Geodesic
Regular half-linear second order differential equations
Archivum mathematicum, Tome 39 (2003) no. 3, pp. 233-245
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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.
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.
Classification : 34C10
Keywords: regular half-linear equation; principal solution; Picone’s identity; Riccati-type equation 
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Došlý, Ondřej; Řezníčková, Jana. Regular half-linear second order differential equations. Archivum mathematicum, Tome 39 (2003) no. 3, pp. 233-245. http://geodesic.mathdoc.fr/item/ARM_2003_39_3_a8/

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