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The existence of 27r-periodic solutions to the system x" + grad G(x) = p(t, x), p being 2w-periodic in /, is established under conditions at infinity on the Hessian matrix of G. The condition used is weaker than earlier known conditions of a similar nature. A MS (MOS) subject classifications (1970). Primary 34C25, 47H15; Secondary 34B15. Key words and phrases. Periodic solutions, boundary value problems, nonlinear system of ordinary differential equations, Hubert space methods.doi:10.2307/2042791 fatcat:ghqlfmeaorctxkapgpl3xoojfy