Oscillation criteria for two-dimensional system of non-linear ordinary differential equations

摘要

New oscillation criteria are established for the system of non-linear equations $$ u'=g(t)|v|^{rac{1}{lpha}}mathrm{sgn},v,qquad v'=-p(t)|u|^{lpha}mathrm{sgn},u, $$ where $lpha>0$, $g:[0,+infty[{}ightarrow[0,+infty[ $, and $p:[0,+infty[{}ightarrow mathbb{R}$ are locally integrable functions. Moreover, we assume that the coefficient $g$ is non-integrable on $[0,+infty]$. Among others, presented oscillatory criteria generalize well-known results of E. Hille and Z. Nehari and complement analogy of Hartman–Wintner theorem for the considered system.