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We study a family of three-point nonlocal boundary value problems (BVPs) for an nthorder linear forward difference equation. In particular, we obtain a maximum principle and determine sign properties of a corresponding Green function. Of interest, we show that the methods used for two-point disconjugacy or right-disfocality results apply to this family of three-point BVPs. 1. Introduction The disconjugacy theory for forward difference equations was developed by Hartman [15] in a landmark paper which has generated so much activity in the study of difference equations. Sturm theory for a second-order finite difference equation goes back to Fort [12], which also serves as an excellent reference for the calculus of finite differences. Hartman considers the nth-order linear finite difference equation Pu(m) =

n
j =0

α j (m)u(m + j) = 0,

(1.1)

αn α0
= 0, m ∈ I = {a,a + 1,a + 2,...}. To illustrate the analogy of (1.1) to an nth-order ordinary differential equation, introduce the finite difference operator ∆ by ∆u(m) = u(m + 1) − u(m), 



∆0 u(m) ≡ u(m),

∆i+1 u(m) = ∆ ∆i u (m),

i ≥ 1.

(1.2)

Clearly, P can be algebraically expressed as an nth-order finite difference operator. Let m1 , b denote two positive integers such that n − 2 ≤ m1 < b. In this paper, we assume that a = 0 for simplicity, and we consider a family of three-point boundary conditions of the form u(0) = 0,...,u(n − 2) = 0, Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 201–210 2000 Mathematics Subject Classification: 39A10, 39A12 URL: http://dx.doi.org/10.1155/S1687183904310083





u m1 = u(b).

(1.3)

202

Nonlocal boundary value problems

Clearly, the boundary conditions (1.3) are equivalent to the boundary conditions ∆i u(0) = 0,



i = 0,...,n − 2,



u m1 = u(b).

(1.4)

There is a current flurry to study nonlocal boundary conditions of the type described by (1.3). In certain sectors of the literature, such boundary conditions are referred to as multipoint boundary conditions. Study was initiated by Il’in and Moiseev [16, 17]. These initial works were motivated by earlier work on nonlocal linear elliptic boundary value problems (BVPs) (see, e.g., [3, 4]). Gupta and coauthors have worked extensively on such problems; see, for example, [13, 14]. Lomtatidze [18] has produced early significant work. We point out that Bobisud [5] has recently developed a nontrivial application of such problems to heat transfer. For the rest of the paper, we will use the term nonlocal boundary conditions, initiated by Il’in and Moiseev [16, 17]. We motivate this paper by first considering the equation Pu(m) = ∆n u(m) = 0,

m = 0,...,b − n.

(1.5)

In this preliminary discussion, we employ the natural family of polynomials, m(k) = m(m − 1) · · · (m − k + 1) so that ∆m(k) = km(k−1) . A Green function, G(m1 ,m,s) for the BVP (1.5), (1.3) exists for (m1 ,m,s) ∈ {n − 2,...,b − 1} × {0,...,b} × {0,...,b − n}. It can be constructed directly and has the form    a m1 ;s m(n−1)    ,    (n − 1)!  (n−1) G m1 ;m,s = 