A posteriori stopping rule for regularized fixed point iterations

Iteratively regularized fixed-point iteration scheme x n + 1 = x n - α n { F ( x n ) - f δ + ε n ( x n - x 0 ) } combined with the generalized discrepancy principle ∥ F ( x N ) - f δ ∥ 2 ⩽ τ δ < ∥ F ( x n ) - f δ ∥ 2 , 0 ⩽ n < N , τ > 1 , for solving nonlinear operator equation F ( x ) = f in a Hilbert space is studied in the paper. It is shown that if F is monotone and Lipschitz-continuous the sequence { N ( δ ) } is admissible, i.e. (1) lim δ → 0 ∥ x N ( δ ) - x * ∥ = 0 , where x * is a solution to F ( x ) = f .