Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-15T05:09:05.105Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

Published online by Cambridge University Press:  20 November 2018

Joseph Bogin
Joseph Bogin*
Affiliation:
Haifa, Israel
Rights & Permissions [Opens in a new window]

Extract

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 [7], Goebel, Kirk and Shimi proved the following:

Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:

(1)

where ai≥0 and Then F has a fixed point in K.

In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 7 - 12
Copyright
Copyright © Canadian Mathematical Society 1976

Footnotes

Technion Preprint series NO. MT-205.

(1)

This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Professor M. Marcus, Dept. of Mathematics, Technion, Israel Institute of Technology, Haifa.

References

1. Belluce, L. P. and Kirk, W. A., Nonexpansive mappings and fixed-points in Banach spaces, Illinois J. Math. 11 (1967), 474–479.Google Scholar
2. Belluce, L. P., Kirk, W. A. and Steiner, E. F., Normal Structure in Banach spaces, Pacific J. Math. 26 (1968), 433–440.Google Scholar
3. Bianchini, R. M. Tiberio, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. Un. Math. Ital. (4) 5 (1972), 103–108.Google Scholar
4. Brodski, M. S. and Milman, D. P., On the center of a convex set (Russian), Dokl. Akad. Nauk. SSSR. 59 (1948), 837–840.Google Scholar
5. Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044.Google Scholar
6. DeMarr, Ralph E., Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139–1141.Google Scholar
7. Goebel, K., Kirk, W. A. and Shimi, T. N., A fixed point theorem in uniformly convex spaces, Boll. Un. Mat. Ital. (4) 7 (1973), 67–75.Google Scholar
8. Gossez, J. P. et Dozo, E. Lami, Structure normale et base de Schauder, Acad. Roy. Belg. Bull. CI. Sci. (5) 55 (1969), 673–681.Google Scholar
9. Kannan, R., Fixed point theorems in reflexive Banach spaces, Proc. Amer. Math. Soc. 38 (1973), 111–118.Google Scholar
10. Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.Google Scholar
11. Lim, Teck-Cheong, Characterization of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313–319.Google Scholar
12. Soardi, P., Su un problema dipunto di S. Reich, Boll. Un. Math. Ital. (4) 4 (1971), 841–845.Google Scholar
13. Reich, S., Remarks on fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 689–697.Google Scholar
14. Roux, D. and Soardi, P., Alcune generalizzazioni del teorema di Browder-Göhde-Kirk, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur (8) 52 (1972), 682–688.Google Scholar