Abstract and Applied Analysis
Volume 1 (1996), Issue 1, Pages 45-64

Iterative solution of unstable variational inequalities on approximately given sets

Y. I. Alber,1 A. G. Kartsatos,2 and E. Litsyn3

1Department of Mathematics, The Technion - Israel Institute of Technology, Haifa 32000, Israel
2Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA
3Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Received 22 October 1995

Copyright © 1996 Y. I. Alber et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The convergence and the stability of the iterative regularization method for solving variational inequalities with bounded nonsmooth properly monotone (i.e., degenerate) operators in Banach spaces are studied. All the items of the inequality (i.e., the operator A, the “right hand side” f and the set of constraints Ω) are to be perturbed. The connection between the parameters of regularization and perturbations which guarantee strong convergence of approximate solutions is established. In contrast to previous publications by Bruck, Reich and the first author, we do not suppose here that the approximating sequence is a priori bounded. Therefore the present results are new even for operator equations in Hilbert and Banach spaces. Apparently, the iterative processes for problems with perturbed sets of constraints are being considered for the first time.