ON THIRD-ORDER LINEAR DIFFERENCE EQUATIONS INVOLVING QUASI-DIFFERENCES ˇ ´ ˇ ZUZANA DOSLA AND ALES KOBZA Received 30 June 2004; Revised 20 September 2004; Accepted 12 October 2004 We study the third-order linear difference equation with quasi-differences and its adjoint equation. The main results of the paper describe relationships between the oscillatory and nonoscillatory solutions of both equations. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction Consider the third-order linear difference equation Δ pn Δ rn Δxn and its adjoint equation Δ rn+1 Δ pn Δun − qn+1 un+2 = 0, + qn xn+1 = 0 (E) (EA ) where Δ is the forward difference. | ON THIRD-ORDER LINEAR DIFFERENCE EQUATIONS INVOLVING QUASI-DIFFERENCES ZUZANA DOSLA AND ALES KOBZA Received 30 June 2004 Revised 20 September 2004 Accepted 12 October 2004 We study the third-order linear difference equation with quasi-differences and its adjoint equation. The main results of the paper describe relationships between the oscillatory and nonoscillatory solutions of both equations. Copyright 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction Consider the third-order linear difference equation A pnA rnAxn qnXn 1 0 E and its adjoint equation A rn iA pnA n - qn 1 n 2 0 EA where A is the forward difference operator defined by Axn xn 1 - xn pn rn and qn are sequences of positive real numbers for n e N. This paper has been motivated by the paper 9 where third-order difference equations A3vn - pn 1Avn 1 qn 1vn 1 0 A A2Wn - pn 1un l - qn 2un 2 0 had been investigated. As it is noted here these equations are not adjoint equations and are referred to as quasi-adjoint equations. Equation E is a special case of linear nth-order difference equations with quasi-differ-ences. Such equations have been widely studied in the literature see for example 6 11 and the references therein. The natural question which arises is to find the adjoint equation to E and to examine the connection between solutions of E and its adjoint one. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 65652 Pages 1-13 DOI ADE 2006 65652 2 Third-order linear difference equations In the continuous case it holds see . 5 Theorem that TrizTrx m q t x t 0 p t r t is oscillatory if and only if the adjoint equation . ỉ x t _ q t x t 0 r t p t has the same property. In addition nonoscillatory solutions of these equations satisfy some interesting relationships see for example 2 5 . The aim of this paper is to investigate oscillatory and asymptotic properties of solutions of E and EA . We will prove that EA is the adjoint .
