The eigenvalue problem in difference equations, ( − 1 ) n − k Δ n y ( t ) = λ ∑ i = 0 k − 1 p i ( t ) Δ i y ( t ) , with Δ t y ( 0 ) = 0 , 0 ≤ i ≤ k , Δ k + i y ( T + 1 ) = 0 , 0 ≤ i < n − k , is examined. Under suitable conditions on the coefficients p i , it is shown that the smallest positive eigenvalue is a decreasing function of T . As a consequence, results concerning the first focal point for the boundary value problem with λ = 1 are obtained.