On the range of certain pendulum-type equations

Let us consider the BVP mx "(t) + g(1)(x'(t)) = f(t), t is an element of [0,T] x(0) = x(T), x'(0) = x'(T), where g(1) is a continuous function. The range R-1 of the operator related to this problem is very well known. In this paper we treat the perturbed pr oblem mx "(t) + g(1)(x'(t)) + g(0)(x(t)) = f(t), t is an element of [0, T] x(0)= x(T), x'(0) = x'(T), where go is of pendulum type, showing that, in general, the range of the pe rturbed operator is not contained in R-1. This points out an important qual itative difference with respect to the case where g(0) is of the Landesmann -Lazer type. On the other hand we prove that if f is small then the mention ed inclusion is true in general. (C) 2000 Academic Press.