Fixed points of certain self maps on an interval

Wong, Chi Song

doi:10.1090/S0002-9939-1974-0325869-5

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fixed points of certain self maps on an interval
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Proc. Amer. Math. Soc. 42 (1974), 234-235 Request permission

Abstract:

Let T be a self map on a bounded interval [a, b] with $a,b \in T([a,b])$. Suppose that for any x, y in [a, b], \[ |T(x) - T(y)| \leqq \tfrac {1}{2}(|x - T(x)| + |y - T(y)|).\] It is shown without the continuity of T that the midpoint of [a, b] is a fixed point of T. A nontrivial example is given.

References

R. L. Franks and R. P. Marzec, A theorem on mean-value iterations, Proc. Amer. Math. Soc. 30 (1971), 324–326. MR 280656, DOI 10.1090/S0002-9939-1971-0280656-9
Chi Song Wong, Approximation to fixed points of generalized nonexpansive mappings, Proc. Amer. Math. Soc. 54 (1976), 93–97. MR 390850, DOI 10.1090/S0002-9939-1976-0390850-9