Existence of Multiple Solutions for Second Order Boundary Value Problems

We give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y″+f(x, y, y′)=0, x [0, 1], y(0)=0=y(1). In the special case f(x, y, y′)=f(y) 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and of Lakshmikantham et al