Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale. (English) Zbl 1290.34109

Ardjouni, Abdelouaheb; Djoudi, Ahcène

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 52, No. 1, 5-19 (2013).

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 \(x^{\Delta}(t) =-a(t) h(x^{\sigma}(t)) +c(t)x^{\tilde{\Delta}}(t-r) +G(t,x(t),x(t-r(t)))\), \(t \in \mathbb {T}\), where \(f^{\Delta}\) is the \(\Delta\)-derivative on \(\mathbb {T}\) and \(f^{\tilde{\Delta}}\) is the \(\Delta\)-derivative on (\(id-r)(\mathbb T)\). We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show that such maps fit very nicely into the framework of Krasnoselskii-Burton’s fixed point theorem so that the existence of periodic solutions is concluded. The results obtained here extend the work of E. Yankson [Opusc. Math. 32, No. 3, 617–627 (2012; Zbl 1248.34105)].

MSC:

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

Keywords:

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

Citations:

Zbl 1248.34105

References:

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