We consider the set of positive solutions (?,u) of the semilinear Sturm-Liouville boundary value problem-u?=?u+fuin 0,p,u0=up=0,where f:[0,8)?R is Lipschitz continuous and ? is a real parameter. We suppose that f(s) oscillates, as s?8, in such a manner that the problem is not linearizable at u=8 but does, nevertheless, have a continuum C of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of C in the ?-|u|0 plane at large |u|0. In particular, we discuss whether C oscillates infinitely often over a single point ?, or over an interval I (of positive length) of ? values. An immediate consequence of such oscillations over I is the existence of infinitely many solutions, of arbitrarily large norm |u|0, of the problem for all values of ? ? I. © 2000 Academic Press.