Continuous dependence on modeling for nonlinear ill-posed problems

The nonlinear ill-posed Cauchy problem du/dt = Au(t) + h(t, u(t)), u(0) = x, where A is a positive self-adjoint operator on a Hilbert space H, chi epsilon H, and h : vertical bar 0, T) x H -> H is a uniformly Lipschitz function, is studied in order to establish continuous dependence results for solutions to approximate well-posed problems. The authors show here that solutions of the problem, if they exist, depend continuously on solutions to corresponding approximate well-posed problems, if certain stabilizing conditions are imposed. The approximate problem is given by dv/dt = f(A)v(t) + h(t. v(t)), v(O) = chi, for suitable functions f. The main result is that parallel to u(t) - v(t)parallel to <= C beta(1-1/TM 1/T), where C and M are computable constants independent of beta and 0 < beta < 1. This work extends to the nonlinear case earlier resufts by the authors and by Ames and Hughes. (c) 2008 Elsevier Inc. All rights reserved.