Continuous regularized Gauss-Newton-type algorithm for nonlinear ill-posed equations with simultaneous updates of inverse derivative

A new continuous regularized Gauss-Newton-type method with simultaneous updates of the operator $(F^{\pr*}(x(t))F'(x(t))+\ep(t) I)^{-1}$ for solving nonlinear ill-posed equations in a Hilbert space is proposed. A convergence theorem is proved. An attractive and novel feature of the proposed method is the absence of the assumptions about the location of the spectrum of the operator $F'(x)$. The absence of such assumptions is made possible by a source-type condition.