Some Oscillation and Non-Oscillation Theorems for Fourth Order Difference Equations

Abstract

Sufficient conditions are established for oscillation of all solutions of the fourth order difference equation

where $\Delta$ is the forward difference operator $\Delta y_n=y_{n+1}–y_n,\{a_n\},\{b_n\},\{c_n\},\{q_n\},\{h_n\}$ are real sequences, and $f$ is a real-valued continuous function. Also, sufficient conditions are provided which ensure that all non-oscillatory solutions of the equation approach zero as $n\to \infty$. Examples are inserted to illustrate the results.