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Existence conditions in general quasimonotone variational inequalities

Published online by Cambridge University Press:  17 April 2009

D. Aussel  and
D. T. Luc
D. Aussel
Affiliation:
Lab. MANO, University of Perpignan, France e-mail: aussel@univ-perp.fr
D. T. Luc
Affiliation:
Lab. Analyse non linéaire et Géométrie, University of Avignon, France e-mail: dtluc@univ-avignon.fr
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In this paper we study a general variational inequality model with set-valued quasimonotone operators, a model which includes several variational inequalities and equilibrium problems. We establish unifying conditions for existence of solutions in a topological vector space setting. Applications to parametric equilibrium models and to a contact problem are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 2 , April 2005 , pp. 285 - 303
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Aussel, D., ‘Subdifferential properties of quasiconvex and pseudoconvex functions: unified approach’, J. Optim. Theory Appl. 97 (1998), 29–45.Google Scholar
[2]Aussel, D. and Hadjisavvas, N., ‘On quasimonotone variational inequalities’, J. Optim. Theory Appl. 121 (2004), 445–450.Google Scholar
[3]Bianchi, M. and Pini, R., ‘A note on equilibrium problems with properly quasimonotone bifunctions’, J. Global Optim. 20 (2001), 67–76.CrossRefGoogle Scholar
[4]Chowdhury, M.S.R. and Tan, K.K., ‘Generalized variational inequalities for quasimonotone operators and applications’, Bull. Polish Acad. Sci. Math. 45 (1997), 25–54.Google Scholar
[5]Daniilidis, A. and Hadjisavvas, N., ‘Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions’, J. Optim. Theory Appl. 102 (1999), 525–536.CrossRefGoogle Scholar
[6]Fan, K., ‘Minimax theorems’, Proc. Natl. Acad. Sci. USA 39 (1953), 42–47.Google Scholar
[7]Flores-Bazan, F., ‘Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case’, SIAM J. Optim. 11 (2000), 675–690.CrossRefGoogle Scholar
[8]Giannessi, F. and Maugeri, A., Variational inequalities and network equilibrium problems (Plenum Press, New York, 1995).Google Scholar
[9]Hadjisavvas, N. and Schaible, S., ‘Quasimonotone variational inequalities in Banach spaces’, J. Optim. Theory Appl. 90 (1996), 95–111.Google Scholar
[10]Harker, P.T. and Pang, J.S., ‘Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications’, Math. Programming 48 (1990), 161–220).Google Scholar
[11]Han, W. and Sofonea, M., Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics 30 (American Mathematical Society, Providence, R.I., 2002).Google Scholar
[12]John, R., ‘A note on Minty variational inequalities and generalized monotonicity’, in Generalized Convexity and Generalized Monotonicity, (Hadjisavvas, N., Martinez-Legaz, J-E. and Penot, J-P., Editors), Lecture Notes in Econom. and Math. Systems 502 (Springer, Berlin, 2001), pp. 240–246.Google Scholar
[13]Karamardian, S., ‘Complementarity over cones with monotone and pseudomonotone maps’, J. Optim. Theory Appl. 18 (1976), 445–454.Google Scholar
[14]Karamardian, S. and Schaible, S., ‘Seven kinds of monotone maps’, J. Optim. Theory Appl. 66 (1990), 37–46.Google Scholar
[15]Kikuchi, N. and Oden, J.T., Contact problems in elasticity: Study of variational inequalities and finite element methods (SIAM, Philadelphia, 1988).Google Scholar
[16]Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications (Academic Press, New York, 1980).Google Scholar
[17]Konnov, I., ‘On quasimonotone variational inequalities’, J. Optim. Theory Appl. 99 (1998), 165–181.Google Scholar
[18]Luc, D.T., ‘Characterizations of quasiconvex functions’, Bull. Austral. Math. Soc. 48 (1993), 393–405.Google Scholar
[19]Luc, D.T., ‘On generalized convex nonsmooth functions’, Bull. Austral. Math. Soc. 49 (1994), 139–149.Google Scholar
[20]Luc, D.T., ‘Generalized monotone maps and support bifunctions’, Acta Math. Vietnam. 21 (1996), 213–252.Google Scholar
[21]Luc, D.T., ‘Existence results for densely pseudomonotone variational inequalities’, J. Math. Anal. Appl. 254 (2001), 291–308.Google Scholar
[22]Noor, M.A., ‘Some recent advances in variational inequalities, Part I, basic concepts’, New Zealand J. Math. 26 (1997), 53–80.Google Scholar
[23]Noor, M.A., ‘Some recent advances in variational inequalities, Part II, other concepts’, New Zealand J. Math. 26 (1997), 229–255.Google Scholar
[24]Penot, J-P. and Quang, P.H., ‘On generalized convexity of functions and generalized monotonicity of set-valued maps’, J. Optim. Theory Appl. 92 (1997), 343–356.Google Scholar
[25]Shih, M.H. and Tan, K.K., ‘A further generalization of Ky Fan's minimax inequality and its applications’, Studia Math. 78 (1984), 279–287.Google Scholar
[26]Stampacchia, G., ‘Formes bilinéaires coercives sur les ensembles convexes’, C. R. Acad. Sci. Paris 258 (1964), 4413–4416.Google Scholar
[27]Yao, J. C., ‘Multivalued variational inequalities with K-pseudomonotone operators’, J. Optim. Theory Appl. 83 (1994), 391–403.Google Scholar
[28]Zhou, J.X. and Chen, G., ‘Diagonally convex conditions for problems in convex analysis and quasi-variational inequalities’, J. Math. Anal. Appl. 132 (1988), 213–225.Google Scholar