Integral characterization of the principal solution to half-linear second-order differential equations. (English) Zbl 1012.34029

Došlý, O.; Elbert, Á.

Stud. Sci. Math. Hung. 36, No. 3-4, 455-469 (2000).

Summary: Under some restrictions on the functions \(r(t)\) and \(c(t)\), the nontrivial solution \(x(t)\) to the nonoscillatory half-linear second-order differential equation \[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0, \qquad \Phi(s):= s ^{p-2}s\quad p>1,\;t\geq \alpha, \] is principal if and only if \[ \int^\infty \frac{dt}{r(t)x^2(t) x'(t) ^{p-2}}=\infty. \tag{\(*\)} \] In case \(p = 2\), the criterion \((*)\) reduces to a Hartman result on the principal solutions to the Sturm-Liouville differential equations.

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

principal solution; half-linear; reciprocal equation; Picone’s identity