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Existence solutions for second order Hamiltonian systems. (English) Zbl 1333.34061

The paper discusses the existence of periodic solutions of the non-autonomous dynamical system \[ -\ddot{x}(t)=B(t)x(t)+\nabla_x V(t,x(t)), \eqno(1) \] where \(x(t)=(x_1(t),\dots,x_n(t))\), and \(V(t,x)\) is continuous on \(\mathbb{R}^{n+1}\) with \[ \nabla_x V(t,x(t))=(\partial V/\partial x_1,\dots, \partial V/\partial x_n)\in C(\mathbb{R}^{n+1}, \mathbb{R}^n). \] For each \(x\in \mathbb{R}^n\), the function \(V(t,x)\) is \(T\)-periodic in \(t\). The matrix \(B(t)\) is symmetric with \(T\)-periodic components from \(L^1(0,T)\). The authors provide conditions which imply that that there exists a \(T\)-periodic weak solution of (1) whose weak second derivative belongs to \(L^1(0,T)\), and conditions giving infinitely many of such solutions. Both subquadratic and superquadratic problems are considered.

MSC:

34C25 Periodic solutions to ordinary differential equations
37C27 Periodic orbits of vector fields and flows
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C60 Nonautonomous smooth dynamical systems
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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