On the existence of generalized solutions and the convergence of difference methods for nonlinear initial-value problems

In the present paper we prove new results for a general perturbation theory for nonlinear mappings between metric spaces. Using these results we are able to establish new principles for the treatment of nonlinear initial-value problems by difference methods. The main results are the characterization of the existence of discrete limits of sequences of mappings and the characterization of the existence of generalized solutions of nonlinear initial-value problems which are limits of solutions of difference equations. As conclusions one obtains generalizations of Lax's equivalence theorem for nonlinear and linear initial-value problems and a convergence theorem for a concrete hyperbolic equation.