The present work leads to the study of the families of existence and uniqueness of bounded linear operators defined on Banach spaces as a tool to handle the abstract Cauchy problem. In this work a more general system of operators is presented, the family  of existence for a linear operator , of bounded linear operators of a Banach space  (can be different from ) in  as a new tool to deal with pProblem Cauchy abstract. This idea is somewhat different from the traditional one. Its companion family of uniqueness will be also studied.