Integral comparison theorems and extremal points for linear differential equations

Comparison theorems of integral type are developed for linear differential equations of the form (1) L_ny + p(t)y = 0 and (2) L_ny + q(t)y = 0. Under the assumption that (2) has no nontrivial solution satisfying given homogeneous two point boundary conditions it follows that the same is true for (1), provided certain sign and integral conditions hold. The criteria may be thought of as rather general extensions of the Hille-Wintner type and also include and extend recent results of Elias.