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A note on almost periodic variational equations

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  • 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.
    Mathematics Subject Classification: Primary: 34C27; Secondary: 26A24, 42A75, 26B05.


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  • [1]

    C. Corduneanu, "Almost Periodic Functions," Interscience Tracts in Pure and Applied Mathematics, no. 22,


    A. M. Fink, "Almost Periodic Differential Equations," Springer Lecture Notes in Mathematics, vol. 377,


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


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

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