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

A note on almost periodic variational equations

Peter Giesl 1, and  Martin Rasmussen 2,

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.
Keywords: almost periodic difference equation, variational equation, almost periodic function, Almost periodic differential equation.
Mathematics Subject Classification: Primary: 34C27; Secondary: 26A24, 42A75, 26B0.
Citation: Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure & 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, ().   Google Scholar

[2]

A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, ().   Google Scholar

[3]

P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 3722-3733.  Google Scholar

[4]

G. R. Sell, Nonautonomous differential equations and dynamical systems - I. The basic theory, Transactions of the American Mathematical Society, 127 (1967), 241-262.  Google Scholar

show all references

References:
[1]

C. Corduneanu, "Almost Periodic Functions,", Interscience Tracts in Pure and Applied Mathematics, ().   Google Scholar

[2]

A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, ().   Google Scholar

[3]

P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 3722-3733.  Google Scholar

[4]

G. R. Sell, Nonautonomous differential equations and dynamical systems - I. The basic theory, Transactions of the American Mathematical Society, 127 (1967), 241-262.  Google Scholar

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