International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 2, Pages 69-76
doi:10.1155/S0161171201006172

Asymptotic decay of nonoscillatory solution of general nonlinear difference equations

E. Thandapani1 , S.Lourdu Marian1 and John R. Graef3

1Department of Mathematics, Periyar University, Salem 636011, Tamil Nadu, India
 3Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga 37403, TN, USA

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.