Báo Cáo Báo cáo hóa học ON THIRD-ORDER LINEAR DIFFERENCE EQUATIONS INVOLVING QUASI-DIFFERENCES

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 ư qn+1 un+2 = 0, + qn xn+1 = 0 (E) (EA ) where Δ is the forward difference operator defined by Δxn = xn+1 ư xn , (pn ), (rn ), and (qn ) are sequences of positive real numbers for n ∈ N. This paper has been motivated by the paper [9], where third-order difference equations Δ3 vn ư pn+1 Δvn+1 + qn+1 vn+1 = 0, Δ Δ2 un ư pn+1 un+1 ư qn+2 un+2 = 0 (1.1) had been