Periodic solutions for neutral functional differential equations with impulses on time scales

Abstract

Let $mathbb{T}$ be a periodic time scale. We use Krasnoselskii's fixed point theorem to show that the neutral functional differential equation with impulses $$displaylines{ x^{Delta}(t)=-A(t)x^sigma(t)+g^Delta(t,x(t-h(t)))+f(t,x(t),x(t-h(t))),quad teq t_j,;tinmathbb{T},cr x(t_j^+)= x(t_j^-)+I_j(x(t_j)), quad jin mathbb{Z}^+ }$$ has a periodic solution. Under a slightly more stringent conditions we show that the periodic solution is unique using the contraction mapping principle.

Keywords

• Positive periodic solution
• neutral functional differential
• equations
• impulses
• Krasnoselskii fixed point
• time scales