Oscillation results for second-order mixed neutral integro-dynamic equations with damping and a nonpositive neutral term on time scales

Abstract

In this work, we are concerned with studying a new class of second-order mixed neutral integro-dynamic equation with damping and a nonpositive neutral term of the form: \begin{equation}\label{h1} (r(t)(z^\Delta(t))^\gamma)^\Delta+ p(t)(z^\Delta(t))^\gamma+ g(t, x(\tau(t)))+\int\limits_{0}^{t}a(t,s)f( s, x(s))\Delta s=0, \end{equation} where \begin{equation}\label{h2} z(t)=x(t)-p_1(t)x(\eta_1(t))+p_2(t)x(\eta_2(t)), \end{equation} on a time scale $\mathbb{T}$. The obtained results not only present some new criteria for such kind of neutral differential equations and neutral difference equations as special cases, but also extend some results obtained on time scales. An example is given to illustrate the importance of our work.