Application of a lemma on bilinear forms to a problem in nonlinear oscillations

In this note we give a simple condition for nondegeneracy of symmetric bilinear forms on infinite dimensional vector spaces. We apply this condition and elementary properties of Fourier series to prove a uniqueness theorem for periodic solutions of a class of second order nonlinear differential systems. In [5] the present author and D. A. S'anchez considered the differential equation (1) x" + grad G(x) = p(t) = p(t + 27r) where peC(R, Rn), GeC2(R', R). The equation may be considered as the Newtonian equations of motion of a mechanical system subject to conservative internal forces and periodic exciting forces. In [5] it was shown that if there exists an integer N and numbers [tN and fN+1 such that (2) N2 < ,uN < YN+1 < (N + 1)2, and for all aeRn, (3) -v < 8G(a)1axiaxj)< YN+1l, where I is the n x n identity matrix, then there exists at least one 27rperiodic solution of (1). The proof of this existence theorem, which was based on a slight modification of a theorem of C. L. Dolph [1] and the Brouwer fixed point theorem, did not imply uniqueness. The purpose of this note is to show that conditions far less restrictive than (2) and (3) imply that there can exist at most one 217-periodic solution of (1). Our proof will be based on two very elementary abstract algebraic lemmas and the most basic properties of Fourier series. LEMMA 1. Let V be a real vector space and let H: Vx V-+R be a real valued symmetric bilinear form on V. If V is the direct sum of subspaces X and Z such that H is positive definite on X and negative definite on Z, i.e. Received by the editors June 21, 1971 and, in revised form, July 21, 1971. AMS 1970 subject classifications. Primary 34C25, 15A63; Secondary 70D05.