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

Asymptotic decay of nonoscillatory solutions of general nonlinear difference equations

E. Thandapani,1 S. Lourdu Marian,1 and John R. Graef2

1Department of Mathematics, Periyar University, Salem 636011, Tamil Nadu, India
2Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga 37403, TN, USA

Received 10 November 2000

Copyright © 2001 E. Thandapani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.