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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 707319, 17 pages
http://dx.doi.org/10.1155/2012/707319
Research Article

Periodic Solutions in Shifts 𝜹 ± for a Nonlinear Dynamic Equation on Time Scales

Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey

Received 16 March 2012; Accepted 27 June 2012

Academic Editor: Elena Braverman

Copyright © 2012 Erbil Çetin and F. Serap Topal. 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

Let 𝕋 be a periodic time scale in shifts 𝛿 ± . We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form 𝑥 Δ ( 𝑡 ) = 𝑎 ( 𝑡 ) 𝑥 𝜎 ( 𝑡 ) + 𝑏 ( 𝑡 ) 𝑥 Δ ( 𝛿 ( 𝑘 , 𝑡 ) ) 𝛿 Δ ( 𝑘 , 𝑡 ) + 𝑞 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝑥 ( 𝛿 ( 𝑘 , 𝑡 ) ) ) , 𝑡 𝕋 , has a periodic solution in shifts 𝛿 ± . We extend and unify periodic differential, difference, -difference, and 𝑞 -difference equations and more by a new periodicity concept on time scales.