Third-order nonlinear difference equations of the form Δ(cnΔ(dnΔxn)) + pnΔxn+1 + qnf (xn−σ )= 0 ,n ≥ n0 are considered. Here, {cn} , {dn} , {pn} ,a nd{qn} are sequences of positive real numbers for n0 ∈ N, f is a continuous function such that f (u) /u ≥ K> 0f oru � 0 . By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.

https://journals.tubitak.gov.tr/math/issues/mat-12-36-3/mat-36-3-8-1010-67.pdf

Oscillation of third-order nonlinear delay difference equations

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria for a certain class of third order nonlinear delay difference equations. Our results extend and improve some previously obtained ones. An example is worked out to demonstrate the validity of the proposed results.

https://www.emis.de/journals/EJQTDE/p527.pdf

On the oscillatory behavior for a certain class of third order nonlinear delay difference equations

The third-order nonlinear difference equations @D(p"n@D(r"n@Dx"n))+q"nf(x"n"+"p)=0,p@?{0,1,2},(E"p) where (P"n), (r"n), and (q"n) are sequences of positive real numbers for n @e @?, @? : @? -> @? is a continuous function such that @?(u)u > 0 for u 0, are investigated. All nonoscillatory solutions of these equations are classified according to the sign of their quasidifferences to classes N"i, i = 0, 1, 2, 3, and sufficient conditions ensuring N"i = , i @e {1,2,3} are given. Special attention is paid to equation (E"1) for which the generalized zeros of solutions are studied and an energy function F is introduced. The relation between the class N"0 and solutions for which F"n < 0 for n @e @? is established.

Global asymptotic properties of third-order difference equations

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form

/pdf/on-the-oscillation-of-third-order-half-linear-neutral-type-232p1ex88i.pdf

B. Smith

Papers

Linear Third-Order Difference Equations: Oscillatory and Asymptotic Behavior