Eigenvalues and eigenfunctions of discrete conjugate boundary value problems

We consider the following boundary value problem: (-1)(n-p)Delta(n)y = lambda F (k, y, Delta y, ..., Delta(n-1)y), n greater than or equal to 2, 0 less than or equal to k less than or equal to m, Delta(i)y(0) = 0, 0 less than or equal to i less than or equal to p - 1; Delta(i)y(m + n - 1) = 0, 0 less than or equal to i less than or equal to n - p - 1, where 1 less than or equal to p less than or equal to n - 1 is fixed and lambda > 0. A characterization of the values of lambda is carried out so that the boundary value problem has a positive solution. Next, for lambda = 1, criteria are developed for the existence of two positive solutions of the boundary value problem. In addition, for particular cases we also offer upper and lower bounds for these positive solutions. Several examples are included to dwell upon the importance of the results obtained. (C) 1999 Elsevier Science Ltd. All rights reserved.