Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)In this work we study the well-posedness for the initial value problem associated to a generalized derivative Schrodinger equation for small size initial data in weighted Sobolev space. The techniques used include parabolic regularization method combined with sharp linear estimates. An important point in our work is that the contraction principle is likely to fail but gives us inspiration to obtain certain uniform estimates that are crucial to obtain the main result. To prove such uniform estimates we assume smallness on the initial data in weighted Sobolev spaces. (C) 2015 Elsevier Inc. All rights reserved.2595 20302060Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)