A sufficient condition for the existence of positive solutions of the nonlinear boundary value problem u ''(t) + f(t, u(t)) = 0, 0 < t < 1, u'(0) = u(1) = 0 is constructed, where f : [0, 1) x (0, infinity) --> (0, infinity) is continuous, f(t, u) is locally Lipschitz continuous, and f(t, u)/u is strictly decreasing in u > 0 for each t is an element of (0, 1).
