BIT 24 (1984), 667-680 ON NONLINEAR DIFFERENCE AND DIFFERENTIAL EQUATIONS GUSTAF SODERLIND Department of Numerical Analysis, Royal Institute of Technolooy, S-10044 Stockholm 70, Sweden Dedicated to my teacher and friend, Professor Germund Dahlquist, on the occasion of his 60th birthday. Abstract. A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results. The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems. 1. Introduction. We shall develop some useful tools for the analysis of the error propagation in nonlinear differential systems. The aim is to introduce a simple unified formalism of equal importance for the analysis of the properties of the differential equation and discretizations thereof. To tl~is end, and to draw attention to the full analogy between difference equations On the one hand and differential equations on the other, we shall consider the following six initial value problems : (LAA) Xn+ 1 = AX n (LAD) 5c = Ax (LNA) x.+l = Anx. (LND) ± = A(t)x (NLA) x,+l = f(x,) (NLD) 5¢ = f(x) Here A stands for difference equation, D for differential equation, and LA, LN, NL for linear autonomous, linear nonautonomous and nonlinear, respectively. We will, for simplicity, not deal with nonautonomous nonlinear problems; Received June 1983. RevgedMarch 1984.