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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)



= 0,

(1.1;δ)

where δ = ±1, n ∈ N = {0,1,2,...}, L0 x(n) = x(n), L2 x(n) =

L1 x(n) =

α 1
∆L1 x(n) 2 , a2 (n)

α 1
∆L0 x(n) 1 , a1 (n)

L3 x(n) = ∆L2 x(n).

(1.2)

In what follows, we will assume that (i) {ai (n)}, i = 1,2, and {q(n)} are positive sequences and ∞ 


ai (n)

1/αi

= ∞,

i = 1,2;

(1.3)

(ii) {g(n)} is a nondecreasing sequence, and limn→∞ g(n) = ∞; (iii) f ∈ Ꮿ(R, R), x f (x) > 0, and f  (x) ≥ 0 for x = 0; (iv) αi , i = 1,2, are quotients of positive odd integers. The domain Ᏸ(L3 ) of L3 is defined to be the set of all sequences {x(n)}, n ≥ n0 ≥ 0 such that {L j x(n)}, 0 ≤ j ≤ 3 exist for n ≥ n0 . A nontrivial solution {x(n)} of (1.1;δ) is called nonoscillatory if it is either eventually positive or eventually negative and it is oscillatory otherwise. An equation (1.1;δ) 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 

∆



α
  1
∆x(n) 1 + δq(n) f x g(n) = 0, a1 (n)

(1.4;δ)

where δ, a1 , q, g, f , and α1 are as in (1.1;δ) and/or related equations has been the subject of intensive study in the last decade. For typical results regarding (1.4;δ), 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;δ), the study of higher-order equations, and in particular third-order equations of type (1.1;δ) has received considerably less attention (see [9, 10, 14]). In fact, not much has been established for equations with deviating arguments. The purpose of this paper is to present a systematic study for the behavioral properties of solutions of (1.1;δ), and therefore, establish criteria for the oscillation of (1.1;δ). 2. Properties of solutions of equation (1.1;1) We will say that {x(n)} is of type B0 if x(n) > 0,

L1 x(n) < 0,

L2 x(n) > 0,

L3 x(n) ≤ 0

eventually,

(2.1)

L1 x(n) > 0,

L2 x(n) > 0,

L3 x(n) ≤ 0

eventually.

(2.2)

it is of type B2 if x(n) > 0,

Clearly, any positive solution of (1.1;1) is either of type B0 or B2 . In what follows, we will present some criteria for the nonexistence of solutions of type B0 for (1.1;1). Theorem 2.1. Let conditions (i)–(iv) hold, g(n) < n for n ≥ n0 ≥ 0, and − f (−xy) ≥ f (xy) ≥ f (x) f (y)

for xy > 0.

(2.3)

Moreover, assume that there exists a nondecreasing sequence {ξ(n)} such that g(n) < ξ(n) < n for n ≥ n0 . If all bounded