Continuous dependence on modeling for nonlinear ill-posed problems
The authors study the nonlinear ill-posed Cauchy problem \[ \frac{du}{dt}=Au(t)+h(t,u(t)),\;\;u(0)=\varkappa, \] where \(A\) is a positive self-adjoint operator on a Hilbert space \(\mathcal{H}\) and \(h:[0,T)\times\mathcal{H}\to\mathcal{H}\) is a uniformly Lipschitz function with respect to both variables. They prove continuous dependence on modeling for this problem. If the solutions exist, they depend continuously on solutions to corresponding approximate well-posed problems \[ \frac{dv}{dt}=f(A)v(t)+h(t,v(t)),\;\;v(0)=\varkappa, \] where \(f\) is a real-valued function that approximates \(A\) in a suitable sense. The results are obtained by extending the solutions into the complex plane and introducing a related holomorphic function whose properties yield the Hölder continuous dependence.
scientific article; zbMATH DE number 5362167