Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-15T23:21:55.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed points and their approximation in Banach spaces for certain commuting mappings

Published online by Cambridge University Press:  18 May 2009

M. S. Khan
M. S. Khan
Affiliation:
Department of Pure Mathematics, La Trobe University, Bundoora, Victoria, Australia3083
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.

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 23 , Issue 1 , January 1982 , pp. 1 - 6
Copyright
Copyright © Glasgow Mathematical Journal Trust 1982

References

REFERENCES

1.Brosowski, B., Fixpunktsätze in der Approximationstheorie, Mathematica (Cluj), 11 (1969), 195–220.Google Scholar
2.Browder, F. E., Nonexpansive nonlinear operators in Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044.CrossRefGoogle ScholarPubMed
3.Dotson, W. G. Jr, Fixed point theorems for nonexpansive mappings onstarshpaed subsets of Banach spaces, J. London Math. Soc., (2) 4 (1972), 408–410.CrossRefGoogle Scholar
4.Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79.CrossRefGoogle Scholar
5.Goebel, K. and Zlotkiewicz, E., Some fixed point theorems in Banach spaces, Colloq. Math., 23 (1971), 103–106.CrossRefGoogle Scholar
6.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
7.Jungck, G., An iff fixed point criterion, Math. Mag., 49 (1976), 32–34.CrossRefGoogle Scholar
8.Kannan, R., Some results on fixed points III, Fund. Math., 70 (1971), 169–177.CrossRefGoogle Scholar
9.Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.CrossRefGoogle Scholar
10.Krasnoselskii, M. A., Two remarks about the method of successive approximations, Uspehi Mat. Nauk., 63 (1955), 123–127.Google Scholar
11.Park, S., Fixed points of f-contractive maps, Rocky Mountain J. Math., 8 (1978), 743–750.CrossRefGoogle Scholar
12.Shimi, T. N., Approximation of fixed points of certain nonlinear mappings, J. Math. Anal. Appl., 65 (1978), 565–571.CrossRefGoogle Scholar
13.Singh, S. P., An application of a fixed-point theorem to approximation theory, J. Approximation Theory, 25 (1979), 89–90.CrossRefGoogle Scholar