We consider the nonlinear Sturm-Liouville problem[formula][formula]whereai,biare real numbers with ai+bi0,i=0,1, ? is a real parameter, and the functionspandaare strictly positive on [0,p]. Suppose that the nonlinearityhsatisfies a condition of the form[formula]as either (?,?)?0 or (?,?)?8, for some constantsM0,M1. Then we show that there exist global continua of nontrivial solutions (?,u) bifurcating fromu=0 or "u=8," respectively. These global continua have properties similar to those of the continua found in Rabonowitz' well-known global bifurcation theorem. © 1998 Academic Press.