Fixed point structures
HTML articles powered by AMS MathViewer
- by T. B. Muenzenberger and R. E. Smithson PDF
- Trans. Amer. Math. Soc. 184 (1973), 153-173 Request permission
Abstract:
A fixed point structure is a triple $(X,\mathcal {P},\mathcal {F})$ where X is a set, $\mathcal {P}$ a collection of subsets of X, and $\mathcal {F}$ a family of multifunctions on X into itself together with a set of axioms which insure that each member of $\mathcal {F}$ has a fixed point. A fixed point structure for noncontinuous multifunctions on semitrees is established that encompasses fixed point theorems of Wallace-Ward and Young-Smithson as well as new fixed point theorems for partially ordered sets and closed stars in real vector spaces. Also two other fixed point structures are presented that subsume fixed point theorems of Tarski-Ward-Smithson on semilattices and, more generally, partially ordered sets. Also the Davis-Ward converse to this last fixed point theorem is obtained.References
- Smbat Abian and Arthur B. Brown, A theorem on partially ordered sets, with applications to fixed point theorems, Canadian J. Math. 13 (1961), 78–82. MR 123492, DOI 10.4153/CJM-1961-007-5
- David P. Bellamy, Composants of Hausdorff indecomposable continua; a mapping approach, Pacific J. Math. 47 (1973), 303–309. MR 331345, DOI 10.2140/pjm.1973.47.303
- C. E. Capel and W. L. Strother, Multi-valued functions and partial order, Portugal. Math. 17 (1958), 41–47. MR 101512
- Orrin Frink Jr., Topology in lattices, Trans. Amer. Math. Soc. 51 (1942), 569–582. MR 6496, DOI 10.1090/S0002-9947-1942-0006496-X J. K. Harris, Order structures for certain acyclic topological spaces, Thesis, University of Oregon, Eugene, Ore., 1962.
- Shizuo Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457–459. MR 4776
- L. Mohler, A fixed point theorem for continua which are hereditarily divisible by points, Fund. Math. 67 (1970), 345–358. MR 261583, DOI 10.4064/fm-67-3-345-358 T. B. Muenzenberger, Fixed point structures, Thesis, University of Wyoming, Laramie, Wy., 1972.
- Raymond E. Smithson, A fixed point theorem for connected multi-valued functions, Amer. Math. Monthly 73 (1966), 351–355. MR 193632, DOI 10.2307/2315393
- Raymond E. Smithson, Topologies generated by relations, Bull. Austral. Math. Soc. 1 (1969), 297–306. MR 257956, DOI 10.1017/S0004972700042167
- R. E. Smithson, Fixed points of order preserving multifunctions, Proc. Amer. Math. Soc. 28 (1971), 304–310. MR 274349, DOI 10.1090/S0002-9939-1971-0274349-1
- R. E. Smithson, Fixed point theorems for certain classes of multifunctions, Proc. Amer. Math. Soc. 31 (1972), 595–600. MR 288750, DOI 10.1090/S0002-9939-1972-0288750-4
- R. E. Smithson, Fixed points in partially ordered sets, Pacific J. Math. 45 (1973), 363–367. MR 316323, DOI 10.2140/pjm.1973.45.363
- Lynn A. Steen and J. Arthur Seebach Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0266131
- Alfred Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309. MR 74376, DOI 10.2140/pjm.1955.5.285
- A. D. Wallace, A fixed-point theorem for trees, Bull. Amer. Math. Soc. 47 (1941), 757–760. MR 4758, DOI 10.1090/S0002-9904-1941-07556-7
- L. E. Ward Jr., A note on dendrites and trees, Proc. Amer. Math. Soc. 5 (1954), 992–994. MR 71759, DOI 10.1090/S0002-9939-1954-0071759-2
- L. E. Ward Jr., Completeness in semi-lattices, Canadian J. Math. 9 (1957), 578–582. MR 91264, DOI 10.4153/CJM-1957-065-3
- L. E. Ward Jr., A fixed point theorem for multi-valued functions, Pacific J. Math. 8 (1958), 921–927. MR 103446, DOI 10.2140/pjm.1958.8.921
- L. E. Ward Jr., Characterization of the fixed point property for a class of set-valued mappings, Fund. Math. 50 (1961/62), 159–164. MR 133122, DOI 10.4064/fm-50-2-159-164
- E. S. Wolk, Order-compatible topologies on a partially ordered set, Proc. Amer. Math. Soc. 9 (1958), 524–529. MR 96596, DOI 10.1090/S0002-9939-1958-0096596-8
- Gail S. Young Jr., The introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479–494. MR 16663, DOI 10.2307/2371828
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 184 (1973), 153-173
- MSC: Primary 54H25; Secondary 54F05, 54F20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0328900-X
- MathSciNet review: 0328900