On the eigenvalues of boundary value problems for higher order difference equations

We consider the boundary value problem Delta(n)y + lambda Q(k,y,Delta y,...,Delta(n-2)y) = lambda P(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 N, Delta(i)y(0) = 0, 0 less than or equal to i less than or equal to n-3, alpha Delta(n-2)y(0) - beta Delta(n-1)y(0) = 0, gamma Delta(n-2)y(N + 1) + delta Delta(n-1)y(N + 1) = 0 where lambda > 0, alpha, beta, gamma and delta are constants satisfying alpha gamma(N + 1) + alpha delta + beta gamma > 0, alpha, gamma > 0, beta greater than or equal to 0 and delta greater than or equal to gamma. Upper and lower bounds for X are established for the existence of positive solutions of this boundary value problem.