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Three periodic solutions for perturbed second order Hamiltonian systems. (English) Zbl 1185.34048
Consider the following problem
$\begin{cases} -\ddot{u}+A(t)u=\nabla F(t,u)+\lambda\nabla G(t,u) \qquad \text{a.e. in } [0,T],\\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{cases}\tag{1}$ where $$\lambda\in \mathbb{R}$$, $$T$$ is a positive real number. $$A : [0,T] \rightarrow \mathbb{R}^{N\times N}$$ is a continuous map from the interval $$[0,T]$$ to the set of $$N$$-order symmetric matrices, $$F,G :[0,T]\times\mathbb{R}^N \rightarrow \mathbb{R}$$ are measurable with respect to $$t$$ for every $$x\in \mathbb{R}^N$$, continuously differentiable in $$x$$ for almost every $$t\in [0,T]$$ and satisfy the following condition:
$\sup_{|x|\leq c}\max\{|F(.,x)|,|G(.,x)|,|\nabla F(.,x)|,|\nabla G(.,x)|\}\in L^1[0,T]$ for all $$c\geq 0$$. Assume that
$$(F_1)$$
$$\lim_{|x| \to +\infty}\left(\frac{1}{2}\lambda_1(A)|x|^2-F(t,x)\right)=+\infty$$ uniformly in [0,T], where
$\lambda_1(A)=\inf_{u\in H_T^1,\|u\|=1}\left(\int_0^T\left(|\dot{u}(t)|^2 +\left(A(t)u(t),u(t)\right)\right)dt\right);$
$$(F_2)$$
there exists $$\delta>0$$ such that $$\frac{1}{2}\lambda_1(A)|x|^2-F(t,x)>0$$ for all $$x\in \mathbb{R}^N \backslash \{0\}$$ with $$|x|<\delta$$ and a.e. $$t\in \text{[0,T]}$$;
$$(F_3)$$
There exists $$x_0\in \mathbb{R}^N$$ such that $$\int_0^T(A(t)x_0,x_0)dt<\int_0^T F(t,x_0)dt$$.
Then the authors prove that there exist $$\lambda^*>0$$ and $$r>0$$ such that, for every $$\lambda\in ]-\lambda^*, \lambda^*[$$, problem (1) admits at least three distinct solutions which belong to $$B_r(0)\subseteq H_T^1$$ by using variational methods.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 47J30 Variational methods involving nonlinear operators
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##### References:
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