Regular half-linear second order differential equations

Došlý, Ondřej ; Řezníčková, Jana

Archivum mathematicum, Tome 39 (2003) no. 3, pp. 233-245.

Résumé

We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation \[ \left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0\,,\quad \Phi (x):=|x|^{p-2}x\,,\quad p>1 \qquad \mathrm {{(*)}}\] and we show that if (*) is regular, a solution $x$ of this equation such that $x^{\prime }(t)\ne 0$ for large $t$ is principal if and only if \[ \int ^\infty \frac{dt}{r(t)x^2(t)|x^{\prime }(t)|^{p-2}}=\infty \,. \] Conditions on the functions $r,c$ are given which guarantee that (*) is regular.

Détail

Classification : 34C10
Keywords: regular half-linear equation; principal solution; Picone’s identity; Riccati-type equation