Oscillatory Behavior of Second Oscillatory behavior of second order integro-dynamic equations with maxima and superlinear or sublinear neutral termsIntegro-Dynamic Equations With Maxima and Superlinear or Sublinear Neutral Terms

Abstract

In this work, we present some new criteria for the oscillatory and the asymptotic behavior of the solutions of the following second-order mixed nonlinear integro-dynamic equations with maxima and superlinear or sublinear neutral terms 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^{\lambda_{1}}(\eta_1(t))+p_2(t)x^{\lambda_{2}}(\eta_2(t)), t \in [0,+\infty)_\mathbb{T}.
\end{equation*}
The obtained results are new for both the discrete and continuous cases. Furthermore, our results extend known ones in the literature. An example is presented to illustrate the relevance of the results.