Asymptotic decay of nonoscillatory solutions of general nonlinear difference equations

Abstract

The authors consider the mth order nonlinear difference equations of the form Dmyn+qnf(yσ(n))=ei, where m≥1, n∈ℕ={0,1,2,…}, ani>0 for i=1,2,…,m−1, anm≡1, D0yn=yn, Diyn=aniΔDi−1yn, i=1,2,…,m, σ(n)→∞ as n→∞, and f:ℝ→ℝ is continuous with uf(u)>0 for u≠0. They give sufficient conditions to ensure that all bounded nonoscillatory solutions tend to zero as n→∞ without assuming that ∑n=0∞1/ani=∞, i=1,2,…,m−1, {qn} is positive, or en≡0 as is often required. If {qn} is positive, they prove another such result for all nonoscillatory solutions.