Abstract and Applied Analysis
Volume 2011 (2011), Article ID 591254, 34 pages
doi:10.1155/2011/591254
Research Article

Nonoscillation of Second-Order Dynamic Equations with Several Delays

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada
2Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 30 December 2010; Accepted 13 February 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Elena Braverman and Başak Karpuz. 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

Existence of nonoscillatory solutions for the second-order dynamic equation ( 𝐴 0 𝑥 Δ ) Δ ( 𝑡 ) + 𝑖 [ 1 , 𝑛 ] 𝐴 𝑖 ( 𝑡 ) 𝑥 ( 𝛼 𝑖 ( 𝑡 ) ) = 0 for 𝑡 [ 𝑡 0 , ) 𝕋 is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case 𝐴 0 ( 𝑡 ) 1 for 𝑡 [ 𝑡 0 , ) and for second-order nondelay difference equations ( 𝛼 𝑖 ( 𝑡 ) = 𝑡 + 1 for 𝑡 [ 𝑡 0 , ) ). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary 𝐴 0 and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.