| Comparison of smallest eigenvalues of three point boundary value problems of second order is established and then the result is extended to the n-th order and 2n-th order in two different ways. For this, the theory of u0-positive operators with respect to a cone in a Banach space, along with sign properties of Green's functions are applied.; Next, sufficient conditions for the comparison of smallest eigenvalues are established for two different types of m-point boundary value problems of second order, and then again in this case, the comparison of smallest eigenvalues is obtained for 2n-th order problems.; Finally, the existence of a smallest interval, such that there exists a nontrivial solution for a two point boundary value problem of fourth order, is established. This is accomplished by obtaining criteria for the existence of extremal points. Then the same is obtained for 2n-th order boundary value problems. |
