Oscillation of third-order nonlinear delay difference equations

Abstract

The third order non linear difference equations of the form Δ(cnΔ(dnΔxn))+pnΔxn+1+qnf(xn−σ)=0 for n ≥ n0 are considered. Here {cn}, {dn}, {pn} and {qn} are sequences of positive real numbers for n0 ∈ N f is a continuous function such that f(u)u≥K>0 for u ≠ 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results. Where n0 ∈ N is a fixed integer, Δ denotes to forward difference operator Δxn = xn+1 xn and σ is a nonnegative integer.