Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable

We consider the nonlinear Sturm-Liouville problem Lu := -(pu')' + qu = lambda au + h(., u, u', lambda), in (0, pi), a(0)u(0) + b(0)u'(0) = 0, a(1)u(pi) + b(1)u'(pi) = 0, where a(i), b(i) are real numbers with \a(i)\ + \b(i)\ > 0, i = 0, 1, lambda is a real parameter, and the functions p and a are strictly positive on [0, pi]. Suppose that the nonlinearity h satisfies a condition of the form \h(x, xi, eta, lambda)\ less than or equal to M-0\xi\ + M-1\eta\, (x, xi, eta, lambda) is an element of [0, pi] x R-3 as either \(xi, eta)\ --> 0 or \(xi, eta)\ --> infinity, for some constants M-0, M-1. Then we show that there exist global continua of nontrivial solutions (lambda, u) bifurcating from u = 0 or "u = infinity," respectively. These global continua have properties similar to those of the continua found in Rabonowitz' well-known global bifurcation theorem. (C) 1998 Academic Press.