# American Institute of Mathematical Sciences

May  2011, 10(3): 983-994. doi: 10.3934/cpaa.2011.10.983

## A note on almost periodic variational equations

 1 Department of Mathematics, University of Sussex, Brighton, BN1 9RF 2 Martin Rasmussen, Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom

Received  April 2009 Revised  August 2009 Published  December 2010

The variational equation of a nonautonomous differential equation $\dot x= F(t,x)$ along a solution $\mu$ is given by $\dot x=D_x F(t,\mu(t))x$. We consider the question whether the variational equation is almost periodic provided that the original equation is almost periodic by a discussion of the following problem: Is the derivative $D_xF$ almost periodic whenever $F$ is almost periodic? We give a negative answer in this paper, and the counterexample relies on an explicit construction of a scalar almost periodic function whose derivative is not almost periodic. Moreover, we provide a necessary and sufficient condition for the derivative $D_xF$ to be almost periodic.
In addition, we also discuss this problem in the discrete case by considering the variational equation $x_{n+1}=D_xF(n,\mu_n)x_n$ of the almost periodic difference equation $x_{n+1}=F(n,x_n)$ along an almost periodic solution $\mu_n$. In particular, we provide an example of a function $F$ which is discrete almost periodic uniformly in $x$ and whose derivative $D_xF$ is not discrete almost periodic.
Citation: Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure and Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983
##### References:
 [1] C. Corduneanu, "Almost Periodic Functions,", Interscience Tracts in Pure and Applied Mathematics, (). [2] A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, (). [3] P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 3722-3733. [4] G. R. Sell, Nonautonomous differential equations and dynamical systems - I. The basic theory, Transactions of the American Mathematical Society, 127 (1967), 241-262.

show all references

##### References:
 [1] C. Corduneanu, "Almost Periodic Functions,", Interscience Tracts in Pure and Applied Mathematics, (). [2] A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, (). [3] P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 3722-3733. [4] G. R. Sell, Nonautonomous differential equations and dynamical systems - I. The basic theory, Transactions of the American Mathematical Society, 127 (1967), 241-262.
 [1] Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263 [2] Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025 [3] Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032 [4] Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857 [5] Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301 [6] Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291 [7] Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315 [8] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [9] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026 [10] Marko Kostić. Almost periodic type functions and densities. Evolution Equations and Control Theory, 2022, 11 (2) : 457-486. doi: 10.3934/eect.2021008 [11] Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 [12] Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345 [13] Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113 [14] Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268 [15] Bin-Guo Wang, Wan-Tong Li, Lizhong Qiang. An almost periodic epidemic model in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 271-289. doi: 10.3934/dcdsb.2016.21.271 [16] Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control and Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101 [17] Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51 [18] Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503 [19] Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022 [20] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

2020 Impact Factor: 1.916