Oscillation criteria for second order nonlinear advanced type difference equations (original) (raw)

Oscillation criteria for certain third order nonlinear difference equations

Applicable Analysis and Discrete Mathematics, 2009

Some new criteria for the oscillation of all solutions of third order nonlinear difference equations of the form ? (a(n)(?2 x(n))? + q(n)f (x[g(n)]) = 0 and ? (a(n)(?2 x(n))? = q(n)f (x[g(n)]) + p(n)h(x[?(n)]) ? -1/? with P a (n) < ? are established.

Oscillation Criteria of Comparison Type for Second Order Difference Equations

Journal of Applied Analysis, 2000

In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms ∆(cn−1∆(xn−1 + pnxσ n )) + qnf (xτ n ) = 0, n = 1, 2, . . . and the corresponding nonhomogeneous equation ∆(cn−1∆(xn−1 + pnxσ n )) + qnf (xτ n ) = rn, n = 1, 2, . . .

Oscillation of solutions of some generalized nonlinear α-difference equations

Advances in Difference Equations, 2014

In this paper, the authors discuss the oscillation of solutions of some generalized nonlinear α-difference equation Δ α ( ℓ ) ( p ( k ) Δ α ( ℓ ) u ( k ) ) + q ( k ) f ( u ( k − τ ( k ) ) ) = 0 , k ∈ [ a , ∞ ) , where the functions p, q, f and τ are defined in their domain of definition and α > 1 , ℓ is a positive real. Further, u f ( u ) > 0 for u ≠ 0 , p ( k ) > 0 and lim k → ∞ ( k − τ ( k ) ) = ∞ , where R k = ∑ r = 0 [ k − ℓ ℓ ] 1 α r p ( r ℓ ) → ∞ as k → ∞ and u ( k ) is defined for k ≥ min i ≥ 0 ( i − τ ( i ) ) for all k ∈ [ a , ∞ ) for some a ∈ [ 0 , ∞ ) . MSC:39A12.

On the oscillation of certain second order difference equations

1998

The present paper is concerned with the oscillation of the second order linear difference equation ∆(c n−1 ∆u n−1 ) + q n u n = 0, n ≥ 1, where ∆ is the usual forward difference operator ∆u n = u n+1 − u n , {c n } and {q n } are sequences of real numbers with c n > 0. Using summation averaging technique, some new oscillation criteria are obtained and the discrete analogue of known results due to Kamenev and Philos for the corresponding differential equations are established.

Oscillation results for nonlinear second order difference equations with mixed neutral terms

Advances in Difference Equations

In this paper, we establish new oscillation criteria for nonlinear second order difference equations with mixed neutral terms. The key idea of our approach is to compare with first order equations whose oscillatory behaviors are already known. The obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations. The results are supported by examples to demonstrate the validity of the theoretical findings.

Comparison theorems for second order nonlinear difference equations

Journal of Mathematical Analysis and Applications, 2005

New oscillation results are obtained for the second order nonlinear difference equation [Delta](rnf([Delta]xn-1))+g(n,xn)=0, and its functional form [Delta](rnf([Delta]xn-1))+g(n,x[tau]n)=0. The role played by the argument [tau]n on the oscillation of the functional equation is explored. In particular, we characterize a class of sequences {[tau]n} which have a harmless effect on the oscillation of this type of equations. Some of our results generalize, improve or unify known fundamental oscillation results for several particular cases of the above equations.