DSM of Newton type for solving operator equations \(F(u) = f\) with minimal smoothness assumptions on \(F\)
Authors' abstract: This paper is a review of the authors' results on the dynamical systems method (DSM) for solving operator equation (*) \(F(u) = f\). It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of \(F\). It is assumed that \(F\) is continuously Fréchet differentiable, but no smoothness assumptions on \(F^{\prime}(u)\) are imposed. The DSM for solving equation \((*)\) is developed. Under weak assumptions global existence of the solution \(u(t)\) is proved, the existence of \(u(\infty )\) is established, and the relation \(F(u(\infty )) = f\) is obtained. The DSM is developed for a stable solution of equation \((*)\) when noisy data \(f\delta \) are given, \(| f - f| \leq \delta \).
scientific article; zbMATH DE number 5771943