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International Journal of Differential Equations
Volume 2010 (2010), Article ID 598068, 14 pages
Review Article

Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

1Department of Mathematics, University of Gjirokastra, 6002 Gjirokastra, Albania
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 December 2009; Accepted 9 January 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 L. K. Kikina and I. P. Stavroulakis. 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.


Consider the second-order linear delay differential equation 𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))=0, 𝑡𝑡0, where 𝑝𝐶([𝑡0,),+), 𝜏𝐶([𝑡0,),), 𝜏(𝑡) is nondecreasing, 𝜏(𝑡)𝑡 for 𝑡𝑡0 and lim𝑡𝜏(𝑡)=, the (discrete analogue) second-order difference equation Δ2𝑥(𝑛)+𝑝(𝑛)𝑥(𝜏(𝑛))=0, where Δ𝑥(𝑛)=𝑥(𝑛+1)𝑥(𝑛), Δ2=ΔΔ, 𝑝+, 𝜏, 𝜏(𝑛)𝑛1, and lim𝑛𝜏(𝑛)=+, and the second-order functional equation 𝑥(𝑔(𝑡))=𝑃(𝑡)𝑥(𝑡)+𝑄(𝑡)𝑥(𝑔2(𝑡)), 𝑡𝑡0, where the functions 𝑃, 𝑄𝐶([𝑡0,),+), 𝑔𝐶([𝑡0,),), 𝑔(𝑡)𝑡 for 𝑡𝑡0, lim𝑡𝑔(𝑡)=, and 𝑔2 denotes the 2th iterate of the function 𝑔, that is, 𝑔0(𝑡)=𝑡, 𝑔2(𝑡)=𝑔(𝑔(𝑡)), 𝑡𝑡0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where liminf𝑡𝑡𝜏(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠1/𝑒 and limsup𝑡𝑡𝜏(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠<1 for the second-order linear delay differential equation, and 0<liminf𝑡{𝑄(𝑡)𝑃(𝑔(𝑡))}1/4 and limsup𝑡{𝑄(𝑡)𝑃(𝑔(𝑡))}<1, for the second-order functional equation, are presented.