A fixed point theorem and attractors

Janos, Ludvik; Solomon, J. L.

doi:10.1090/S0002-9939-1978-0482716-2

A fixed point theorem and attractors
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by Ludvik Janos and J. L. Solomon

Proc. Amer. Math. Soc. 71 (1978), 257-262

DOI: https://doi.org/10.1090/S0002-9939-1978-0482716-2

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Abstract:

We investigate attractors for compact sets by considering a certain quotient space. The following theorem is included. Let $f:G \to G$, G a closed convex subset of a Banach space, f a mapping satisfying (i) there exists $M \subset G$ which is an attractor for compact sets under f; (ii) the family $\{ {f^n}\} _{n = 1}^\infty$ is equicontinuous. Then f has a fixed point.

References

Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
Ludvík Janoš, On representations of selfmappings, Proc. Amer. Math. Soc. 26 (1970), 529–533. MR 270346, DOI 10.1090/S0002-9939-1970-0270346-X
Ludvik Janos, On the Edelstein contractive mapping theorem, Canad. Math. Bull. 18 (1975), no. 5, 675–678. MR 420589, DOI 10.4153/CMB-1975-118-8
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
Roger D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375. MR 310719, DOI 10.1090/S0002-9947-1972-0310719-6
J. L. Solomon, A note on attractors for compact sets, Proc. Amer. Math. Soc. 65 (1977), no. 2, 293–296. MR 637100, DOI 10.1090/S0002-9939-1977-0637100-4