Hailu Bikila (PhD)Teka Gidreta2024-04-022024-04-022021-08This 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 dy dx = f(x, y), y(x0) = y0,where f is Lipschitz continuous function.So that to show the exis tence and 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 applied to prove the Picard’s Existence and Uniqueness Theorem.This theorem also provides a constructive procedures(called iteration) by which to get a better approximation to the solution of ODE.enThesis