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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 \(\mathbb R^n\), \(b(\cdot)\) is a continuous, \(T\)-periodic real function and \(V(\cdot) \in C^2(\mathbb R^n, \mathbb R)\) 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.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 0894.34036
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References:

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