On certain minimizing the discrepancy method for operator regularization of equations of first kind (Q1902804)

The author studies an approximation scheme for the nonlinear operator equation \[ F_{(u)}= f_0,\quad u\in D(F)\tag{1} \] with \(f_0\in H\), \(F: D(F)\subset H\to H\) being a maximal monotone operator on a real Hilbert space. For this he considers a regularized equation \[ F(u)+ \emptyset(\alpha u)= f_\delta,\quad u\in D(F),\quad \alpha> 0, \] where \(\emptyset\) is a smooth strongly convex functional on \(H\), and \(f_\delta\) is such that \[ |f_0- f_\delta|\leq \delta,\quad \delta\geq 0. \] The results obtained are interpreted in connection with certain variational problems arising in plasticity theory.