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Uniformly Lipschitzian Families of Transformations in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

K. Goebel ,
W. A. Kirk  and
R. L. Thele
K. Goebel
Affiliation:
Maria Curie Sklodowska University, Lublin, Poland
W. A. Kirk
Affiliation:
The University of Iowa, Iowa City, Iowa
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The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ K

where {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1245 - 1256
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Belluce, L. P. and Kirk, W. A., Fixed point theorems for families of contractive mappings, Pacific J. Math. 18 (1966), 213–217.Google Scholar
2. Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54 (1965), 1041–1044.Google Scholar
3. De Marr, R., Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139–1141.Google Scholar
4. Edelstein, M., The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc. 78 (1972), 206–208.Google Scholar
5. Edelstein, M., Fixed point theorems in uniformly convex Banach spaces (to appear).Google Scholar
6. Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288.Google Scholar
7. Freudenthal, H. and W. Hurewicz, Dehnungen, Verkürzungen, Isometrien, Fund. Math. 26 (1930), 120–122.Google Scholar
8. Goebel, K., Convexity of balls and fixed-point theorems for mappings with nonexpansive square, Compositio Math. 22 (1970), 269–274.Google Scholar
9. Goebel, K., On the minimal displacement of points under lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163.Google Scholar
10. Goebel, K. and Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174.Google Scholar
11. Goebel, K. and Kirk, W. A., A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140.Google Scholar
12. Göhde, D., Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258.Google Scholar
13. Holmes, R. D. and Lau, A. T., Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc. 5 (1972), 330–336.Google Scholar
14. Gurarii, V. I., On the differential properties of the modulus of convexity in a Banach space (in Russian), Mat. Issled. 2 (1967), 141–148.Google Scholar
15. James, R. C., Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.Google Scholar
16. Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.Google Scholar
17. Kirk, W. A., Fixed point theorems for non-lip schitzian mappings of asymptotically nonexpansive type, Israel J. Math, (to appear).Google Scholar
18. Lim, T.-C., A fixed point theorem for families of nonexpansive mappings, Pacific J. Math. (to appear).Google Scholar
19. Lim, T.-C., Characterizations of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313–319.Google Scholar
20. Milman, Ju. I., Geometric theory of Banach spaces, II. Geometry of the unit ball, Uspehi Mat. Nauk. 26 (1971), 73–150.Google Scholar
21. Mitchell, T., Fixed points of reversible semigroups of nonexpansive mappings, Ködai Math. Sem. Rep. 22 (1970), 322–323.Google Scholar
22. Opial, Z., Lecture notes on nonexpansive and monotone mappings in Banach spaces, Center for Dynamical Systems, Brown University, Providence, R.I., USA, 1967.Google Scholar
23. Schaefer, H., Uber die Méthode sukessiver Approximationen, Jber, Deutsch. Math.-Verein 59 (1957), 131–140.Google Scholar