Abstract. The paper is devoted to the theoretical aspects of constructing a family of single-stage multi-step methods for solving the Cauchy problem with prehistory for ordinary differential equations. The paper includes general issues related to discretization, approximation, convergence, and stability. The problem of improving the accuracy of numerical solutions is analyzed in detail. Many of the results presented in the paper go beyond its title and are suitable for the numerical solution of partial differential equations.

Keywords: ordinary differential equations, one-stage multi-step method, discretization, approximation, convergence, strong stability.