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Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign. (English) Zbl 0894.34036

The author studies the system \(\ddot x= W'(z,t)= 0\), with \(x(0)= x(T)\), \(\dot x(0)= \dot x(T)\), where \(W(x,t)= \langle A(t)x,x\rangle+ b(t)V(x)\). Here, \(b\) is a continuous \(T\)-periodic function, \(V\in C^2(\mathbb{R}^N\times \mathbb{R})\) has superquadratic behavior, \(A\) is a continuous \(T\)-periodic symmetric matrix function, with \(\langle A(t)\xi,\xi\rangle> 0\), for all \(\xi\in \mathbb{R}^N\), and \(|\xi|= 1\).
The author shows that if \(\lim_{x\to 0} V(x)/| x^2|= 0\), and \(\int^T_0 b(t)dt> 0\) then the system has a \(T\)-periodic solution. With some additional assumptions including \(\int^T_0 b(t)dt< 0\) (reversing the sign) it also follows that a \(T\)-periodic solution exists. Using a variational approach, she proves that the corresponding Lagrangian action integral \[ {1\over 2}\int^T_0| \dot x^2| dt-{1\over 2}\int^T_0 [\langle A(t)x,x\rangle+ b(t)V(x)]dt \] has a critical point in an appropriately chosen finite subspace of the original Hilbert space \(H^1\).
Reviewer: V.Komkov (Roswell)

MSC:

34C25 Periodic solutions to ordinary differential equations
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