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

Huff, Robert E.

doi:10.1090/S0002-9947-1970-0267433-3

Existence and uniqueness of fixed-points for semigroups of affine maps.
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Trans. Amer. Math. Soc. 152 (1970), 99-106 Request permission

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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