(n, p) Boundary Value Problems

Here, we shall obtain results similar to those in Sections 29 and 30 for the difference equation (29.1) satisfying the (n, p) boundary conditions 31.1 $$\begin{array}{*{20}{c}} {{\Delta ^i}y\left( 0 \right) = 0,0 \leqslant i \leqslant n - 2} \\ {{\Delta ^p}y\left( {J + n - p - 1} \right) = 0} \end{array}$$ where λ > 0, n ≥ 2 and 0 ≤ p ≤ n − 1 is fixed. We begin with the characterization of λ so that the boundary value problem (29.1), (31.1) has positive solutions. By a positive solution y of (29.1), (31.1), we mean a non-trivial y : N(0, J +n − 1) → [0, ∞) satisfying (29.1) and (31.1).