Expanding the applicability of Lavrentiev regularization methods for ill-posed equations under general source condition (Q2923278)

This is another paper on the analysis of Lavrentiev regularization for the stable approximate solution of nonlinear ill-posed operator equations \(F(x)=y\) in the Hilbert space \(X\) with monotone forward operators \(F\). It is a remarkable fact that, in the period from 2011 to 2014, the first author together with coauthors has published 14 papers on various problems in the same journal `Numerical Functional Analysis and Applications'. As already done by \textit{P. Mahale} and \textit{M. T. Nair} [``Lavrentiev regularization of nonlinear ill-posed equations under general source condition'', J. Nonlinear Anal. Optim. 4, No. 2, 193--204 (2013)], the authors have formulated error estimates leading to convergence rates when general source conditions are supposed, i.e., the solution \(x_0\) shifted by some initial guess \(\hat x\) belongs to the range of \(\varphi(F^\prime(x_0))\) with some (in general concave) index functions \(\varphi\) and a sufficiently small source element. The focus here is preferably on one detail concerning the usually used nonlinearity condition NEWLINE\[NEWLINE[F^\prime(x_0)-F^\prime(u_\theta)]v=F^\prime(u)P(x,u,v) \;\; \text{with} \;\; \|P(x,u,v)\| \leq K\|v\| \|x-u\|, NEWLINE\]NEWLINE valid for all \(v \in X\) and \(x,u\) in a neighbourhood of \(\hat x\) with some constant \(K>0\) and elements \(P(x,u,v) \in X\). Such condition was used by Mahale and Nair [loc. cit.] and in a similar way also by \textit{U. Tautenhahn} [Inverse Probl. 18, No. 1, 191--207 (2002; Zbl 1005.65058)]. Now this condition can be weakened to the pair of conditions NEWLINE\[NEWLINE[F^\prime(x_0)-F^\prime(u_\theta)]v=F^\prime(u_\theta)\phi(x_0,u_\theta,v) \quad \text{with} \quad \phi(x_0,u_\theta,v) \in X NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|\phi(x_0,u_\theta,v)\| \leq K_0\|v\| \|x_0-u_\theta\|, NEWLINE\]NEWLINE valid for all \(u_\theta\) from a ball around \(x_0\) and \(v \in X\), in the sense that there exists \(K_0>0\) such that, for all \(u_\theta=u+\theta(x_0-u)\) from that ball and for \(\theta \in [0,1]\), there is an element \(\phi(x_0,u_\theta,v) \in X\) satisfying both conditions. At the end of the paper, examples (partially referring only to \(X=\mathbb{R}\)) are given in order to show the utility of the new nonlinearity condition.