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July  2019, 18(4): 1695-1709. doi: 10.3934/cpaa.2019080

Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

Pablo Amster 1,, , Mariel Paula Kuna 1, and  Gonzalo Robledo 2,

1. 

IMAS – CONICET, Universidad de Buenos Aires, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria - Pabellón I - (C1428EGA), Buenos Aires, Argentina
2. 

Universidad de Chile, Departamento de Matemáticas, Facultad de Ciencias, Casilla 653, Santiago, Chile

* Corresponding author

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author is supported by projects CONICET PIP 11220130100006CO and UBACyT 20020160100002BA.

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $ T $-periodic solutions lying inside a bounded domain $ \Omega\subset \mathbb{R}^{N} $ is, generically, at least $ |\chi \pm 1|+1 $, where $ \chi $ denotes the Euler characteristic of $ \Omega $. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.

Keywords: Delay differential systems, multiple periodic solutions, Poincaré operator, fixed points, topological degree.
Mathematics Subject Classification: Primary: 34K13, 47H10; Secondary: 47H11.
Citation: Pablo Amster, Mariel Paula Kuna, Gonzalo Robledo. Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1695-1709. doi: 10.3934/cpaa.2019080
References:
[1]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, First edition, Birkhäuser, Boston, 2004. doi: 10.1007/978-0-8176-8124-1.  Google Scholar

[2]

J. Haddad, Topología y geometría aplicada al estudio de algunas ecuaciones diferenciales de segundo orden, (Spanish) [Topology and Geometry Applied to the Study of Some Second Order Differential Equations] Ph.D thesis, Universidad de Buenos Aires, Argentina, 2012. Available from: cms.dm.uba.ar/academico/carreras/doctorado/2012/tesisHaddad.pdf Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer–Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann., 96 (1927), 225-250.  doi: 10.1007/BF01209164.  Google Scholar

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R. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence RI, 1994. doi: 10.1090/gsm/004.  Google Scholar

[6]

J. Liu, G. N'Guérékata and Nguyen Van Minh, Topics on Stability and Periodicity in Abstract Differential Equations, World Scientific, Singapore, 2008. doi: 10.1142/9789812818249.  Google Scholar

[7]

M. A. Krasnoselskii, The Operator of Translation along the Trajectories of Differential Equations, American Mathematical Society, Providence RI, 1968.  Google Scholar

[8]

M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerVerlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[9] J. Milnor, Topology from a Differential Viewpoint, University of Virginia Press, 1965.   Google Scholar
[10]

R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 42 (1990), 505-516.  doi: 10.1112/jlms/s2-42.3.505.  Google Scholar

[11]

M. Pinto, Pseudo-almost periodic solutions of neutral integral and differential equations with applications, Nonlinear Anal., 72 (2010), 4377-4383.  doi: 10.1016/j.na.2009.12.042.  Google Scholar

[12]

S. Smale, An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 87 (1965), 861-866.  doi: 10.2307/2373250.  Google Scholar

[13]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer–Verlag, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

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F. Wecken, Fixpunktklassen Ⅲ: Mindestzahlen von Fixpunkten, Math. Ann., 118 (1941/1943), 544-577.  doi: 10.1007/BF01487386.  Google Scholar

show all references

References:
[1]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, First edition, Birkhäuser, Boston, 2004. doi: 10.1007/978-0-8176-8124-1.  Google Scholar

[2]

J. Haddad, Topología y geometría aplicada al estudio de algunas ecuaciones diferenciales de segundo orden, (Spanish) [Topology and Geometry Applied to the Study of Some Second Order Differential Equations] Ph.D thesis, Universidad de Buenos Aires, Argentina, 2012. Available from: cms.dm.uba.ar/academico/carreras/doctorado/2012/tesisHaddad.pdf Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer–Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[4]

H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann., 96 (1927), 225-250.  doi: 10.1007/BF01209164.  Google Scholar

[5]

R. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence RI, 1994. doi: 10.1090/gsm/004.  Google Scholar

[6]

J. Liu, G. N'Guérékata and Nguyen Van Minh, Topics on Stability and Periodicity in Abstract Differential Equations, World Scientific, Singapore, 2008. doi: 10.1142/9789812818249.  Google Scholar

[7]

M. A. Krasnoselskii, The Operator of Translation along the Trajectories of Differential Equations, American Mathematical Society, Providence RI, 1968.  Google Scholar

[8]

M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerVerlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[9] J. Milnor, Topology from a Differential Viewpoint, University of Virginia Press, 1965.   Google Scholar
[10]

R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 42 (1990), 505-516.  doi: 10.1112/jlms/s2-42.3.505.  Google Scholar

[11]

M. Pinto, Pseudo-almost periodic solutions of neutral integral and differential equations with applications, Nonlinear Anal., 72 (2010), 4377-4383.  doi: 10.1016/j.na.2009.12.042.  Google Scholar

[12]

S. Smale, An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 87 (1965), 861-866.  doi: 10.2307/2373250.  Google Scholar

[13]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer–Verlag, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[14]

F. Wecken, Fixpunktklassen Ⅲ: Mindestzahlen von Fixpunkten, Math. Ann., 118 (1941/1943), 544-577.  doi: 10.1007/BF01487386.  Google Scholar

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