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)...