Lagrange Stability for Duffing-Type Equations

In this paper the Duffing-type equation (d2x/dt2)+x2n+1+p2n(t) x2n+…+p1(t) x+p0(t)=0 is studied where the pjʹs are 1-periodic. It is shown that all solutions of this equation are bounded, provided that the pj(t) (0 j n) are of bounded variation in [0, 1] and that the derivatives of pj(t) (n j 2n) are Lipschitzian. It is also shown that there exist pjʹs being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of pjʹs is not necessary for the boundedness of solutions of the equation