Perturbation theory and the Banach–Steinhaus theorem for regularization of the linear equations of the first kind

The regularizing equations with a vector parameter of regularization are constructed for the linear equations with closed operator acting in Banach spaces. Range of the operator can be an open, and the homogeneous equation may have a non-trivial solution. It is assumed that only approximations of operator and source are known. The conditions of solution uniqueness for the auxiliary regularized equation are derived. The convergence of regularized solution to B-normal solution of the exact equation is proved. The bounds estimates are derived for both deterministic and stochastic cases. The choice of the stabilizing operator and vector regularization parameter are provided. The method is applied to the problem of stable differentiation.