Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T18:02:13.963Z Has data issue: false hasContentIssue false
  • English
  • Français

A fixed point theorem and its applications to a system of variational inequalities

Published online by Cambridge University Press:  17 April 2009

Qamrul Hasan Ansari  and
Jen-Chih Yao
Qamrul Hasan Ansari
Affiliation:
Department of MathematicsAligarh Muslim UniversityAligarh - 202 002, India
Jen-Chih Yao
Affiliation:
Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiung 804, Taiwan, Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we first prove a fixed point theorem for a family of multivalued maps defined on product spaces. We then apply our result to prove an equilibrium existence theorem for an abstract economy. We also consider a system of variational inequalities and prove the existence of its solutions by using our fixed point theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 433 - 442
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bianchi, M., Pseudo P-monotone operators and variational inequalities (Universita Cat-tolica del Sacro Cuore 6, 1993).Google Scholar
[2]Borglin, A. and Keiding, H., ‘Existence of equilibrium actions and of equilibrium: a note on the ‘new’ existence theorem’, J. Math. Econom. 3 (1976), 313–316.CrossRefGoogle Scholar
[3]Browder, F.E., ‘The fixed point theory of multivalued mappings in topological vector spaces’, Math. Ann. 177 (1968), 283–301.CrossRefGoogle Scholar
[4]Chang, S.S., Lee, B.S., Wu, X., Cho, Y.J. and Lee, G.M., ‘On the generalized quasi-variational inequality problems’, J. Math. Anal. Appl. 203 (1996), 686–711.CrossRefGoogle Scholar
[5]Cohen, G. and Chaplais, F., ‘Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms’, J. Optim. Theory Appl. 59 (1988), 360–390.CrossRefGoogle Scholar
[6]Ding, X.P., Kim, W.K. and Tan, K.K., ‘A selection theorem and its applications’, Bull. Austral. Math. Soc. 46 (1992), 205–212.CrossRefGoogle Scholar
[7]Fan, K., ‘A generalization of Tychonoff's fixed point theorem’, Math. Ann. 142 (1961), 305–310.CrossRefGoogle Scholar
[8]Lan, K.Q. and Webb, J., ‘New fixed point theorems for a family of mappings and applications to problems on sets with convex sections’, Proc. Amer. Math. Soc. 126 (1998), 1127–1132.CrossRefGoogle Scholar
[9]Pang, J.S., ‘Asymmetric variational inequality problems over product sets: applications and iterative methods’, Math. Programming 31 (1985), 206–219.CrossRefGoogle Scholar
[10]Tarafdar, E., ‘On nonlinear variational inequalities’, Proc. Amer. Math. Soc. 67 (1977), 95–98.CrossRefGoogle Scholar
[11]Tarafdar, E. and Yuan, G.X.-Z., ‘Generalized variational inequalities and its applications’, Nonlinear Anal. 30 (1997), 4171–4181.CrossRefGoogle Scholar
[12]Tychonoff, A., ‘Ein Fixpunktsatz’, Math. Ann. 111 (1935), 767–776.CrossRefGoogle Scholar
[13]Wu, X., ‘A new fixed point theorem and its applications’, Proc. Amer. Math. Soc. 125 (1997), 1779–1783.CrossRefGoogle Scholar
[14]Wu, X. and Shen, S., ‘A further generalization of Yannelis-Prabhakar's continuous selection theorem and its applications’, J. Math. Anal. Appl. 196 (1996), 61–74.CrossRefGoogle Scholar
[15]Yannelis, N.C. and Prabhakar, N.D., ‘Existence of maximal elements and equilibria in linear topological spaces’, J. Math. Econom. 12 (1983), 233–245.CrossRefGoogle Scholar
[16]Zhu, D.L. and Marcotte, P., ‘Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities’, SIAM J. Optim. 6 (1996), 714–726.CrossRefGoogle Scholar