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Radial solutions for a class of Hénon type systems with partial interference with the spectrum
Stability problems in nonautonomous linear differential equations in infinite dimensions
1. | Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil |
2. | Departament de Matemàtiques, Universitat Politècnica de Catalunya, ETSEIB, Av. Diagonal 647, 08028 Barcelona, Spain |
In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [
As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [
References:
[1] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Syustems, Springer-Verlag Berlin, 2011.
doi: 10.1007/978-1-4614-4581-4. |
[2] |
W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath & Co., Boston, 1965. |
[3] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag Berlin Heidelberg New York, 1970. |
[4] |
Ju. L. Dalekĭi and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1974. |
[5] |
A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathmatics, Vol. 377, Springer-Verlag, Berlin-Heidelberg-New York, 1974. |
[6] |
P. R. Halmos, A Hilbert Space Problem Book, Graduate texts in Mathematics, Vol. 19, Springer-Verlag, New York, Heidelberg, Berlin, 1974. |
[7] |
J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Krieger Publishing Co., Huntington, New York, 1980. |
[8] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math., Vol 840, Springer-Verlag, Berlin, 1981. |
[9] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980. |
[10] |
P. E. Kloeden and H. M. Rodrigues,
Dynamics of a Class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.
doi: 10.1016/j.na.2010.12.025. |
[11] |
H. M. Rodrigues, Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo (São Carlos), 1970. |
[12] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[13] |
Ch. E. Rickart, General Theory of Banach Algebras, Princeton, D. van Nostrand, 1960. |
[14] |
H. M. Rodrigues, T. Caraballo and M. Gameiro,
Dynamics of a class of ODEs via wavelets, Commun. Pure Appl. Anal., 16 (2017), 2337-2355.
doi: 10.3934/cpaa.2017115. |
[15] |
H. M. Rodrigues, M. A. Teixeira and Ma. Gameiro,
On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications to the Cima system, J. Dyn. Differ. Equ., 30 (2018), 1199-1219.
doi: 10.1007/s10884-017-9598-y. |
[16] |
H. M. Rodrigues and J. G. Ruas-Filho,
Evolution equations: dichotomies and the Fredholm alternative for bounded solutions, J. Differ. Equ., 119 (1995), 263-283.
doi: 10.1006/jdeq.1995.1091. |
[17] |
H. M. Rodrigues and J. Solà-Morales,
Linearization of Class $C^1$ for Contractions on Banach Spaces, J. Differ. Equ., 201 (2004), 351-382.
doi: 10.1016/j.jde.2004.02.013. |
[18] |
H. M. Rodrigues and J. Solà-Morales,
On the Hartman-Grobman Theorem with Parameters, J. Dyn. Differ. Equ., 22 (2010), 473-489.
doi: 10.1007/s10884-010-9160-7. |
[19] |
H. M. Rodrigues and J. Solà-Morales,
Invertible Contractions and Asymptotically Stable ODE's that are not $C^1$-Linearizable, J. Dyn. Differ. Equ., 18 (2006), 961-973.
doi: 10.1007/s10884-006-9050-1. |
[20] |
H. M. Rodrigues and J. Solà-Morales,
Smooth Linearization for a Saddle on Banach Spaces, J. Dyn. Differ. Equ., 16 (2004), 767-793.
doi: 10.1007/s10884-004-6116-9. |
[21] |
H. M. Rodrigues, J. Solà-Morales, An example on Lyapunov stability and linearization, preprint, arXiv: 1902.02111, (2019). |
show all references
References:
[1] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Syustems, Springer-Verlag Berlin, 2011.
doi: 10.1007/978-1-4614-4581-4. |
[2] |
W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath & Co., Boston, 1965. |
[3] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag Berlin Heidelberg New York, 1970. |
[4] |
Ju. L. Dalekĭi and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1974. |
[5] |
A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathmatics, Vol. 377, Springer-Verlag, Berlin-Heidelberg-New York, 1974. |
[6] |
P. R. Halmos, A Hilbert Space Problem Book, Graduate texts in Mathematics, Vol. 19, Springer-Verlag, New York, Heidelberg, Berlin, 1974. |
[7] |
J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Krieger Publishing Co., Huntington, New York, 1980. |
[8] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math., Vol 840, Springer-Verlag, Berlin, 1981. |
[9] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980. |
[10] |
P. E. Kloeden and H. M. Rodrigues,
Dynamics of a Class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.
doi: 10.1016/j.na.2010.12.025. |
[11] |
H. M. Rodrigues, Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo (São Carlos), 1970. |
[12] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[13] |
Ch. E. Rickart, General Theory of Banach Algebras, Princeton, D. van Nostrand, 1960. |
[14] |
H. M. Rodrigues, T. Caraballo and M. Gameiro,
Dynamics of a class of ODEs via wavelets, Commun. Pure Appl. Anal., 16 (2017), 2337-2355.
doi: 10.3934/cpaa.2017115. |
[15] |
H. M. Rodrigues, M. A. Teixeira and Ma. Gameiro,
On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications to the Cima system, J. Dyn. Differ. Equ., 30 (2018), 1199-1219.
doi: 10.1007/s10884-017-9598-y. |
[16] |
H. M. Rodrigues and J. G. Ruas-Filho,
Evolution equations: dichotomies and the Fredholm alternative for bounded solutions, J. Differ. Equ., 119 (1995), 263-283.
doi: 10.1006/jdeq.1995.1091. |
[17] |
H. M. Rodrigues and J. Solà-Morales,
Linearization of Class $C^1$ for Contractions on Banach Spaces, J. Differ. Equ., 201 (2004), 351-382.
doi: 10.1016/j.jde.2004.02.013. |
[18] |
H. M. Rodrigues and J. Solà-Morales,
On the Hartman-Grobman Theorem with Parameters, J. Dyn. Differ. Equ., 22 (2010), 473-489.
doi: 10.1007/s10884-010-9160-7. |
[19] |
H. M. Rodrigues and J. Solà-Morales,
Invertible Contractions and Asymptotically Stable ODE's that are not $C^1$-Linearizable, J. Dyn. Differ. Equ., 18 (2006), 961-973.
doi: 10.1007/s10884-006-9050-1. |
[20] |
H. M. Rodrigues and J. Solà-Morales,
Smooth Linearization for a Saddle on Banach Spaces, J. Dyn. Differ. Equ., 16 (2004), 767-793.
doi: 10.1007/s10884-004-6116-9. |
[21] |
H. M. Rodrigues, J. Solà-Morales, An example on Lyapunov stability and linearization, preprint, arXiv: 1902.02111, (2019). |

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