Manel Gouasmia, Abdelouaheb Ardjouni, Ahcene Djoudi
abstract:
Let $\mathbb{T}$ be an unbounded above and below time scale such that $0\in
\mathbb{T}$. Let $id-\tau :[ 0,\infty)\cap \mathbb{T} $ be such that
$(id-\tau)([0,\infty)\cap\mathbb{T})$ is a time scale. We use
Krasnoselskii-Burton's fixed point theorem to obtain stability results about the
zero solution for the following nonlinear neutral dynamic equation with a
variable delay:
\begin{equation*}
x^{\Delta }(t)=-a(t)h(x^{\sigma }(t))+Q(t,x(t-\tau
(t)))^{\Delta}+G\big(t,x(t),x(t-\tau (t))\big).
\end{equation*}
The stability of the zero solution of this equation is provided by $h(0) =Q(
t,0) =G( t,0,0) =0$. The Carathéodory
condition is used for the functions $Q$ and $G$. The results obtained here
extend the work of Mesmouli, Ardjouni and Djoudi (Study of stability in
nonlinear neutral differential equations with variable delay using
Krasnoselskii-Burton's fixed point. Acta Univ. Palack. Olomuc. Fac. Rerum
Natur. Math. 55 (2016), no. 2, 129-142).
Mathematics Subject Classification: 34K20, 34K30, 34K40
Key words and phrases: Krasnoselskii-Burton's theorem, large contraction, neutral dynamic equation, integral equation, stability, time scales