Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T05:30:54.863Z Has data issue: false hasContentIssue false
  • English
  • Français

On D-KKM theorem and its applications

Published online by Cambridge University Press:  17 April 2009

H. K. Pathak  and
M. S. Khan
H. K. Pathak
Affiliation:
Department of Mathematics, Kalyan Mahavidyalaya, Bhilai Nagar (C.G.)490006, India
M. S. Khan
Affiliation:
Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Al-Khod, Muscat, Sultanate of Oman
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 introduce a new class of set-valued mappings in a non-convex setting called D-KKM mappings and prove a general D-KKM theorem. This extends and improves the KKM theorem for several families of set-valued mappings, such as (X, Y), C(X, Y), C (X, Y), C (X, Y) and C (X, Y). In the sequel, we apply our theorem to get some existence results for maximal elements, generalised variational inequalities, and price equilibria.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 1 , February 2003 , pp. 67 - 77
Copyright
Copyright © Australian Mathematical Society 2003

References

REFERENCES

[1]Ben-El-Mixhaiekh, H. and Dequire, P., ‘Approachability and fixed points for non-convex set valued maps’, J. Math. Anal. Appl. 170 (1992), 477–500.CrossRefGoogle Scholar
[2]Border, K.C., Fixed point theorems with applications to economics and game theory (Cambridge University Press, Cambridge, 1989).Google Scholar
[3]Chang, T.H., ‘Generalized KKM theorm and its application’, Far East J. Math 4 (1996), 137–147.Google Scholar
[4]Fan, K., ‘Some properties of convex sets related fixed point theorems’, Math. Ann. 26 (1984), 519–537.CrossRefGoogle Scholar
[5]Lassoude, M., ‘Fixed point Kakutani factorizable multifunctions’, J. Math. Anal. Appl. 152 (1990), 46–60.CrossRefGoogle Scholar
[6]Park, S., ‘Generalizations of Ky Fan's matching theorems and their application’, J. Math. Anal. Appl. 141 (1989), 164–176.CrossRefGoogle Scholar
[7]Park, S., ‘Convex spaces and KKM families of subsets’, Bull. Korean Math. Soc. 27 (1990), 11–14.Google Scholar
[8]Park, S., ‘Foundations of KKM theory via coincidence of composites of upper semi continuous maps’, J. Korean Math. Soc. 31 (1994), 493–519.Google Scholar
[9]Park, S., Singh, S.P. and Watson, B., ‘Some fixed point theorems for composite of a cyclic maps’, Proc. Amer. Math. Soc. 121 (1994), 1151–1158.CrossRefGoogle Scholar
[10]Shioji, N., ‘A further generalization of the Knaster—Kuratowski—Mazur Kiewicz Theorem’, Proc. Amer. Math. Soc. 111 (1991), 187–195.CrossRefGoogle Scholar
[11]Tian, G.Q., ‘Generalized KKM theorem, mini-max inequalities, and their application’, J. Optim. Theory Appl. 83 (1994), 375–389.CrossRefGoogle Scholar
[12]Yannelis, N.C. and Prabhakar, N.D., ‘Existence of maximal elements and equilibria in linear topological spaces’, J. Math. Econ. 12 (1983), 233–245.CrossRefGoogle Scholar
[13]Yao, J.C., ‘On the general variational inequality’, J. Math. Anal. Appl. 174 (1993), 550–555.CrossRefGoogle Scholar