AbstractWe consider the evolution equationu'=Au+Nuu(0) = ϕuis a function on a real interval [0,T] with values in a Hilbert spaceH, Ais a linear operator inHgenerating a continuous semigroupeAtandNis a nonlinear operator inH.We show that1Existence of the exact solution implies existence of the Faedo‐Galerkin Approximations.2Existence of the Faedo‐Galerkin Approximations implies existence of the exact solution.3Uniform convergence of the Faedo‐Galerkin Approximations to the exact solution.The Paper consists of two parts. In the first five sections we require thatApossesses a complete orthonormal system of eigenfunctions, in section 6 we drop this requi