Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. (English) Zbl 1254.34128

Ardjouni, Abdelouaheb; Djoudi, Ahcene

Commun. Nonlinear Sci. Numer. Simul. 17, No. 7, 3061-3069 (2012).

Summary: Let \(\mathbb{T}\) be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay \[ x^\Delta(t)= -a(t) x^3(\sigma(t))+ c(t) x^{\widetilde\Delta}(t- r(t))+ G(t, x^3(t), x^3(t- r(t))),\quad t\in\mathbb{T}, \] where \(f^\Delta\) is the \(\Delta\)-derivative on \(\mathbb{T}\) and \(f^{\widetilde\Delta}\) is the \(\Delta\)-derivative on \((\mathrm{id}-r)(\mathbb{T})\). We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the Burton-Krasnoselskii’s theorem.

MSC:

34N05	Dynamic equations on time scales or measure chains
34K13	Periodic solutions to functional-differential equations
34K40	Neutral functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations

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