On the convergence of multistep methods for the Cauchy problem for ordinary differential equations

The general form of a quasilinear nonstationary k-step method for solving of the Cauchy problem for ordinary differential equations is discussed. The convergence theorem is stated under rather weak conditions. It is not assumed that the increment function is Lipschitz-continuous but only that it satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function. The result established in the paper is an extension of the theory given by G. Dahlquist and the recent result of K. Taubert.