Asymptotic decay of nonoscillatory solutions of general nonlinear difference equations

The authors consider themth order nonlinear difference equations of the formDmyn+qnf(yσ(n))=ei, wherem≥1,n∈ℕ={0,1,2,…},ani>0fori=1,2,…,m−1,anm≡1,D0yn=yn,Diyn=aniΔDi−1yn,i=1,2,…,m,σ(n)→∞asn→∞, andf:ℝ→ℝis continuous withuf(u)>0foru≠0. They give sufficient conditions to ensure that all bounded nonoscillatory solutions tend to zero asn→∞without assuming that∑n=0∞1/ani=∞,i=1,2,…,m−1,{qn}is positive, oren≡0as is often required. If{qn}is positive, they prove another such result for all nonoscillatory solutions.