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ON THE OSCILLATION OF CERTAIN THIRD-ORDER DIFFERENCE EQUATIONS RAVI P. AGARWAL, SAID R. GRACE, AND

ON THE OSCILLATION OF CERTAIN THIRD-ORDER DIFFERENCE EQUATIONS RAVI P. AGARWAL, SAID R. GRACE, AND DONAL O’REGAN Received 28 August 2004 We establish some new criteria for the oscillation of third-order difference equations of the form ∆((1/a2 (n))(∆(1/a1 (n))(∆x(n))α1 )α2 ) + δq(n) f (x[g(n)]) = 0, where ∆ is the forward difference operator defined by ∆x(n) = x(n + 1) − x(n). 1. Introduction In this paper, we are concerned with the oscillatory behavior of the third-order difference equation L3 x(n) + δq(n) f x g(n) where δ = ±1, n ∈ N = {0,1,2,.}, L0 x(n) = x(n), L2 x(n). | ON THE OSCILLATION OF CERTAIN THIRD-ORDER DIFFERENCE EQUATIONS RAVI P. AGARWAL SAID R. GRACE AND DONAL O REGAN Received 28 August 2004 We establish some new criteria for the oscillation of third-order difference equations of the form A 1 a2 n A 1 a1 n Ax n a1 2 8q n f x g n 0 where A is the forward difference operator defined by Ax n x n 1 - x n . 1. Introduction In this paper we are concerned with the oscillatory behavior of the third-order difference equation Í3x n 8q n f x g n 0 1.1 8 where 8 1 n e N 0 1 2 . L0x n x n L1 x n - AL0x n a1 a1 n 1 _ 1.2 L2x n AL1x n a2 L3x n AL2x n . 2 n In what follows we will assume that i a.ị n i 1 2 and q n are positive sequences and 00 ữi n 1 ai o i 1 2 1.3 ii g n is a nondecreasing sequence and lim o.g n o iii f e R R xf x 0 and f x 0 for x 0 iv ai i 1 2 are quotients of positive odd integers. The domain L3 of L3 is defined to be the set of all sequences x n n n0 0 such that Ljx n 0 j 3 exist for n n0. A nontrivial solution x n of 1.1 8 is called nonoscillatory if it is either eventually positive or eventually negative and it is oscillatory otherwise. An equation 1.1 8 is called oscillatory if all its nontrivial solutions are oscillatory. Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 3 2005 345-367 DOI 10.1155 ADE.2005.345 346 On the oscillation of certain third-order difference equations The oscillatory behavior of second-order half-linear difference equations of the form a Ax n Sq n f x g n 0 1.4 S a1 n where s a1 q g f and a1 are as in 1.1 d and or related equations has been the subject of intensive study in the last decade. For typical results regarding 1.4 S we refer the reader to the monographs 1 2 4 8 12 the papers 3 6 11 15 and the references cited therein. However compared to second-order difference equations of type 1.4 S the study of higher-order equations and in particular third-order equations of type 1.1 S has received considerably less attention see 9 10 14 . In fact not .