Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T12:27:29.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous selection theorem, coincidence theorem and intersection theorems concerning sets with H-convex sections

Part of: Connections with other structures, applications General convexity Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

Xie-Ping Ding
Xie-Ping Ding
Affiliation:
Sichuan Normal University Chengdu, Sichuan People's, 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.

A continuous selection and a coincidence theorem are proved in H-spaces which generalize the corresponding results of Ben-El-Mechaiekh-Deguire-Granas, Browder, Ko-Tan, Lassonde, Park, Simon and Takahashi to noncompact and/or nonconvex settings. By applying the two theorems, some intersection theorems concerning sets with H-convex sections are obtained which generalize the corresponding results of Fan, Lassonde and Shih-Tan to H-spaces. Some applications to minimax principle are given.

Keywords

Continuous selectioncoincidenceH-spacecontractibleweakly H-convexH – KKMcompactly closed (open)upper semi-continuousminimεx principle

MSC classification

Secondary: 54C65: Selections 54H25: Fixed-point and coincidence theorems 52A07: Convex sets in topological vector spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 1 , February 1992 , pp. 11 - 25
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Barbaro, C. and Ceppitelli, R., ‘Minimax inequalities in Riesz spaces’, Atti Sem. Mat. Fis. Univ. Modena 35 (1987), 63–69.Google Scholar
[2]Bardaro, C. and Ceppitelli, R., ‘Some further generalizations of Knanster-Kuratowski-Mazurkiewicz theorem and minimax inequalities’, J. Math. Anal. Appl. 132 (1988), 484–490.CrossRefGoogle Scholar
[3]Barbaro, C. and Ceppitelli, R., ‘Fixed point theorem and vector-valued minimax theorems’, J. Math. Anal. Appl. 146 (1990), 363–373.CrossRefGoogle Scholar
[4]Ben-El-Mechaiekh, H., Deguire, P. and Granas, A., ‘Une alternative non lineaire en analyse convexe et applications’, C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), 257–259.Google Scholar
[5]Ben-El-Mechaiekh, H., Deguire, P. and Granas, A., ‘Points fixes et coincidences pour les fonctions multivocues II (Applications de type ϕ et ϕ*)’, C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), 391–384.Google Scholar
[6]Browder, F. E., ‘The fixed point theory of multi-valued mappings in topological vector spaces’, Math. Ann. 177 (1968), 283–301.CrossRefGoogle Scholar
[7]Ding, X. P., Kim, W. K. and Tan, K. K., ‘A new minimax inequality on H-spaces with applications’, Bull. Austrail. Math. Soc. 41 (1990), 457–473.CrossRefGoogle Scholar
[8]Ding, X. P., Kim, W. K. and Tan, K. K., ‘Applications of a minimax inequality on H-spaces’, Bull. Austral. Math. Soc. 41 (1990), 475–485.CrossRefGoogle Scholar
[9]Ding, X. P. and Tan, K. K., ‘Matching theorems, fixed point theormes and minimax inequalities without convexity’, J. Austral. Math. Soc. (Series A) 49 (1990), 111–128.CrossRefGoogle Scholar
[10]Ding, X. P. and Tan, K. K., ‘Generalizations of KKM theorem and applications to best approximations and fixed point theorems’, submitted.Google Scholar
[11]Fan, K., ‘Sur un theoreme minimax’, C. R. Acad. Sci. Paris Groups, 250 (1964), 3925–3928.Google Scholar
[12]Fan, K., ‘Fixed-point and related thoerems for non-compact convex sets’, Game theory and related topics edited by Moeschlin, O. and Pallaschke, D., pp. 151–156 (North Holland, Amsterdam, 1979).Google Scholar
[13]Fan, K., ‘Some properties of convex sets related to fixed poin theorems’, Math. Ann. 226 (1984), 519–537.CrossRefGoogle Scholar
[14]Ha, C. A., ‘A non-compact minimax theorem’, Pacific J. Math. 97 (1981), 115–117.CrossRefGoogle Scholar
[15]Horvath, C., ‘Some results on multivalued mappings and inequalities without convexity’, Nonlinear and convex analysis edited by Lin, B. L. and Simons, S., pp. 96–106 (Marcel Dekker, 1987).Google Scholar
[16]Ko, H. M. and Tan, K. K., ‘A coincidence theorem with applications to minimiax inequalities and fixed point theorems’, Tamkang J. Math. 17 (1986), 37–45.Google Scholar
[17]Lassonde, M., ‘One the use of KKM multifunctions in fixed point theory and related topics’, J. Math. Anal. Appl. 97 (1983), 151–201.CrossRefGoogle Scholar
[18]Liu, F. C., ‘A note on the von Neumann-Sion minimax principle’, Bull. Inst. Math. Acad. Sinica, 6 (1978), 517–524.Google Scholar
[19]Park, S., ‘Generalizations of Ky Fan's matching theorems and their application’, J. Math. Anal. Appl. 141 (1989), 164–176.CrossRefGoogle Scholar
[20]Simons, S., Two function minimax theorems and variational inequalities for functions on compact and noncompact sets, with some comments on fixed point theorems, (Proc. Sympos. Pure Math. 45 (2) (1986), 377–392, Amer. Math. Soc., Providence, R.I.).Google Scholar
[21]Sion, M., ‘On nonlinear variational inequalities’, Pacific J. Math. 8 (1958), 171–176.CrossRefGoogle Scholar
[22]Shih, M. H. and Tan, K. K., ‘Non-compact sets with convex sections II’, J. Math. Anal. Appl. 120 (1986), 264–270.CrossRefGoogle Scholar
[23]Takahashi, W., Fixed point, minimax, and Hahn-Banach theorems, (Proc. Sympos. Pure Math. 45 (2) (1986), 419–427, Amer. Math. Soc., Providence, R.I.).Google Scholar