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On a Theorem of Mitchell

Published online by Cambridge University Press:  20 November 2018

E. R. Bishop
E. R. Bishop*
Affiliation:
Acadia University, Wolfville, Nova Scotia
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In this note we apply a theorem of T. Mitchell on fixed points of contraction maps to a particular set of contraction maps of metric spaces. Also, part of Mitchell's theorem is extended to permit application, under suitable conditions, when the space involved is not compact.

We employ the following terminology due to Mitchell [1]. Let S be a semigroup; m(S) the space of all bounded real functions on S with sup norm; and X a subset of m(S). For all s ∊ S, the left translation ls {right translation rs} of m(s) by s is given by (lsf)(s') = f(ss') {(rsf)(s')=f(s's)} where f ∊ m(S) and s' ∊ S.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 215 - 217
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Mitchell, T., Function algebras, means and fixed points, Trans. Amer. Math. Soc. 130 (1968), 117-126.Google Scholar
2. Edelstein, M., On fixed and periodic points under contractive mappings J. London Math. Soc. 37 (1962), 74-79.Google Scholar
3. Royden, H. L., Function algebras, Bull. Amer. Math. Soc. (4) 69 (1963), 281-298.Google Scholar