Teka Gidreta2024-04-052024-04-052021-08This project work is mainly concerned with the question of the existence and unique ness solution of the IVP in linear first order ODE’s in Banach space.That is dydx =f(x, y), y(x0) = y0,where f is Lipschitz continuous function. So that to show the exist tenceand uniqueness solution of IVP of ODE’s we will use Picard’s Theorem and iter ation method.Firstly, state and prove Banach- Cacciopoli theorem that has been appliedto prove the Picard’s Existence and Uniqueness Theorem.This theorem also provides aconstructive procedures(called iteration) by which to get a better approximation to thesolution of ODE.enThesis