Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale

Ardjouni, Abdelouaheb ; Djoudi, Ahcène

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 1, pp. 5-19.

Résumé

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 $x^{\triangle }\left( t\right) =-a\left( t\right) h\left( x^{\sigma }\left( t\right) \right) +c(t)x^{\widetilde{\triangle }}\left( t-r\left( t\right) \right) +G\left( t,x\left( t\right) ,x\left( t-r\left( t\right) \right) \right)$, $t\in \mathbb {T}$, where $f^{\triangle }$ is the $\triangle $-derivative on $\mathbb {T}$ and $f^{\widetilde{\triangle }}$ is the $\triangle $-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 Yankson [Yankson, E.: Existence of periodic solutions for totally nonlinear neutral differential equations with functional delay Opuscula Mathematica 32, 3 (2012), 617–627.].

Détail

Classification : 06E30, 34K13, 34K30, 34L30
Keywords: fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations