Iterative solution of unstable variational inequalities on approximately given sets. (English) Zbl 0932.49014

Alber, Y. I.; Kartsatos, A. G.; Litsyn, E.

Abstr. Appl. Anal. 1, No. 1, 45-64 (1996).

Summary: 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 \(\Omega\)) 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.

MSC:

49J40	Variational inequalities
47J20	Variational and other types of inequalities involving nonlinear operators (general)
65K10	Numerical optimization and variational techniques
47A55	Perturbation theory of linear operators
47H05	Monotone operators and generalizations
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

convex sets; Lyapunov functionals; monotone operators; Hausdorff distances; convergence; stability; iterative regularization method; variational inequalities; perturbations