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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 $H:[0,T]\times \Bbb R^{2N}\to\Bbb R$ 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 $u\in\Bbb R^{2N}$. 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 $\beta,\gamma\in L^q(0,T;\Bbb R^+)$ such that $$\delta|u|-\beta(t)\le H(t,u)\le \tfrac\alpha q |u|^q+\gamma(t)$$ for all $u\in\Bbb R^{2N}$ 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 $\widetilde W^{1,p}_T=\{v\in W^{1,p}_T:\int^T_0 v(t)\,dt=0\}$.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34C25 Periodic solutions of ODE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
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