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

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
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References:

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