Bounds for positive solutions for a focal boundary value problem

We will be concerned with the focal boundary value problem (−1)nΔn[p(t)Δny(t − n)] = f(h,t,y(t), ..., Δn−1y(t)),Δiy(0) = Δn+iy(b + 1) = 0, 0 → i → n − 1. Using cone theory in a Banach space, we show that under certain assumptions on f, this focal boundary value problem has two positive solutions. In the special case −Δ2y(t − 1) = h2[yα(t) + yβ(t)], y(0) = δ(b + 1) = 0, where 0 < α < 1 < β, we are able to exhibit upper and lower bounds for these two positive solutions.