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Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. (English) Zbl 1082.34012
Here, the following boundary value problem for Hamiltonian systems is studied \[ J\dot u(t)+\nabla H(t,u(t))=0\quad \text{a.e. on } [0,T],\quad u(0)=u(T), \] where the function \(H:[0,T]\times \mathbb R^{2N}\to\mathbb R\) is called Hamiltonian and \(J\) is a symplectic \(2N\times 2N\)-matrix. Special attention is given to the case in which the Hamiltonian \(H\), besides being measurable on \(t\in [0,T]\), is convex and continuously differentiable with respect to \(u\in\mathbb R^{2N}\). The basic assumption is that the Hamiltonian \(H\) satisfies the following growth condition:
Let \(p\in(1,2)\) and \(q=\frac {p}{p-1}\). There exist positive constants \(\alpha\), \(\overline\alpha\) and functions \(\beta,\gamma\in L^q(0,T;\mathbb R^+)\) such that \[ \delta|u|-\beta(t)\leq H(t,u)\leq \tfrac\alpha q |u|^q+\gamma(t) \] for all \(u\in\mathbb R^{2N}\) and a.e. \(t\in[0,T]\). The main result assures that under suitable bounds on \(\alpha,\delta\) and the functions \(\beta,\gamma\), the problem above has at least a solution that belongs to \(W^{1,p}_T\). Such a solution corresponds, in the duality, to a function that minimizes the dual action restricted to a subset of \(\widetilde W^{1,p}_T=\{v\in W^{1,p}_T:\int^T_0 v(t)\,dt=0\}\).

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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