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MAXIMUM PRINCIPLES FOR A FAMILY OF NONLOCAL BOUNDARY VALUE PROBLEMS PAUL W. ELOE Received 21 October 2003 and in revised form 16 February 2004 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. | MAXIMUM PRINCIPLES FOR A FAMILY OF NONLOCAL BOUNDARY VALUE PROBLEMS PAUL W. ELOE Received 21 October 2003 and in revised form 16 February 2004 We study a family of three-point nonlocal boundary value problems BVPs for an nth-order 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 n Pu m aj m u m j 0 j 0 1.1 ana0 0 m e 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 A by Au m u m 1 - u m A0u m u m Ai 1u m A A m 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 mj u b . 1.3 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 202 Nonlocal boundary value problems Clearly the boundary conditions 1.3 are equivalent to the boundary conditions Aiu 0 0 i 0 . n - 2 mj 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 .
