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Conley index for manifold-valued retarded functional differential equations without uniqueness of solutions

Part of: Functional-differential and differential-difference equations Topological dynamics

Published online by Cambridge University Press:  19 April 2021

M. C. Carbinatto  and
K. P. Rybakowski
M. C. Carbinatto
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, 13.560-970, São CarlosSP, Brazil ([email protected])
K. P. Rybakowski
Affiliation:
Universität Rostock, Institut für Mathematik, Ulmenstraße 69, Haus 3, 18057, Rostock, Germany ([email protected])
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Abstract

We develop a Conley index theory for retarded functional differential equations $\dot x=f(x_{t})$ with values in a differentiable manifold and (merely) continuous nonlinearities f. We use this index to establish an existence result for nonconstant full solutions of such equations.

Keywords

infinite-dimensional dynamical systemsConley indexfunctional differential equations on manifolds

MSC classification

Primary: 37B30: Index theory, Morse-Conley indices
Secondary: 34K05: General theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 428 - 449
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Benevieri, P., Calamai, A., Furi, M. and Pera, M. P.. Retarded functional differential equations on manifolds and applications to motion problems for forced constrained systems. Adv Nonlinear Stud. 9 (2009), 199–214.CrossRefGoogle Scholar
Berger, M. and Gostiaux, B.. Differential geometry: manifolds, curves, and surfaces (New York: Springer Verlag, 1988).CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. On a general Conley index continuation principle for singular perturbation problems. Ergodic Theory Dyn. Syst. 22 (2002), 729–755.CrossRefGoogle Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Conley index and homology index braids in singular perturbation problems without uniqueness of solutions. Topol. Methods Nonlinear Anal. 35 (2010), 1–32.Google Scholar
Carbinatto, M. C. and Rybakowski, K. P.. Conley index continuation for some classes of RFDEs on manifolds. Fundam. Math. 250 (2020), 41–62.CrossRefGoogle Scholar
Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M. and Walther, H.-O.. Delay equations, functional-, complex- and nonlinear analysis (New York: Springer-Verlag, 1995).Google Scholar
Hale, J. and Verduyn Lunel, S. M.. Introduction to functional differential equations (New York: Springer-Verlag, 1993).CrossRefGoogle Scholar
Hale, J., Magalhães, L. T. and Oliva, W.. An introduction to infinite dimensional dynamical systems - geometric theory — with an appendix by K. P. Rybakowski (New York: Springer-Verlag, 2002).Google Scholar
Izydorek, M. and Rybakowski, K. P.. On the Conley index in Hilbert spaces in the absence of uniqueness. Fundam. Math. 171 (2002), 31–52.CrossRefGoogle Scholar
Lasota, A. and Yorke, J. A.. The generic property of existence of solutions of differential equations in Banach space. J. Diff. Equ. 13 (1973), 1–12.CrossRefGoogle Scholar
Oliva, W. M.. Functional differential equations on compact manifolds and an approximation theorem. J. Diff. Equ. 5 (1969), 483–496.CrossRefGoogle Scholar
Rybakowski, K. P.. On the homotopy index for infinite-dimensional semiflows. Trans. AMS 269 (1982), 351–382.CrossRefGoogle Scholar
Rybakowski, K. P.. The homotopy index and partial differential equations (Berlin: Springer-Verlag, 1987).CrossRefGoogle Scholar