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 systemx″+∑j=0lbj(t)x2j+1x′+x2n+1+∑i=0n−1ai(t)x2i+1=0. It is shown that all the solutions are bounded provided that the ai(t)ai(t)(0⩽i⩽[(n−1)/2])(0⩽i⩽[(n−1)/2]) are of bounded variation in [0,1][0,1] and the derivatives of bj(t)bj(t) and ai(t)ai(t)([(n−1)/2]+1⩽i⩽n−1,0⩽j⩽l) are Lipschitzian. It is also shown that there exist aiai's being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of aiai's is not necessary for the boundedness of solutions of the equation.