Interval Oscillation Criteria for Second-Order Damped Differential Equations with Mixed Nonlinearities

Abstract

In this paper, we consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where \(\gamma \) is a quotient of odd positive integers, \(\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n\) with \(r,\ p,\ e\), and \(q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}\) are nondecreasing continuous functions on \(\mathbb {R}\) and \(\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.\) Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.