In this thesis, we study the existence and uniqueness of weak solutions gov erned by first order nonlinear evolution equations on Banach spaces of the form where A is a nonlinear operator, EC(1,V), feL(1,V*) and V in a Banach space with dual V*. We split the proof of the existence of solutions into three steps. In Step 1: we derived a priori estimates for the solutions leading to the iden tification of an appropriate spaces to the solution. In Step 2: we constructed an approximate solutions by using Galerkin's method which converts an infinite dimensional problem to finite one. In Step 3: we derived a priori estimates for the approximed solution in similar way as the first step and have shown that the limit of the approximated solution converges to the solution of the original problem. For illustration, we worked out in details an example modelling a reaction diffusion equation,