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Limit and integral properties of principal solutions for half-linear differential equations

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Mariella Cecchi, Zuzana Dosla and Mauro Marini
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** Address.**
Department of Electronics and Telecommunications, University of Florence, Via S. Marta 3, 50139 Florence, Italy

Department of Mathematics, Masaryk University, Janackovo nam. 2a, 602 00 Brno, Czech Republic

Department of Electronics and Telecommunication, University of Florence, Via S. Marta 3, 50139 Florence, Italy

** E-mail:**
mariella.cecchi@unifi.it

dosla@math.muni.cz

mauro.marini@unifi.it

**Abstract.**
Some asymptotic properties of principal solutions of the half-linear differential equation \begin{equation}
(a(t)\Phi(x^{\prime}))^{\prime}+b(t)\Phi(x)=0\,, \tag{*}\end{equation} $\Phi(u)=|u|^{p-2}u$, $p>1$, is the
$p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of \eqref{H}, which completes previous results, are presented as well.

**AMSclassification. **34C10, 34C11.

**Keywords. **Half-linear equation, principal solution, limit characterization, integral characterization.