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 & 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

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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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