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

A Perron-type theorem for nonautonomous differential equations with different growth rates

Yongxin Jiang 1,, , Can Zhang 1, and  Zhaosheng Feng 2,

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.

Keywords: Perron-type theorem, Lyapunov exponent, stability, dichotomy, nonautonomous differential equations.
Mathematics Subject Classification: Primary: 34D08, 34D10.
Citation: Yongxin Jiang, Can Zhang, Zhaosheng Feng. A Perron-type theorem for nonautonomous differential equations with different growth rates. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 995-1008. doi: 10.3934/dcdss.2017052
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, 1965.  Google Scholar

[12]

P. Hartman and A. Wintner, Asymptotic integrations of linear differential equations, Amer. J. Math., 77 (1955), 45-86.  doi: 10.2307/2372422.  Google Scholar

[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.  Google Scholar

[14]

F. Lettenmeyer, Üer das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgleichungssystemen, Verlag d. Bayr. Akad. d. Wiss, 1929. Google Scholar

[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.  Google Scholar

[16]

O. Perron, Üer Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, 1965.  Google Scholar

[12]

P. Hartman and A. Wintner, Asymptotic integrations of linear differential equations, Amer. J. Math., 77 (1955), 45-86.  doi: 10.2307/2372422.  Google Scholar

[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.  Google Scholar

[14]

F. Lettenmeyer, Üer das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgleichungssystemen, Verlag d. Bayr. Akad. d. Wiss, 1929. Google Scholar

[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.  Google Scholar

[16]

O. Perron, Üer Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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