Oscillation theorems for nonlinear second-order differential equations
The authors study the second-order nonlinear differential equation \[ [r(t)\Phi(x(t))\varphi\bigl(x'(t)\bigr)]' +c(t)\varphi(x(t))=0\tag{1} \] on the interval \([t_0,\infty)\). The functions \(r:[t_0,\infty)\to(0,\infty)\), \(c:[t_0,\infty)\to \mathbb{R}\) and \(\Phi:\mathbb{R}\to(0,\gamma]\) are supposed to be continuous, \(\gamma\) is a real number. The function \(\varphi\) is defined as a generalized power function \(\varphi(x)=| x| ^{p-2}x\) for fixed real \(p>1\). The authors study the proper solutions of equation (1) and by using the Riccati-type transformation and integral averaging technique with the averaging \(H\)-function (originally due to \textit{Ch. G. Philos} [Arch. Math. 53, 482--492 (1989; Zbl 0661.34030)]), they prove integral oscillation criteria for equation (1).
scientific article; zbMATH DE number 2116486
scientific article