Stability of difference approximations to differential equations

AbstractConsider the differential equation (1) ẋ = f(x) in a Banach space and let x∗ be an equilibrium. The basic question treated is the following: if x∗ is asymptotically stable and if (2) xk + 1 = xk + hϑ(xk, h) is a one-step method, with fixed step size h, for integrating (1), then does the sequence xk converge to x∗? It is shown that uniform asymptotic stability of (1) implies stability of (2) and that exponential asymptotic stability of (1) implies asymptotic stability of (2)