Roots of Unity and Unbounded Motions of an Asymmetric Oscillator

Abstract

(R2?Z).The general questions that we have in mind are: under what conditions ona, b and f(t) can we say that (1) has unbounded solutions? are there equationsin the class (1) where unbounded and periodic solutions can coexist?Equation (1) was first considered by Dancer in [2, 3, and 4] and byFucik in [7]. They studied the periodic and Dirichlet boundary valueproblems for (1) and looked at this equation as a model of the so called‘‘equations with jumping nonlinearities’’. We sum up some of the results in[2, 3]. Consider the family of curves in the (a, b)-parameter space,1-a+1-b=2p, p=1,2,.... (2)When (a, b) is not in any of these curves then (1) has at least one 2?-periodic solution. However, if (a, b) lies in one of these curves there existfunctions f(t) for which (1) has no periodic solution. More results on theperiodic problem for (1) have been obtained by Lazer and McKenna [8],Fabry [5] and Fabry and Fonda [6]. The known results on the existenceof periodic solutions already shed some light on the boundedness problem.Actually, the second theorem of Massera implies that if (1) has no periodic