On the Banach mapping theorem and a related conjecture. (English) Zbl 07524619
Summary: We extend, using an elementary method, results of Banach (1924), Fan (1952), and Sanders (1961), which concern a finite collection \(\{ f_i : A_i \to A_{i + 1} \}_{i = 1}^n\) of mappings with \(A_{n + 1} = A_1\) which is decomposable as \(f_i (B_i) = A_{i + 1} \backslash B_{i + 1} \), where \(B_i \subseteq A_i\) for all \(i\) and \(B_{n + 1} = B_1\). Our theorem determines when such a collection is decomposable. We also show that such a set \(B_1\) is unique up to an addition of a certain set, which was conjectured by Sanders.
MSC:
| 03E30 | Axiomatics of classical set theory and its fragments |
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 47H10 | Fixed-point theorems |
References:
| [1] | S. Banach, “Un théorème sur les transformations biunivoques”, Fund. Math. 6 (1924), 236-239. · JFM 50.0136.01 · doi:10.4064/fm-6-1-236-239 |
| [2] | F. P. Callahan and S. G. Kneale, “A note on the Schroeder-Bernstein theorem”, Amer. Math. Monthly 64 (1957), 423-424. · Zbl 0077.26807 · doi:10.2307/2310168 |
| [3] | K. Fan, “Note on a theorem of Banach”, Math. Z. 55 (1952), 308-309. · Zbl 0047.28505 · doi:10.1007/BF01181129 |
| [4] | S. G. Krantz, Elements of advanced mathematics, CRC Press, Boca Raton, FL, 2012. · Zbl 1243.03001 |
| [5] | M.-C. Li, “An elementary proof of a generalization of Banach’s mapping theorem”, Amer. Math. Monthly 121:5 (2014), 445-446. · Zbl 1478.03076 · doi:10.4169/amer.math.monthly.121.05.445 |
| [6] | B. L. Sanders, “Concerning a theorem of Ky Fan’s”, Math. Z. 76 (1961), 51-55 · Zbl 0100.04706 · doi:10.1007/BF01210960 |
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