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Two fixed point theorems and invariant integrals

Published online by Cambridge University Press:  17 April 2009

T.J. Cooper  and
J.H. Michael
T.J. Cooper
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia.
J.H. Michael
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia.
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Abstract

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Two fixed point theorems for a subset C of a normed vector space X are established by using the concept of centre. These results differ from previous fixed point theorems in that X is assumed to have a topology T as well as a norm. The norm is required to be lower semi-continuous with respect to T and C is required to be convex, bounded with respect to the norm and compact with respect to T.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 1 , August 1974 , pp. 15 - 30
Copyright
Copyright © Australian Mathematical Society 1974

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]Бродский, M.C. и Мильман, Д.П [ Brodskiĭ, M.S. and Mil'man, O.P.], “О центре выпуклого множества” [On the centre of a convex set], Dokl. Akad. Nauk SSSR (NS) 59 (1948), 837–840.Google ScholarPubMed
[3]Clifford, A.M. and Preston, G.B., The algebraic theory of semigroups, Volume I (Math. Surveys 7 (I), Amer. Math. Soc., Providence, Rhode Island, 1961).Google Scholar
[4]DeMarr, Ralph, “Common fixed points for commuting contraction mappings”, Pacific J. Math. 13 (1963), 1139–1141.Google Scholar
[5]Dunford, Nelson and Schwartz, Jacob T., Linear operators, Part I (Interscience [John Wiley & Sons], New York, London, 1958).Google Scholar
[6]Edelstein, Michael, “The construction of an asymptotic centre with a fixed-point property”, Bull. Amer. Math. Soc. 78 (1972), 206–208.Google Scholar
[7]Edelstein, Michael, “Fixed point theorems in uniformly convex Banach spaces”, submitted.Google Scholar
[8]Gaal, Steven A., Point set topology (Pure and Applied Mathematics, 16. Academic Press, New York and London, 1964).Google Scholar
[9]Michael, J.H., “Right invariant integrals on locally compact semigroups”, J. Austral. Math. Soc. 4 (1964), 273–286.Google Scholar
[10]Mitchell, Theodore, “Fixed points of reversible semigroups of non-expansive mappings”, Kōdai Math. Sem. Rep. 22 (1970), 322–323.Google Scholar
[11]Rosen, William G., “On invariant means over compact semigroups”, Proc. Amer. Math. Soc. 7 (1956), 1076–1082.Google Scholar
[12]Шнеперман, Л.Б. [Šneperman, L.B.], “Неподвижная точка полугрупы преобразований и инвариантное инвариантное интегрирование на бикомпактной полугруппе” [A fixed point of a semigroup of transformations and invariant integration on a bicompact semigroup], Vesci Akad. Navuk BSSR Sev. Fis.-Mat. Navuk 1966, no. 4, 30–36.Google Scholar
[13]Taylor, Angus E., Introduction to functional analysis (John Wiley & Sons, New York; Chapman & Hall, London; 1958).Google Scholar
[14]Wallace, A.D., “The structure of topological semigroups”, Bull. Amer. Math. Soc. 61 (1955), 95–112.CrossRefGoogle Scholar