Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

Abstract

We present new oscillation criteria for the differential equation of the form ${\left[r\left(t\right)U\left(t\right)\right]}^{\prime }+p\left(t\right)k_2\left(x\left(t\right),\hspace{0.17em}{x}^{\prime }\left(t\right)\right){\left|x\left(t\right)\right|}^{\nu }$$U\left(t\right)+q\left(t\right)\varphi \left(x(g_1\left(t\right)\right),x^{\prime }\left(g_2\left(t\right))\right)f\left(x\left(t\right)\right)=0$, where $U\left(t\right)=k_1\left(x\left(t\right),x^{\prime }\left(t\right)\right){\left|x^{\prime }\left(t\right)\right|}^{\alpha -1}{x}^{\prime }\left(t\right)$, $\alpha \le \beta ,\hspace{0.17em}\hspace{0.17em}\nu =\left(\beta -\alpha \right)/\left(\alpha +1\right)$. Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking $\beta =\alpha$, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.