Boundedness of solutions for a class of second-order differential equations

In this paper we study the boundedness of solutions for the second-order differential equation (F-p(x') + psi(x'))' + alpha F-p(x(+)) - beta F-p(x(-)) = phi (t, x), where F-p(s) = vertical bar s vertical bar(p-2)s, P > 1 and alpha, beta are strictly positive constants satisfying a resonant relation pi/alpha(1/p) sin pi/p + pi/beta(1/p) sin pi/p = 2 pi/n with n being a positive integer, and phi(t, x) is a 2 pi-periodic function in t. There exists a function W(theta) such that if it is of constant sign, under some increasing restrictions on psi(x') and phi(t, x), by a small twist theorem we prove that all the solutions of the above equation are bounded in the sense of Lagrangian stability. (c) 2007 Elsevier Ltd. All rights reserved.