We consider the following boundary value problem: (−1)n−pΔny=λF(k,y,Δy,…,Δn−1y), n≪2, 0≤k≤m, Δiy(0)=0, 0≤i≤p−1; Δiy(m+n−i)=0, 0≤i≤n−p−1, where 1 ≤ p ≤ n − 1 is fixed and λ > 0. A characterization of the values of λ is carried out so that the boundary value problem has a positive solution. Next, for λ = 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.