Positive solutions for a class of nonlinear boundary value problems with Neumann-Robin boundary conditions

We consider the two point boundary value problem -u"(x) = lambda f(u(x)); 0 <x <1 u'(0) = 0; u'(1) + alpha u(1) = 0 where lambda > 0 and alpha > 0 are parameters, and f is an element of C-2[0, 1]. We discuss the existence of nonnegative solutions for superlinear nonlinearities by developing a quadrature method. We study the positone (f(0)> 0) case as well as the semipositone (f(0) < 0) case, and note a drastic difference in the respective bifurcation diagrams for positive solutions. (C) 1999 Academic Press.