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Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $
Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium
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 |
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.
References:
[1] |
R. F. Brown, A Topological Introduction to Nonlinear Analysis, First edition, Birkhäuser, Boston, 2004.
doi: 10.1007/978-0-8176-8124-1. |
[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. |
[4] |
H. Hopf,
Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann., 96 (1927), 225-250.
doi: 10.1007/BF01209164. |
[5] |
R. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence RI, 1994.
doi: 10.1090/gsm/004. |
[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. |
[7] |
M. A. Krasnoselskii, The Operator of Translation along the Trajectories of Differential Equations, American Mathematical Society, Providence RI, 1968. |
[8] |
M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerVerlag, Berlin, 1984.
doi: 10.1007/978-3-642-69409-7. |
[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. |
[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. |
[12] |
S. Smale,
An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 87 (1965), 861-866.
doi: 10.2307/2373250. |
[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. |
[14] |
F. Wecken,
Fixpunktklassen Ⅲ: Mindestzahlen von Fixpunkten, Math. Ann., 118 (1941/1943), 544-577.
doi: 10.1007/BF01487386. |
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. |
[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. |
[4] |
H. Hopf,
Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann., 96 (1927), 225-250.
doi: 10.1007/BF01209164. |
[5] |
R. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence RI, 1994.
doi: 10.1090/gsm/004. |
[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. |
[7] |
M. A. Krasnoselskii, The Operator of Translation along the Trajectories of Differential Equations, American Mathematical Society, Providence RI, 1968. |
[8] |
M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerVerlag, Berlin, 1984.
doi: 10.1007/978-3-642-69409-7. |
[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. |
[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. |
[12] |
S. Smale,
An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 87 (1965), 861-866.
doi: 10.2307/2373250. |
[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. |
[14] |
F. Wecken,
Fixpunktklassen Ⅲ: Mindestzahlen von Fixpunkten, Math. Ann., 118 (1941/1943), 544-577.
doi: 10.1007/BF01487386. |
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