Journal of Applied Mathematics
Volume 2012 (2012), Article ID 516476, 28 pages
http://dx.doi.org/10.1155/2012/516476
Research Article

Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Yongkun Li and Chao Wang

Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China

Received 20 October 2011; Revised 19 December 2011; Accepted 25 December 2011

Academic Editor: Laurent Gosse

Copyright © 2012 Yongkun Li and Chao Wang. 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

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale ∑ ( 𝑑 / 𝑑 𝑡 ) ( 𝑥 ( 𝑡 ) + 𝑐 ( 𝑡 ) 𝑥 ( 𝑡 − 𝛼 ) ) = 𝑎 ( 𝑡 ) 𝑔 ( 𝑥 ( 𝑡 ) ) 𝑥 ( 𝑡 ) − 𝑛 𝑗 = 1 𝜆 𝑗 𝑓 𝑗 ( 𝑡 , 𝑥 ( 𝑡 − 𝑣 𝑗 ( 𝑡 ) ) ) , ( 𝑡 , 𝑥 ) ∈ 𝕋 0 ( 𝑥 ) , Δ 𝑡 | ( 𝑡 , 𝑥 ) ∈ 𝒮 2 𝑖 = Π 1 𝑖 ( 𝑡 , 𝑥 ) − 𝑡 , Δ 𝑥 | ( 𝑡 , 𝑥 ) ∈ 𝒮 2 𝑖 = Π 2 𝑖 ( 𝑡 , 𝑥 ) − 𝑥 , where Π 1 𝑖 ( 𝑡 , 𝑥 ) = 𝑡 2 𝑖 + 1 + 𝜏 2 𝑖 + 1 ( Π 2 𝑖 ( 𝑡 , 𝑥 ) ) and Π 2 𝑖 ( 𝑡 , 𝑥 ) = 𝐵 𝑖 𝑥 + 𝐽 𝑖 ( 𝑥 ) + 𝑥 , 𝑖 = 1 , 2 , … . 𝜆 𝑗 ( 𝑗 = 1 , 2 , … , 𝑛 ) are parameters, 𝕋 0 ( 𝑥 ) is a variable time scale with ( 𝜔 , 𝑝 ) -property, 𝑐 ( 𝑡 ) , 𝑎 ( 𝑡 ) , 𝑣 𝑗 ( 𝑡 ) , and 𝑓 𝑗 ( 𝑡 , 𝑥 ) ( 𝑗 = 1 , 2 , … , 𝑛 ) are 𝜔 -periodic functions of 𝑡 , 𝐵 𝑖 + 𝑝 = 𝐵 𝑖 , 𝐽 𝑖 + 𝑝 ( 𝑥 ) = 𝐽 𝑖 ( 𝑥 ) uniformly with respect to 𝑖 ∈ ℤ .