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Existence of periodic solutions to second-order Hamiltonian systems with potential indefinite in sign. (English) Zbl 1157.37329
From the text: The authors study second-order systems of differential equations of the form $\ddot x + W'(t,x) = 0$, where $W(t,x) = (A(t)x,x) + b(t)V(x)$ with $A(\cdot)$ a continuous, $T$-periodic matrix-valued function, $(\cdot,\cdot)$ denotes the scalar product in $\Bbb R^n$, $b(\cdot)$ is a continuous, $T$-periodic real function and $V(\cdot) \in C^2(\Bbb R^n, \Bbb R)$ is a nonnegative, superquadratic function. Using variational methods and applying a linking theorem, the authors prove the existence of a nontrivial $T$-periodic solution in a case when $A(t)$ is not negative definite and several additional technical conditions are satisfied.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C25Periodic solutions of ODE
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
Full Text: DOI
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