Eigenvalues of boundary value problems for higher order differential equations

We shall consider the boundary value problem y((n)) + lambda Q(t,y,y',...,y((n-2))) = lambda P(t,y,y',...,y((n-1))), n greater than or equal to 2, t is an element of (0,1), y((i))(0) = 0, 0 less than or equal to i less than or equal to n - 3, alpha y((n-2))(0) - beta y((n-1))(0) = 0, gamma y((n-2))(1) + delta y((n-1))(1) = 0, where lambda > 0, alpha, beta, gamma and delta are constants satisfying alpha gamma + alpha delta + beta gamma > 0, beta, delta greater than or equal to, 0, beta + alpha > 0 and delta + gamma > 0 to characterize the values of lambda so that it has a positive solution. For the special case lambda = 1, sufficient conditions are also established for the existence of positive solutions.