u on the torus satisfying the first-order differential equation (2) A u + g (x , u) = 0. We shall give necessary and sufficient conditions for a solution to exist. In the study of such problems one usually encounters difficulty with small divisors-in trying to invert A using Fourier series. It is because of (1) that this difficulty can be avoided. Let N (A) = { u E L2 I A u = 0) (to be understood in the distribution sense). Let P denote the L z projection on N (A). P has the important property that Pf 2 0 when f Z 0 (this follows from the fact that ~=lim,,+, (I + A A)-~). Since ~ (1) = I, P is a contraction in L". Our main result is the following. THEOREM 1. Equation (2) has a (unique) C" solution if and only if (3) there exist constants 6 > 0 , A4, such that Pg(x, fi)2 6 , Pg(x, A4) 5-6, at every point of R.