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Zbl 0936.34034
Carminati, Carlo
Forced systems with almost periodic and quasiperiodic forcing term.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 32, No.6, 727-739 (1998). ISSN 0362-546X

Bounded solutions to $\ddot x-\nabla V(x)= h(t)$ are investigated where the forcing term is bounded. The main theorem is the following: if $V\in C^1(\Omega_0; \bbfR)$ has a local minimum at $x_0= \Omega_0$, and for all $x\in B_\rho(x_0)= \{x\in \bbfR^N:|x- x_0|\le\rho\}$ the potential $V(x)$ is of the form $V(x)= {1\over 2}Ax\cdot x+ U(x)$, where $A$ is a symmetric matrix such that $Ax\cdot x\ge \sigma|x|^2$ and $U(x)\in C^1(\Omega:\bbfR)$ is a convex function defined on some open set $\Omega\supset B_\rho(x_0)$. It is also assumed that $h(t)$ is a bounded continuous function and that $\sup_{t\in\bbfR}|h(t)|\le \sigma\rho$. Then the system above has one bounded solution $u(t)$ such that $\sup_{t\in\bbfR}|u(t)- x_0|\le {1\over\sigma}\sup_{t\in \bbfR}|h(t)|$.
[P.Smith (Keele)]
MSC 2000:
*34C27 Almost periodic solutions of ODE

Keywords: forced systems; bounded solutions; ordinary differential equations

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