Green’s Functions for Nonhomogeneous Second Order Difference Equations

Abstract

In this chapter we will be concerned with second order nonhomogeneous difference equations. We will be studying equations of the form

$${L_y}\left( t \right) = \Delta \left[ {P\left( t \right)\Delta y\left( {t - 1} \right)} \right] + Q\left( t \right)y\left( t \right) = h\left( t \right)$$

where P(t) and Q(t) are given n × n Hermitian matrix functions on the discrete intervals [a + 1, b + 2] and [a + 1, b + 1], respectively. We also assume P(t) is nonsingular for t ∈ [a + 1, b + 2]. In Section 2 of this chapter we will derive a variation of constants formula for the above nonhomogeneous problem. In Sections 3 and 4 we will be concerned with Green’s matrix functions for the conjugate boundary value problem and for the right focal boundary value problem. In Section 5 we will consider a Green’s matrix function for a more general two point boundary value problem.