On the embedding of variational inequalities

Authors:
B. Djafari Rouhani and A. A. Khan

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3861-3871

MSC (2000):
Primary 47A52; Secondary 47H14

DOI:
https://doi.org/10.1090/S0002-9939-03-07000-X

Published electronically:
May 8, 2003

MathSciNet review:
1999935

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Abstract: This work is devoted to the approximation of variational inequalities with pseudo-monotone operators. A variational inequality, considered in an arbitrary real Banach space, is first embedded into a reflexive Banach space by means of linear continuous mappings. Then a strongly convergent approximation procedure is designed by regularizing the embedded variational inequality. Some special cases have also been discussed.

**1.**Y. I. Alber, A. D. Butnariu and I. Ryazantseva:*Regularization methods for ill-posed inclusions and variational inequalities with domain perturbations*, J. Nonlinear Convex Anal., Vol. 2, pp. 53-79 (2001). MR**2002d:47087****2.**F. E. Browder and B. A. Ton:*Nonlinear functional equations in Banach spaces and elliptic super regularization*, Math. Zeitschr., Vol. 105, pp. 177-195 (1968). MR**38:582****3.**B. Djafari Rouhani, E. Tarafdar and P. J. Watson:*Fixed point theorems, coincidence theorems and variational inequalities*, Lecture Notes in Economics and Mathematical Systems, Vol. 502, pp. 183-188, Springer, Berlin, 2001. MR**2002b:47129****4.**F. Giannessi:*Embedding variational inequalities and their generalizations into a separation scheme*, J. Inequal. Appl. 1, pp. 139-147 (1997). MR**2000j:49017****5.**F. Giannessi and A. A. Khan:*Regularization of non-coercive quasi variational inequalities*, Control and Cybernetics, Vol. 29, pp. 91-110 (2000). MR**2001f:49019****6.**J. Gwinner:*A note on pseudomonotone functions, regularization, and relaxed coerciveness,*Nonlinear Analysis, Vol. 30, pp. 4217-4227 (1997).**7.**P.T. Harker and J.S. Pang:*Finite dimensional variational inequality and complementarity problems; A survey of theory algorithms and applications*, Mathematical Programming, Vol. 48, pp. 161-220 (1990). MR**91g:90166****8.**G. Isac:*Tikhonov regularization and the complementarity problem in Hilbert spaces*, J. Math. Anal. Appl., Vol. 174, pp. 53-66 (1993). MR**94b:49014****9.**V. V. Kalashnikov and A. A. Khan:*A regularization approach for variational inequalities with pseudo-monotone operators*, Inderfurth, K. (ed.) et al., Operations research proceedings 1999, pp. 19-22, Springer, Berlin, 2000.**10.**A. A. Khan:*A regularization approach for variational inequalities*, Comput. Math. Appl., Vol. 42, pp. 65-74 (2001). MR**2002b:49020****11.**D. Kinderlehrer and G. Stampacchia:*An introduction to variational inequalities and their applications*, Academic Press, New York, 1980. MR**81g:49013****12.**O. A. Liskovets:*Regularization of problems with discontinuous monotone, arbitrarily perturbed operators*, Soviet Math. Dokl., Vol. 28, pp. 324-327 (1983). MR**85e:47092****13.**O. A. Liskovets:*Regularization of ill-posed mixed variational inequalities*, Sov. Math. Dokl., Vol. 43, pp. 384-387 (1991). MR**92i:47085****14.**U. Mosco:*Convergence of convex sets and of solutions of variational inequalities*, Adv. Math. Vol. 3, pp. 510-585 (1969). MR**45:7560****15.**M. Z. Nashed and F. Liu:*On nonlinear ill-posed problems II: Monotone operator equations and monotone variational inequalities*, Lecture Notes in Pure and Applied Mathematics, Vol. 178, pp. 223-240, Marcel Dekker, Inc. New York, 1996. MR**97e:47115****16.**M. Z. Nashed and O. Scherzer:*Stable approximation of nondifferentiable optimization problems with variational inequalities*, Contemporary Mathematics, Vol. 204, pp. 155-170 (1997). MR**98j:49015****17.**M. A. Noor, K. I. Noor and T. M. Rassias:*Some aspects of variational inequalities*, J. Comput. Appl. Math. 47, 285-312 (1993). MR**94h:49017****18.**B. A. Ton:*Nonlinear operators on convex subsets of Banach spaces,*Math. Ann., Vol. 181, pp. 35-44 (1969). MR**39:4717****19.**M.M. Vainberg:*Variational methods and method of monotone operators in the theory of nonlinear equations*, John Wiley, New York, 1973. MR**57:7286b****20.**E. Zeidler:*Nonlinear Functional Analysis and its Applications*, Vol. II/B, Springer-Verlag, New York, 1990. MR**91b:47002**

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Additional Information

**B. Djafari Rouhani**

Affiliation:
Institute for Studies in Nonlinear Analysis, School of Mathematical Sciences, Shahid Beheshti University, P.O. Box 19395-4716 Evin, 19834 Tehran, Iran

Email:
b-rohani@cc.sbu.ac.ir

**A. A. Khan**

Affiliation:
Institute of Applied Mathematics, University of Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany

Address at time of publication:
Department of Mathematical Sciences, Michigan Technological University, 319 Fisher Hall, 1400 Townsend Drive, Houghton, Michigan 49931-1295

Email:
khan@am.uni-erlangen.de, aakhan@mtu.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07000-X

Keywords:
Variational inequalities,
regularization,
pseudo-monotone,
embedding

Received by editor(s):
October 22, 2001

Received by editor(s) in revised form:
August 1, 2002

Published electronically:
May 8, 2003

Additional Notes:
The first author’s research was supported by a grant from Shahid Beheshti University

The second author’s research was supported by the German Science Foundation (DFG)

Dedicated:
Dedicated to Jochem Zowe on the occasion of his sixtieth birthday

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
American Mathematical Society