On the Range of Certain Pendulum-Type Equations

Let us consider the BVP mx Žt. g1Žx Žt.. fŽt., t 0, T xŽ0. xŽT., x Ž0. x ŽT., where g1is a continuous function. The range R1of the operator related to this problem is very well known. In this paper we treat the perturbed problem mx Žt. g1Žx Žt.. g0ŽxŽt.. fŽt., t 0, T xŽ0. xŽT., x Ž0. x ŽT., where g0 is of pendulum type, showing that, in general, the range of the perturbed operator is not contained in R1. This points out an important qualitative difference with respect to the case where g0 is of the Landesmann Lazer type. On the other hand we prove that if f is small then the mentioned inclusion is true in general