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Existence and uniqueness of fixed-points for semigroups of affine maps.


Author: Robert E. Huff
Journal: Trans. Amer. Math. Soc. 152 (1970), 99-106
MSC: Primary 47.85
DOI: https://doi.org/10.1090/S0002-9947-1970-0267433-3
MathSciNet review: 0267433
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Abstract: The Day fixed-point theorem is extended to include both existence and uniqueness. For uniqueness of fixed-points, continuity for pointwise limits of a semigroup of continuous affine maps is needed ; necessary and sufficient conditions for this are obtained and compared with the stronger condition of equicontinuity. The comparison is between, on the one hand, the above condition, separate continuity, and weak compactness, and, on the other hand, equicontinuity, joint continuity, and strong compactness. An extension of the Kakutani fixed-point theorem results. Also as a corollary, known necessary and sufficient conditions for continuity of the convolution operation are obtained.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0267433-3
Keywords: Fixed-points, equicontinuity, separate continuity, joint continuity, weak compactness, almost periodic function, weakly almost periodic functions, convolutions, amenable semigroups, affine space, functions spaces, invariant means
Article copyright: © Copyright 1970 American Mathematical Society

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