A note on the existence and construction of Dulac functions

We provide sucient conditions on the components of a vector eld, which en sure theexistence of Dulac functions depending on special functions for such vector eld. Wealso present some applications and examples in order to illustrate our results.Keywords: Bendixson-Dulac criterion, Dulac functions, limit cycles2000, Classications numbers AMS . 34A34, 34C25.1. IntroductionMany problems of the qualitative theory of dierential equations in the plane arerelated to closed orbits, this fact motivates their study. But deciding whether an arbi-trary dierential equation has periodic orbits or not is a dicult question that remainsopen.There are some criteria that allow us to rule out the existence of periodic orbitsin the plane; between them, we will take particular interest in studying the BendixsonDulac criterion. It is well known that Bendixson-Dulac criterion is a very useful toolfor investigation of limit cycles (see [1], [2], [3], [5]).For convenience, we recall the Bendixson-Dulac criterion,see ([4], pag. 137).Theorem 1.1. (Bendixson-Dulac criterion)Let f