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International Journal of Differential Equations
Volume 2010 (2010), Article ID 598068, 14 pages
http://dx.doi.org/10.1155/2010/598068
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.

Abstract

Consider the second-order linear delay differential equation ๐‘ฅ ๎…ž ๎…ž ( ๐‘ก ) + ๐‘ ( ๐‘ก ) ๐‘ฅ ( ๐œ ( ๐‘ก ) ) = 0 , ๐‘ก โ‰ฅ ๐‘ก 0 , where ๐‘ โˆˆ ๐ถ ( [ ๐‘ก 0 , โˆž ) , โ„ + ) , ๐œ โˆˆ ๐ถ ( [ ๐‘ก 0 , โˆž ) , โ„ ) , ๐œ ( ๐‘ก ) is nondecreasing, ๐œ ( ๐‘ก ) โ‰ค ๐‘ก for ๐‘ก โ‰ฅ ๐‘ก 0 and l i m ๐‘ก โ†’ โˆž ๐œ ( ๐‘ก ) = โˆž , the (discrete analogue) second-order difference equation ฮ” 2 ๐‘ฅ ( ๐‘› ) + ๐‘ ( ๐‘› ) ๐‘ฅ ( ๐œ ( ๐‘› ) ) = 0 , where ฮ” ๐‘ฅ ( ๐‘› ) = ๐‘ฅ ( ๐‘› + 1 ) โˆ’ ๐‘ฅ ( ๐‘› ) , ฮ” 2 = ฮ” โˆ˜ ฮ” , ๐‘ โˆถ โ„• โ†’ โ„ + , ๐œ โˆถ โ„• โ†’ โ„• , ๐œ ( ๐‘› ) โ‰ค ๐‘› โˆ’ 1 , and l i m ๐‘› โ†’ โˆž ๐œ ( ๐‘› ) = + โˆž , and the second-order functional equation ๐‘ฅ ( ๐‘” ( ๐‘ก ) ) = ๐‘ƒ ( ๐‘ก ) ๐‘ฅ ( ๐‘ก ) + ๐‘„ ( ๐‘ก ) ๐‘ฅ ( ๐‘” 2 ( ๐‘ก ) ) , ๐‘ก โ‰ฅ ๐‘ก 0 , where the functions ๐‘ƒ , ๐‘„ โˆˆ ๐ถ ( [ ๐‘ก 0 , โˆž ) , โ„ + ) , ๐‘” โˆˆ ๐ถ ( [ ๐‘ก 0 , โˆž ) , โ„ ) , ๐‘” ( ๐‘ก ) โ‰ข ๐‘ก for ๐‘ก โ‰ฅ ๐‘ก 0 , l i m ๐‘ก โ†’ โˆž ๐‘” ( ๐‘ก ) = โˆž , and ๐‘” 2 denotes the 2th iterate of the function ๐‘” , that is, ๐‘” 0 ( ๐‘ก ) = ๐‘ก , ๐‘” 2 ( ๐‘ก ) = ๐‘” ( ๐‘” ( ๐‘ก ) ) , ๐‘ก โ‰ฅ ๐‘ก 0 . 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 l i m i n f ๐‘ก โ†’ โˆž โˆซ ๐‘ก ๐œ ( ๐‘ก ) ๐œ ( ๐‘  ) ๐‘ ( ๐‘  ) ๐‘‘ ๐‘  โ‰ค 1 / ๐‘’ and l i m s u p ๐‘ก โ†’ โˆž โˆซ ๐‘ก ๐œ ( ๐‘ก ) ๐œ ( ๐‘  ) ๐‘ ( ๐‘  ) ๐‘‘ ๐‘  < 1 for the second-order linear delay differential equation, and 0 < l i m i n f ๐‘ก โ†’ โˆž { ๐‘„ ( ๐‘ก ) ๐‘ƒ ( ๐‘” ( ๐‘ก ) ) } โ‰ค 1 / 4 and l i m s u p ๐‘ก โ†’ โˆž { ๐‘„ ( ๐‘ก ) ๐‘ƒ ( ๐‘” ( ๐‘ก ) ) } < 1 , for the second-order functional equation, are presented.