On the convexity of the Tikhonov functional and iteratively regularized methods for solving irregular nonlinear operator equations

Domains in a Hilbert space are localized where the Tikhonov functional of an irregular nonlinear operator equation is either strongly convex or has other similar properties. Depending on the sourcewise representability conditions imposed on the solution, four such domains are detected, and their size is estimated. These results are used to substantiate the general scheme for the design of nonlocal two-level iterative processes for solving irregular equations.