## Abstract and Applied Analysis

### Iterative solution of unstable variational inequalities on approximately given sets

#### Abstract

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.

#### Article information

Source
Abstr. Appl. Anal., Volume 1, Number 1 (1996), 45-64.

Dates
First available in Project Euclid: 7 April 2003

https://projecteuclid.org/euclid.aaa/1049725991

Digital Object Identifier
doi:10.1155/S1085337596000024

Mathematical Reviews number (MathSciNet)
MR1390559

Zentralblatt MATH identifier
0932.49014

Subjects
Secondary: 47A55 47H05 65L20

#### Citation

Alber, Y. I.; Kartsatos, A. G.; Litsyn, E. Iterative solution of unstable variational inequalities on approximately given sets. Abstr. Appl. Anal. 1 (1996), no. 1, 45--64. doi:10.1155/S1085337596000024. https://projecteuclid.org/euclid.aaa/1049725991

#### References

• Y. I. Alber, The solution of nonlinear equations with monotone operators in Banach spaces, Siberian Math. J. 16 (1975), 1-8.
• Y. I. Alber, The regularization method for variational inequalities with nonsmooth unbounded operators in Banach space, Appl. Math. Lett. 6 (1993), 63-68.
• Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, (to appear).
• Y. I. Alber, The monotonicity method and the approximate computation of the value of the nonlinear unbounded operator, Siberian Math. J. 19 (1978), 179-183.
• Y. I. Alber, A continuous regularization of linear operator equations in Hilbert spaces, Math. Notes 9 (1968), 42-54.
• Y. I. Alber, Iterative regularization in Banach spaces, Soviet Math. (Izv. VUZ) 30 (1986), 1-8.
• Y. I. Alber, The solution of equations and variational inequality with maximal monotone operators, Soviet Math. Dokl. 20 (1979) 871-876.
• Y. I. Alber and O. A. Liskovets, The principle of the smoothing functional for solution of equations of the first kind with monotone operators, Differential Equations 20 (1984), 603-608.
• Y. I. Alber and A. I. Notik, Parallelogram inequalities in Banach spaces and some properties of the duality mapping, Ukrainian Math. J. 40 (1988), 650-652.
• Y. I. Alber and A. I. Notik, Geometric properties of Banach spaces and approximate methods for solving nonlinear operator equations, Soviet Math. Dokl. 29 (1984), 611-615.
• Y. I. Alber and A. I. Notik, Perturbed unstable variational inequalities with unbounded operator on approximately given sets, Set-Valued Anal. 1 (1993), 393-402.
• Y. I. Alber and A. I. Notik, Iterative processes in Orlicz spaces, Methods of Optimization and Operational Research, Irkutsk, 1984, 114-123 (in Russian).
• Y. I. Alber and A. I. Notik, On minimization of functionals and solution of variational inequalities in Banach spaces, Soviet Math. Dokl. 34 (1986), 296-300.
• Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4 (1994), 39-54.
• Y. I. Alber and I. P. Ryazantseva, The residual principle in nonlinear problems with discontinuous monotone mappings is a regularizing algorithm, Soviet Math. Dokl. 19 (1978), 437-440.
• Y. I. Alber and I. P. Ryazantseva, Variational inequalities with discontinuous monotone mappings, Soviet Math. Dokl. 25 (1982), 206-210.
• A. B. Bakushinskii, Methods for solution of monotone variational inequalities that are based on the principle of iterative regularization, Zh. Vychisl. Mat. i Mat. Fiz. 17 (1977), 1350-1362 (in Russian).
• A. B. Bakushinskii, On the principle of iterative regularization, Comput. Math. Math. Phys. 19 (1979), 256-260.
• F. E. Browder, Nonlinear maximal monotone operators in Banach space, Math. Ann. 175 (1968), 89-113.
• F. E. Browder, Existence and approximation of solution of nonlinear variational inequalities, Proc. Nat. Acad. Sci. USA 56 (1966), 1080-1086.
• R. E. Bruck, Jr., A strongly convergent iterative solution of $0 \in U x$ for a maximal monotone operator $U$ in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114--126.
• J. Diestel, The geometry of Banach spaces, Lecture Notes in Math. 485, Springer, 1975.
• V.K. Ivanov, Approximate solution of operator equations of the first kind, Zh. Vychisl. Mat. i Mat. Fiz. 6 (1966), 1089-1094. (in Russian).
• A. Kartsatos, New results in the perturbation theory of maximal monotone and $m$-accretive operators in Banach spaces, Trans. Amer. Math. Soc. (to appear)
• D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
• R. Lattes and J.-L. Lions, Méthode de quasiriversibilité et applications, Dunod, Paris, 1967.
• M. M. Lavrentjev, On some incorrect problems of mathematical physics, Novosibirsk, 1962.
• J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
• O. A. Liskovets, External approximations for regularization of monotone variational inequalities, Soviet Math. Dokl. 36 (1988), 220-224.
• O. A. Liskovets, Discrete convergence of elements and operators for ill-posed problems with a monotone operator, Soviet Math. Dokl. 31 (1985), 202-206.
• U. Mosco, Perturbation of variational inequalities, in Nonlinear Functional Analysis, Proc. Sympos. Pure. Math. 18, Part 1, AMS, Providence, 1970, 182-184.
• V. A. Morozov, Linear and nonlinear ill-posed problems, J. Soviet Math. 4 (1975), 706-736.
• G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 329-350.
• S. Reich, Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, Academic Press, New York, 1979, 335-345.
• I. P. Ryazantseva, Solution of variational inequalities with monotone operators by the regularization method, Comput. Math. Math. Phys. 23 (1983), 145-148.
• I. P. Ryazantseva, Variational inequalities with monotone operators on sets which are specified approximately, Comput. Math. Math.Phys. 24 (1984), 194-197.
• G. M. Vainikko, The residual principle for one class of regularization methods, Comput. Math. Math.Phys. 22 (1982), 1-19.