Regularization methods for ill-posed inclusions and variational inequalities with domain perturbations (Q2720309)

The problems under consideration are the next. In a reflexive, smooth, strictly convex Banach space \(X\), having Efimov's property, there are defined a maximal monotone point-to-set operator \(A:X \to X^{\ast} \), a nonempty closed convex \(\Omega \subset \text{Dom}(A)\). Given \(f \in X^{\ast} \), NEWLINENEWLINENEWLINE1) Find \(x \in \Omega \) such that \(f \in Ax\), or NEWLINENEWLINENEWLINE2) Find \(x \in \Omega \) such that, for some \(y\in Ax\), \(\langle{y-f, x-z}\rangle\leq 0\), for all \(z \in \Omega \). NEWLINENEWLINENEWLINEBoth problems have nonempty solution sets on assumption and real computational problems are to approximate these solutions having some disturbed data \(({A^{h}, f^{\delta}})\) in the first problem and \(({A^{h}, f^{\delta}, \Omega _{\sigma}})\) in the second. The classical Tikhonov regularization method is investigated. Here some duality mapping \(J^{\mu} :X \to X^{\ast} \) (of weight \(\mu :R_{+}\to R_{_{ +} }\)) is introduced. In the first problem the disturbed operator \(A^{h}\) is replaced by \(A^{h} + \alpha J^{\mu} \) and in the second the vector \(y\) is replaced by \(y + \alpha J^{\mu} w\), \(w \in \Omega _{\sigma} \). The main generalization of similar previous investigations by \textit{U. Mosco} [Adv. Math. 3, 510-585 (1969; Zbl 0192.49101)], \textit{O. A. Liskovets} [Sov. Math. Dokl. 36, No. 2, 220-224 (1988; Zbl 0651.49002)] and the authors is assumption about disturbances of the sets \(\text{Dom}(A^h)\) and \(\Omega _{\sigma} \). Authors eliminate the previous limitations \(\text{Dom}(A^h)= \text{Dom}(A)\) and \(\Omega \subseteq \text{Dom}(A)\) but entered another less restrictive ones ``locally uniform boundedness conditions''. Under this each element of \(\text{Dom}(A)\) can be approximated by some vector of \(\text{Dom}(A^h)\). Two theorems about regularization property of the Tikhonov method are proved.