October  2017, 10(5): 995-1008. doi: 10.3934/dcdss.2017052

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

* Corresponding author: Yongxin Jiang

Received  January 2016 Revised  February 2017 Published  June 2017

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.

Citation: Yongxin Jiang, Can Zhang, Zhaosheng Feng. A Perron-type theorem for nonautonomous differential equations with different growth rates. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 995-1008. doi: 10.3934/dcdss.2017052
References:
[1]

L. BarreiraJ. 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. MatsuiH. 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. ZhuC. 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. BarreiraJ. 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. MatsuiH. 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. ZhuC. 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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