Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale

Abdelouaheb Ardjounia,* and Ahcene Djoudib

Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale

Authors :

Abdelouaheb Ardjouni^a,* and Ahcene Djoudi^b

Author Address :

^a,bDepartment of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12 Annaba, Algeria.

*Corresponding author.

Abstract :

Let $\mathbb{T}$ be a periodic time scale. The purpose of this paper is to use Krasnoselskii’s fixed point theorem to prove the existence of positive periodic solutions on time scale of the nonlinear neutral dynamic equation with variable delay
\[
\left( x\left( t\right) -g\left( t,x\left( t-\tau\left( t\right)
\right) \right) \right) ^{\triangle}=r\left( t\right) x\left( t\right)
-f\left( t,x\left( t-\tau\left( t\right) \right) \right) .
\]
We invert this equation to construct a sum of a contraction and a compact map which is suitable for applying the Krasnoselskii’s theorem. The results obtained here extend the works of Raffoul [17] and Ardjouni and Djoudi [3].

Keywords :

Positive periodic solutions, nonlinear neutral dynamic equations, fixed point theorem, time scales.

DOI :

Article Info :

Received : December 25, 2012; Accepted : March 23, 2013.