We consider an abstract inverse Cauchy problem with a sectorial operator A in a Banach space. For solving the problem, a class of regularization methods is used and their rate of convergence is studied in terms of the regularization parameter α → 0+. It is shown that the methods provide an approximation error of order O((– ln α)−p) (p > 0) if and only if the initial discrepancy has a sourcewise representation with respect to the operator A−p.