Periodic solutions for a neutral nonlinear dynamical equation on a time scale. (English) Zbl 1096.34057

Kaufmann, Eric R.; Raffoul, Youssef N.

J. Math. Anal. Appl. 319, No. 1, 315-325 (2006).

Summary: Let \(\mathbb{T}\) be a periodic time scale. We use a fixed-point theorem due to Krasnosel’skiĭ to show that the nonlinear neutral dynamic system with delay \[ x^\Delta(t)=-a(t)x^\sigma(t)+c(t)x^\Delta(t-k)+q \bigl(t,x(t),x(t-k)\bigr),\;t \in\mathbb{T}, \] has a periodic solution. We assume that \(k\) is a fixed constant if \(\mathbb{T}=\mathbb{R}\) and is a multiple of the period of \(\mathbb{T}\) if \(\mathbb{T}\neq\mathbb{R}\). Under a slightly more stringent inequality, we show that the periodic solution is unique using the contraction mapping principle.

MSC:

34K40	Neutral functional-differential equations
34K13	Periodic solutions to functional-differential equations
39A12	Discrete version of topics in analysis

Keywords:

Krasnosel’skiĭ; Neutral; Nonlinear; Time scales; Periodic solution; Unique solution

References:

[1]	Bohner, M.; Peterson, A., Dynamic Equations on Time Scales, An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[2]	Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[4]	Raffoul, Y. N., Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations, 102, 1-7 (2003) · Zbl 1054.34115
[6]	Smart, D. R., Fixed Points Theorems (1980), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0427.47036