Boundedness of solutions for reversible system via Moser's twist theorem

In this paper we consider the problem of the boundedness of all solutions for the reversible system x '' + Sigma(l)(j=0)b(j)(t)x(2j+1)x' + x(2n+1) + Sigma(n-1)(i=0)a(i)(t)x(2i+1) = 0. It is shown that all the solutions are bounded provided that the a(i)(t) (0 <= i <= [(n - 1)/2]) are of bounded variation in [0, 1] and the derivatives of b(j)(t) and a(i)(t) ([(n - 1)/2] + 1 <= i <= n - 1, 0 <= j <= 1) are Lipschitzian. It is also shown that there exist a(i)'s being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of a(i)'s is not necessary for the boundedness of solutions of the equation. (C) 2007 Elsevier Inc. All rights reserved.