Bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances

This paper deals with the third order nonlinear difference equations with several delays and advances Δ(anΔ2xn) + f (n, xn, xn−d1n , . . . , xn−dkn ) = 0, n ≥ n0 (∗) and Δ(anΔ2xn) + bng(xn, xn−d1n , . . . , xn−dkn ) = 0, n ≥ n0. (∗∗) Sufficient conditions of the existence of uncountably many bounded positive, negative and nonoscillatory solutions, respectively, and necessary and sufficient conditions of the bounded oscillation of (∗) and (∗∗) with certain asymptotic behavior are given. In addition, a sufficient condition of the oscillatory behavior of all bounded solutions of (∗) with a special of asymptotic property is also presented. Nine nontrivial examples which dwell upon the importance of the results obtained in this paper are also included.