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

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
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