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On the topological structure of the set of generalized solutions of the catenary problem

Published online by Cambridge University Press:  14 November 2011

J. C. Alexander  and
Michael Reeken
J. C. Alexander
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
Michael Reeken
Affiliation:
Fachbereich Mathematik, Bergische Universität-Gesamthochschule Wuppertal, Gaußstraße 20, D-5600 Wuppertal 1, Federal Republic of Germany
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Synopsis

The global topological structure of the space of configurations of a non-rotating elastic string under compression and tension is studied. The part of the string under tension is specified by a measurable subset of the interval. The set of such intervals, with the Hausdorff topology, is considered a parameter space for the equation satisfied by the string, and the solutions are shown to form an infinitedimensional continuum over this parameter space. A new global topological theorem is needed, since the parameter space is not Euclidean. The topological theorem is based on the fixed-point transfer.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 339 - 348
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Alexander, J. C.. A primer on connectivity. In Proc. Conf. on Fixed Point Theory, 1980, Faddell, E. and Fournier, G. (eds). Lecture Notes in Mathematics 886 (Berlin: Springer, 1981).Google Scholar
2Alexander, J. C. and Antman, S. S.. Global behavior of nonlinear equations depending on infinite-dimensional parameters. Indiana Univ. Math. J. 32 (1983), 39–62.CrossRefGoogle Scholar
3Alexander, J. C., Massab, I.ò and Pejsachowicz, J.. On the connectivity properties of the solution set of infinitely-parametrized families of vector fields. Bull. Un. Mat. Ital.(6) 1A (1982), 309–312.Google Scholar
4Antman, S. S.. Multiple equilibrium states of nonlinearly elastic strings. SIAM J. Appf. Math. 37, 588–604.CrossRefGoogle Scholar
5Dold, A.. The fixed-point transfer of fibre-preserving maps. Math. Z. 148 (1976), 215–244.CrossRefGoogle Scholar
6Jurewicz, W. and Wallman, H.. Dimension Theory (Princeton: Princeton Univ. Press, 1941).Google Scholar
7Massab, I.ò and Pejsachowicz, J.. On the connectivity properties of the solution set of parametrized families of compact vector fields. J. Funct. Anal, in press.Google Scholar
8Reeken, M.. Exotic equilibrium states of the elastic string. Math. Ann. 268 (1984), 59–84.CrossRefGoogle Scholar