Regular half-linear second order differential equations. (English) Zbl 1119.34029

Došlý, Ondřej; Řezníčková, Jana

Arch. Math., Brno 39, No. 3, 233-245 (2003).

The authors investigate principal solutions of half-linear second order differential equations (=HLDE) of the form
\[ (r(t)\Phi (x'))'+c(t)\Phi (x)=0, \]
where \(\Phi (x)=| x| ^{p-2}x\) with \(p>1\). The concept of a regular HLDE is introduced, which enables to prove that the divergence of the integral \[ \int ^\infty \frac {\text dt}{r(t)x^2(t)| x'(t)| ^{p-2}} \] is necessary and sufficient for a (nonoscillatory) solution \(x\) to be principal, provided HLDE is regular and \(x'(t)\neq 0\). Sufficient conditions are given which guarantee that HLDE is regular and some open problems are posed.

Reviewer: Pavel Řehák (Brno)

MSC:

34C10

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

Keywords:

principal solution; Picone’s identity; Riccati-type equation