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International Journal of Differential Equations
Volume 2010 (2010), Article ID 596350, 15 pages
doi:10.1155/2010/596350
Research Article

Oscillatory Nonautonomous Lucas Sequences

José M. Ferreira1 and Sandra Pinelas2

1Department of Mathematics, Instituto Superior Técnico, Avenue Rovisco Pais, 1049-001 Lisboa, Portugal
2Department of Mathematics, Universidade dos Açores, R. Mãe de Deus, 9500-321 Ponta Delgada, Portugal

Received 18 September 2009; Accepted 3 December 2009

Academic Editor: Elena Braverman

Copyright © 2010 José M. Ferreira and Sandra Pinelas. 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 oscillatory behavior of the solutions of the second-order linear nonautonomous equation 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) 𝑥 ( 𝑛 ) − 𝑏 ( 𝑛 ) 𝑥 ( 𝑛 − 1 ) , 𝑛 ∈ ℕ 0 , where 𝑎 , 𝑏 ∶ ℕ 0 → ℝ , is studied. Under the assumption that the sequence 𝑏 ( 𝑛 ) dominates somehow 𝑎 ( 𝑛 ) , the amplitude of the oscillations and the asymptotic behavior of its solutions are also analized.