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,     

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.     

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.     

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

Sufficient conditions are obtained under which all solutions of

(∗)     

一类拟线性二阶微分方程解的振动与非振动的判定

本文讨论了拟线性微分方程（ｐ（ｘ＇））＇＋ｑ（ｔ）ｐ（ｘ）＝０,ｔ≥ｔ０,ｑ（ｔ）≥０（这里 ｐ（ｕ）＝｜ｕ｜ｐ－１ｕ,ｐ＞０是常数）的解的振动与非振动条件,并改进了文献［２］的结果．     

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

     

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

     

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 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.     

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

     

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.     

Oscillation and nonoscillation of solutions of second order linear dynamic equations with integrable coefficients on time scales

We obtain Willett-Wong-type oscillation and nonoscillation theorems for second order linear dynamic equations with integrable coefficients on a time scale. The results obtained extend and are motivated by oscillation and nonoscillation results due to Willett [20] and Wong [21] for the second order linear differential equation. As applications of the new results obtained, we give the complete classification of oscillation and nonoscillation for the difference equations

     

Wong's comparison theorem for second order linear dynamic equations on time scales

We obtain Wong-type comparison theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by Wong's comparison theorems. As a particular application of our results, we show that the difference equation

     

Positive solutions for a scalar differential equation with several delays

For a scalar delay differential equation , we obtain new explicit conditions for the existence of a positive solution.     

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 < ∞.     

脉冲时滞差分方程的振动性

讨论脉冲时滞差分方程xn+1- xn +pnxn-2 =0 ,n≥ 0 ,n≠ nkxnk+1- xnk =bkxnk,k =1,2 ,3,…给出了方程所有解振动的充分条件     

Oscillation of delay difference equations with several delays

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

     

“一类拟线性二阶微分方程解的振动与非振动的判定”的注记

本文给出了一类拟线性二阶微分方程解振动与非振动的新的判定准则,完善和改进了最近一些文献中的某些结论。