Stability of nonlinear neutral mixed Liven-Nohel integro-dynamic equations on time scales

Abstract

   

Let $\mathbb{T}$ be a time scale not bounded below and above such that $%
t_{0}\in \mathbb{T}$. We provide new conditions and we use the fixed point
theorem to establish the stability results for the nonlinear neutral mixed
Levin-Nohel intego-dynamic equation
\begin{align*}
\varkappa ^{\Delta }(\mathfrak{t})& =-\sum_{k=1}^{\mathfrak{m}}\int_{%
\mathfrak{t}-r_{k}(\mathfrak{t})}^{\mathfrak{t}}\mathfrak{a}_{k}(\mathfrak{t}%
,\mathsf{s})\varkappa (\mathsf{s})\Delta \mathsf{s}-\sum_{k=1}^{\mathfrak{m}%
}\int_{\mathfrak{t}}^{\mathfrak{t}+\lambda _{k}\left( \mathfrak{t}\right)
}b_{k}\left( \mathfrak{t},\mathsf{s}\right) \varkappa \left( \mathsf{s}%
\right) \Delta s \\
& +g^{\Delta }\left( \mathfrak{t},\varkappa _{1}\left( \mathfrak{t}-r_{1}(%
\mathfrak{t})\right) ,...,\varkappa _{\mathfrak{m}}\left( \mathfrak{t}-r_{%
\mathfrak{m}}(\mathfrak{t})\right) \right) ,
\end{align*}%
where $x^{\Delta }$ is $\Delta $-derivative on $\mathbb{T}$. The results
obtained here extend the work of Bessioud, Ardjouni, Djoudi \cite{k}.