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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 $$\widetilde W^{1,p}_T=\{v\in W^{1,p}_T:\int^T_0 v(t)\,dt=0\}$$.

##### MSC:
 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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