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A Perron-type theorem for nonautonomous differential equations with different growth rates
| 1. | Department of Mathematics, College of Science, Hohai University, Nanjing, Jiangsu 210098, China |
| 2. | School of Mathematical and Statistical Sciences, University of Texas-Rio Grande valley, Edinburg, Texas 78539, USA |
We show that if the Lyapunov exponents associated to a linear equation $x'=A(t)x$ are equal to the given limits, then this asymptotic behavior can be reproduced by the solutions of the nonlinear equation $x'=A(t)x+f(t, x)$ for any sufficiently small perturbation $f$. We consider the linear equation with a very general nonuniform behavior which has different growth rates.
References:
| [1] |
L. Barreira, J. Chu and C. Valls,
Robustness of nonuniform dichotomies with different growth rates, São Paulo J. Math. Sci., 5 (2011), 203-231.
doi: 10.11606/issn.2316-9028.v5i2p203-231. |
| [2] |
L. Barreira, D. Dragičevič and C. Valls, Tempered exponential dichotomies and Lyapunov exponents for perturbations, Commun. Contemp. Math., 18 (2016), 1550058, 16 pp.
doi: 10.1142/S0219199715500583. |
| [3] |
L. Barreira and Ya. Pesin,
Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23 American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/ulect/023. |
| [4] |
L. Barreira and C. Valls,
Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, 1926 Springer, Berlin, 2008.
doi: 10.1007/978-3-540-74775-8. |
| [5] |
L. Barreira and C. Valls,
Nonautonomous equations with arbitrary growth rates: A Perron-type theorem, Nonlinear Anal., 75 (2012), 6203-6215.
doi: 10.1016/j.na.2012.06.027. |
| [6] |
L. Barreira and C. Valls,
A Perron-type theorem for nonautonomous difference equations, Nonlinearity, 26 (2013), 855-870.
doi: 10.1088/0951-7715/26/3/855. |
| [7] |
L. Barreira and C. Valls,
A Perron-type theorem for nonautonomous differential equations, J. Differential Equations, 258 (2015), 339-361.
doi: 10.1016/j.jde.2014.09.012. |
| [8] |
A. J. G. Bento and C. Silva,
Nonuniform $(μ,ν)$
-dichotomies and local dynamics of difference equations, Nonlinear Anal., 75 (2012), 78-90.
doi: 10.1016/j.na.2011.08.008. |
| [9] |
J. Chu,
Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math., 137 (2013), 1031-1047.
doi: 10.1016/j.bulsci.2013.03.003. |
| [10] |
C. Coffman,
Asymptotic behavior of solutions of ordinary difference equations, Trans. Amer. Math. Soc., 110 (1964), 22-51.
doi: 10.1090/S0002-9947-1964-0156122-9. |
| [11] |
W. Coppel,
Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, 1965. |
| [12] |
P. Hartman and A. Wintner,
Asymptotic integrations of linear differential equations, Amer. J. Math., 77 (1955), 45-86.
doi: 10.2307/2372422. |
| [13] |
Y. X. Jiang and F. F. Liao, Admissibility for nonuniform $(μ, ν)$ contraction and dichotomy, Abstr. Appl. Anal., 2012 (2012), Article ID 741696, 23pp.
doi: 10.1155/2012/741696. |
| [14] |
F. Lettenmeyer,
Üer das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgleichungssystemen, Verlag d. Bayr. Akad. d. Wiss, 1929. |
| [15] |
K. Matsui, H. Matsunaga and S. Murakami,
Perron type theorem for functional differential equations with infinite delay in a Banach space, Nonlinear Anal., 69 (2008), 3821-3837.
doi: 10.1016/j.na.2007.10.017. |
| [16] |
O. Perron,
Üer Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.
doi: 10.1007/BF01180524. |
| [17] |
M. Pituk,
A Perron type theorem for functional differential equations, J. Math. Anal. Appl., 316 (2006), 24-41.
doi: 10.1016/j.jmaa.2005.04.027. |
| [18] |
M. Pituk,
Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl., 322 (2006), 1140-1158.
doi: 10.1016/j.jmaa.2005.09.081. |
| [19] |
H. L. Zhu, C. Zhang and Y. X. Jiang,
A Perron-type theorem for nonautonomous difference equations with nonuniform behavior, Electron. J. Qual. Theory Differ. Equ, 36 (2015), 1-15.
doi: 10.14232/ejqtde.2015.1.36. |
show all references
References:
| [1] |
L. Barreira, J. Chu and C. Valls,
Robustness of nonuniform dichotomies with different growth rates, São Paulo J. Math. Sci., 5 (2011), 203-231.
doi: 10.11606/issn.2316-9028.v5i2p203-231. |
| [2] |
L. Barreira, D. Dragičevič and C. Valls, Tempered exponential dichotomies and Lyapunov exponents for perturbations, Commun. Contemp. Math., 18 (2016), 1550058, 16 pp.
doi: 10.1142/S0219199715500583. |
| [3] |
L. Barreira and Ya. Pesin,
Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23 American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/ulect/023. |
| [4] |
L. Barreira and C. Valls,
Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, 1926 Springer, Berlin, 2008.
doi: 10.1007/978-3-540-74775-8. |
| [5] |
L. Barreira and C. Valls,
Nonautonomous equations with arbitrary growth rates: A Perron-type theorem, Nonlinear Anal., 75 (2012), 6203-6215.
doi: 10.1016/j.na.2012.06.027. |
| [6] |
L. Barreira and C. Valls,
A Perron-type theorem for nonautonomous difference equations, Nonlinearity, 26 (2013), 855-870.
doi: 10.1088/0951-7715/26/3/855. |
| [7] |
L. Barreira and C. Valls,
A Perron-type theorem for nonautonomous differential equations, J. Differential Equations, 258 (2015), 339-361.
doi: 10.1016/j.jde.2014.09.012. |
| [8] |
A. J. G. Bento and C. Silva,
Nonuniform $(μ,ν)$
-dichotomies and local dynamics of difference equations, Nonlinear Anal., 75 (2012), 78-90.
doi: 10.1016/j.na.2011.08.008. |
| [9] |
J. Chu,
Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math., 137 (2013), 1031-1047.
doi: 10.1016/j.bulsci.2013.03.003. |
| [10] |
C. Coffman,
Asymptotic behavior of solutions of ordinary difference equations, Trans. Amer. Math. Soc., 110 (1964), 22-51.
doi: 10.1090/S0002-9947-1964-0156122-9. |
| [11] |
W. Coppel,
Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, 1965. |
| [12] |
P. Hartman and A. Wintner,
Asymptotic integrations of linear differential equations, Amer. J. Math., 77 (1955), 45-86.
doi: 10.2307/2372422. |
| [13] |
Y. X. Jiang and F. F. Liao, Admissibility for nonuniform $(μ, ν)$ contraction and dichotomy, Abstr. Appl. Anal., 2012 (2012), Article ID 741696, 23pp.
doi: 10.1155/2012/741696. |
| [14] |
F. Lettenmeyer,
Üer das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgleichungssystemen, Verlag d. Bayr. Akad. d. Wiss, 1929. |
| [15] |
K. Matsui, H. Matsunaga and S. Murakami,
Perron type theorem for functional differential equations with infinite delay in a Banach space, Nonlinear Anal., 69 (2008), 3821-3837.
doi: 10.1016/j.na.2007.10.017. |
| [16] |
O. Perron,
Üer Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.
doi: 10.1007/BF01180524. |
| [17] |
M. Pituk,
A Perron type theorem for functional differential equations, J. Math. Anal. Appl., 316 (2006), 24-41.
doi: 10.1016/j.jmaa.2005.04.027. |
| [18] |
M. Pituk,
Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl., 322 (2006), 1140-1158.
doi: 10.1016/j.jmaa.2005.09.081. |
| [19] |
H. L. Zhu, C. Zhang and Y. X. Jiang,
A Perron-type theorem for nonautonomous difference equations with nonuniform behavior, Electron. J. Qual. Theory Differ. Equ, 36 (2015), 1-15.
doi: 10.14232/ejqtde.2015.1.36. |
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