Electron. J. Diff. Equ., Vol. 2012 (2012), No. 57, pp. 1-14.

Periodic solutions for neutral functional differential equations with impulses on time scales

Yongkun Li, Xiaoyan Dou, Jianwen Zhou

Abstract:
Let $\mathbb{T}$ be a periodic time scale. We use Krasnoselskii's fixed point theorem to show that the neutral functional differential equation with impulses
$$\displaylines{ x^{\Delta}(t)=-A(t)x^\sigma(t)+g^\Delta(t,x(t-h(t))) +f(t,x(t),x(t-h(t))),\quad t\neq t_j,\;t\in\mathbb{T},\cr x(t_j^+)= x(t_j^-)+I_j(x(t_j)), \quad j\in \mathbb{Z}^+ }$$
has a periodic solution. Under a slightly more stringent conditions we show that the periodic solution is unique using the contraction mapping principle.

Submitted August 22, 2011 Published April 10, 2012.
Math Subject Classifications: 34N05, 34K13, 34K40, 34K45.
Key Words: Positive periodic solution; neutral functional differential equations; impulses; Krasnoselskii fixed point; time scales.

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Yongkun Li
Department of Mathematics, Yunnan University
Kunming, Yunnan 650091, China
email: yklie@ynu.edu.cn
Xiaoyan Dou
Department of Mathematics, Yunnan University
Kunming, Yunnan 650091, China
email: douxy21@163.com
Jianwen Zhou
Department of Mathematics, Yunnan University
Kunming, Yunnan 650091, China
email: zhoujianwen2007@126.com

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