The existence of positive solutions and oscillation of solutions of higher-order difference equations with forcing terms

Sufficient conditions are established for the existence of positive solutions and oscillation of all bounded solutions of the neutral difference equation Δp[xn−cxn−1]+qnf(xn−k)=hn,n≥n0 where Δ is the forward difference operator Δxn = xn+1 − xn, l and k are integers, c ≠ ± 1 is a real number, and “qn” and “hn” are real sequences. It is also shown that some of these sufficient conditions are necessary.