#### Glasnik Matematicki, Vol. 39, No.2 (2004), 287-299.

### OSCILLATION OF HIGHER ORDER DIFFERENCE
EQUATIONS VIA COMPARISON

### Ravi P. Agarwal, Said R. Grace and
Donal O'Regan

R. P. Agarwal, Department of Mathematical Sciences,
Florida Institute of Technology, Melbourne, FL 32901, U.S.A.

*e-mail:* `agarwal@fit.edu`

S. R. Grace, Department of Engineering Mathematics, Faculty of Engineering,
Cairo University, Orman, Giza 12221, Egypt

*e-mail:* `srgrace@eng.cu.edu.eg`

D. O'Regan, Department of Mathematics, National University of Ireland,
Galway, Ireland

*e-mail:* `donal.oregan@nuigalway.ie`

**Abstract.** In this paper we shall present some new oscillation criteria for difference
equations of the form

Δ^{m}
*x*(*n*) +
*q*(*n*) *f*(*x*[*n* -
τ]) = 0

and
Δ^{m}
*x*(*n*) =
*q*(*t*) *f*(*x*[*n* -
τ]) +
*p*(*n*) *F*(*x*[*n* +
σ])

via comparison with some difference equations of lower order whose
oscillatory behavior are known.
**2000 Mathematics Subject Classification.**
39A10.

**Key words and phrases.** Oscillation, nonoscillation,
difference, nonlinear, comparison.

**Full text (PDF)** (free access)

*Glasnik Matematicki* Home Page