Free Vibrations for a Nonlinear Wave Equation and a Theorem of

A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation utt-ux, + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u+m as Iu(+w. The proof uses a modified form (PS), of the Palais-Smale condition (PS). Let g : R-+R be a continuous nondecreasing function such that g(0) = 0. Set G (t) = g(s) ds. We seek a nontrivial solution of the equation 6' under the boundary conditions (2) u(0, t) = U(T, t) = 0 and periodicity condition (3) u(x, t + 2 7 r) = u(x, t) .