Triple positive solutions of conjugate boundary value problems

We consider the following boundary value problems:(−1)n−py(n) = a(t)h(y), n ⩽ 2, t ∈(0,1),y(i)(0) = 0, 0 ⩾ i ⩾ p − 1, y(i)(1) = 0, 0 ⩾ i ⩾ n − p − 1,and(−1)n−pΔny = F(k,y,Δy,…,Δn−1y), n ⩽ 2, 0 ⩾ k ⩾ m,Δiy(0), 0 ⩾ i ⩾ p − 1, Δiy(m+p+1)=0, 0 ⩾ i ⩾ n − p − 1,where 1 ≤ p ≤ n − 1 is fixed. By employing fixed-point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problems. As an application, we also establish the existence of radial solutions of certain partial difference equations. Several examples are included to dwell upon the importance of the results obtained.