Discrete Second Order Boundary Value Problems

Abstract

Let a, b (b > a) be nonnegative integers. We define the discrete interval [a, b] = {a, a + 1,..., b}. All other intervals will carry its standard meaning, e.g. [0, ∞) denotes the set of nonnegative real numbers. The symbol ∆ denotes the forward difference operator with step size 1, i.e. Δy(k) = y(k + 1) − y(k). Further for a positive m, Δ^m is defined as Δ^m y(k) = Δ^{m −1}(Δy(k)). In this chapter we shall study positive solutions of the second order discrete boundary value problem

$$\begin{array}{*{20}{c}} {{{\Delta }^{2}}y(k - 1) + \mu f(k,y(k)) = 0,\quad k \in [1,T]} \\ {y(0) = 0 = y(T + 1)} \\ \end{array}$$

(17.1)

where μ > 0 is a constant and T > 0 is a positive integer. In fact, all the results we shall prove in this chapter are the discrete analogs of some of those established in Chapters 3, 4 and 7.