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Periodic solutions of a class of singular Hamiltonian systems. (English) Zbl 0648.34048
The author investigates the existence of periodic solutions, with a prescribed period, of the system of N second order differential equations $d\sp 2u/dt\sp 2=-V\sp 1(t,u)$ where $V\sp 1(t,.)$ denotes the gradient of the function V(t,.) defined on ${\bbfR}\sp N-\{0\}$. V(t,x) is supposed to be T-periodic int. With suitable restriction on V, basically related to its singular properties at $x=0$ and its behaviour at $\vert x\vert \to \infty$, theorems are proved pertaining to the existence of at least one non-constant T-periodic $C\sp 2$ solution, as well as the existence of infinitely many non-constant T-periodic $C\sp 2$ solutions. The proofs are heavily based on functional analysis, in particular of the nature and behaviour of the critical points. The paper mostly deals with $N>2$ when the set of singularities of V is simple.
Reviewer: N.D.Sengupta

MSC:
34C25Periodic solutions of ODE
70H05Hamilton’s equations
34A34Nonlinear ODE and systems, general
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References:
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