2014-07-292010-09-012010-03-29SOUZA, Wender José de. A injectividade condition and the global asymptotic estability on the plane. 2010. 99 f. Dissertação (Mestrado em Ciências Exatas e da Terra) - Universidade Federal de Goiás, Goiânia, 2010.http://repositorio.bc.ufg.br/tede/handle/tde/1926In this work we are interested in the solution of the following problem: Let Y = ( f ,g) be a vector field of class C1 in R2. Suppose that (x, y) = (0,0) is a singular point of Y and assume that for any q &#8712; R2, the eigenvalues of DY have negative real part, this is, det(DY) > 0 and tr(DY) < 0. Then, the solution (x, y) = (0,0) of Y is globally asymptotically stable. To this end, we show that this problema is equivalent to the following: Let Y : R2 &#8594;R2 be a C1 vector field. If det(DY) > 0 and tr(DY) < 0, then Y is globally injective. This equivalence was proved by C. Olech [1]. So we show the injectivity of the vector field Y under the conditions det(DY) > 0 and tr(DY)<0. In fact, we present a more stronger result, which was obtained by C. Gutierrez and can be found in [4]. This result is given by: Any planar vector field X of class C2 satisfying the r-eigenvalue condition for some r &#8712; [0,¥) is injective.application/pdfAcesso AbertoEstabilidade globalAtrator globalConjectura jacobianaComponentes ReebCondições de injetividadeGlobal estabilityGlobal attractorJacobian conjectureReeb componentInjective conditionEstabilidade global; Atrator global; Conjectura jacobiana; Componentes Reeb; Condições de injetividadeCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIADissertação