Continuous dependence on modeling for nonlinear ill-posed problems

Abstract

The nonlinear ill-posed Cauchy problem d u d t = A u ( t ) + h ( t , u ( t ) ) , u ( 0 ) = χ , where A is a positive self-adjoint operator on a Hilbert space H , χ ∈ H , and h : [ 0 , T ) × 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 d v d t = f ( A ) v ( t ) + h ( t , v ( t ) ) , v ( 0 ) = χ , for suitable functions f. The main result is that ‖ u ( t ) − v ( t ) ‖ ⩽ C β 1 − t T M t T , where C and M are computable constants independent of β and 0 < β < 1 . This work extends to the nonlinear case earlier results by the authors and by Ames and Hughes.