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On Asymptotically Nonexpansive Semigroups of Mappings

Published online by Cambridge University Press:  20 November 2018

R. D. Holmes  and
P. P. Narayanaswami
R. D. Holmes
Affiliation:
Dalhousie University, Halifax, Nova Scotia
P. P. Narayanaswami
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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A selfmapping f of a metric space (X, d) is nonexpansive (ε-nonexpansive) if d(f(x), f(y)) ≤ d(x, y) for all x, y ∊ X (respectively if d(x, y) < ε). In [1], M. Edelstein proved that a nonexpansive mapping f of En admits a fixed point provided the f-closure of En (i.e. the set of all points which are cluster points of {fn(x)} for some x) is nonempty. R. D. Holmes [2] considered commutative semigroups of selfmappings of a metric space and obtained fixed point theorems for such semigroups under certain contractivity conditions.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 209 - 214
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Edelstein, M., On nonexpansive mappings, Proc. Amer. Math. Soc. 15 (1964), 689-695.Google Scholar
2. Holmes, R. D., Contributions to the theory of contraction mappings, Doctoral Dissertation, Dalhousie University, 1969.Google Scholar