Iterative solution of unstable variational inequalities on approximately given sets

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.

[1]  Variational inequalities with monotonic operators on sets which are specified approximately , 1984 .

[2]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[3]  J. Diestel.  Geometry of Banach Spaces: Selected Topics , 1975 .

[4]  M. A. Krasnoselʹskii,et al.  Approximate Solution of Operator Equations , 1972 .

[5]  A. Bakushinskii.  On the principle of iterative regularization , 1979 .

[6]  Y. Alber.  Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

[7]  F. Browder.  Nonlinear maximal monotone operators in Banach space , 1968 .

[8]  A. Kartsatos.  New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces , 1996 .

[9]  Robert Lattès,et al.  Méthode de quasi-réversibilbilité et applications , 1967 .

[10]  G. Pisier.  Martingales with values in uniformly convex spaces , 1975 .

[11]  Ya. I. Al'ber.  Continuous regularization of linear operator equations in a Hilbert space , 1968 .

[12]  F. Browder.  Existence and approximation of solutions of nonlinear variational inequalities. , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[13]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[14]  Y. Alber,et al.  Perturbed unstable variational inequalities with unbounded operators on approximately given sets , 1993 .

[15]  Ronald E. Bruck.  A strongly convergent iterative solution of 0 ϵ U(x) for a maximal monotone operator U in Hilbert space , 1974 .

[16]  Simeon Reich,et al.  CONSTRUCTIVE TECHNIQUES FOR ACCRETIVE AND MONOTONE OPERATORS , 1979 .

[17]  On the solution of variational inequalities with monotonic operators by the regularization method , 1983 .

[18]  Ya.I. Alber.  The regularization method for variational inequalities with nonsmooth unbounded operators in Banach space , 1993 .

[19]  Ya. I. Al'ber,et al.  Parallelogram inequalities in banach spaces and some properties of a dual mapping , 1988 .

[20]  Approximate solution of integro-operator Volterra equations of the first kind , 1986 .

[21]  V. A. Morozov,et al.  Linear and nonlinear ill-posed problems , 1975 .