On the oscillatory properties of certain fourth order nonlinear difference equations

Some new criteria for the oscillation of fourth order nonlinear difference equations of the form

     

Oscillation and nonoscillation criteria for second order quasilinear difference equations

New oscillation and nonoscillation theorems are obtained for the second order quasilinear difference equation

Δ(|Δxn−1|^ρ−1Δxn−1)+pn|xn|^ρ−1xn=0,     

Oscillation of a class of nonlinear neutral difference equations of higher order

Sufficient conditions are obtained under which all solutions of

(∗)     

Remarks on oscillation and nonoscillation for second-order linear difference equations

In this remark, we shall show the main results of the earlier work [W.T. Li, S.S. Cheng, Remarks on two recent oscillation theorems for second-order linear difference equations, Appl. Math. Lett. 16 (2003) 161–163] are incorrect.     

Nonoscillation and oscillation of solutions of a class of third order difference equations

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form

     

Oscillation of some nonlinear difference equations

We investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations. Several explicit oscillation criteria are established. Our main results are supported by illustrative examples.     

Comparison theorems for second order nonlinear difference equations

New oscillation results are obtained for the second order nonlinear difference equation

Δ(rnf(Δxn−1))+g(n,xn)=0,     

The oscillation of higher-ordernonlinear difference equations

The authors discuss the relation of the oscillation of the following two difference equations,

where m ≥ 2, τ : N → N, N isthe set of integers, |n − τ(n)| ≤ Mfor n N₀, M is a positive integer, is nondecreasing in x, xf(n, x)> 0, as x ≠ 0. Wewill show some relations of the oscillation of the above two equations. Especially, for m to be even, we establish the equivalenceof the oscillation of the above two difference equations.     

On the Behaviour of Solutions of a Class of Third Order Difference Equations

In this paper, the structure of the solution space of y n +3 + ry n +2 + qy n +1 + py n =0, n S 0, is studied, keeping oscillatory/nonoscillatory behaviour of solutions of the equation in view, where p , q and r are constants. Some of these results are generalized partially to hold for y n +3 + r n y n +2 + q n y n +1 + p n y n =0, n S 0, where { p n }, { q n } and { r n } are sequences of real numbers.     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

Oscillation of higher-order nonlinear difference equations of neutral type

We shall establish some new criteria for the oscillation of all solutions of higher-order difference equations of the form

δ^m(x_n-x_n-r)+q_nf(x_n-g=0, m1

     

Oscillation of second-order quasilinear neutral delay difference equations

By using the Riccati transformation and mathematical analytic methods,some sufficient conditions are obtained for oscillation of the second-order quasilinear neutral delay difference equations Δ[r n |Δz n | α-1 Δ z n ] + q n f (x n-σ)=0,where z n=x n + p n x n τ and ∞ Σ n=0 1 /r n 1/α < ∞.     

On oscillation and nonoscillation properties of Emden-Fowler difference equations

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation

Oscillation of delay difference equations with several delays

In this paper, we obtain some new oscillation criteria for the difference equation with several delays