Oscillatory and asymptotic behavior of solutions for second-order mixed nonlinear integro-dynamic equations with maxima on time scales

Abstract

This paper is concerned with the oscillatory and asymptotic behavior for solutions of the following second-order mixed nonlinear integro-dynamic equations with maxima on time scales
\begin{equation*}
(r(t)(z^\Delta(t))^\gamma)^\Delta+\int\limits_{0}^{t}a(t,s)f( s, x(s))\Delta s+\sum_{i=1}^{n}q_{i}(t) \max_{s\in [\tau_{i}(t), \xi_{i}(t)]}x^{\alpha}(s)=0,
\end{equation*}
where
\begin{equation*}
z(t)=x(t)+p_1(t)x(\eta_1(t))+p_2(t)x(\eta_2(t)).
\end{equation*}
The oscillatory behavior of this equation hasn't been discussed before, also our results improve and extend some results established by Grace et al. \cite{diir8} and \cite{diir7}.% We also give some examples to illustrate our main results.