Positive solutions and eigenvalues of conjugate boundary value problems

We consider the following boundary value problem (-1)(n-p)y((n)) + lambda H(t, y) =lambda K(t, y), n less than or equal to 2 , t epsilon (0, 1) y((i))(0)= 0, 0 less than or equal to i less than or equal to p -1 y((i))(1) = 0, 0 less than or equal to i n - p - 1 where lambda > 0 and 1 less than or equal to p less than or equal to n -1 i s fixed. The values of lambda are characterized so that the boundary value problem has a positive solution. Further, for the case lambda = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also estab lished for special cases. Several examples are included to dwell upon the i mportance of the results obtained.