The index function and Tikhonov regularization for ill-posed problems

In this paper, we study the regularizing properties of the conditional stability estimates in ill-posed problems. First, we analyze how conditional stability estimates occur, and which properties the corresponding index functions must obey. In addition, we adapt the convergence analysis for the Tikhonov regularization in Banach spaces where the difference between the approximated solution and the exact one in metric measure is taken into account. We conclude this study with a comparison of stability estimates and variational inequalities, another emerging tool in Banach space regularization.