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Solutions of a second-order Hamiltonian system with periodic boundary conditions. (English) Zbl 1207.34027
The authors are concerned with second-order Hamiltonian systems $$\ddot{u}(t)=\nabla\,F(t,u(t))\quad \text{ a.e. }\,0<t<T$$ with periodic boundary conditions $$u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,$$ where $F: [0,T]\times\mathbb{R}^N\longrightarrow\mathbb{R}$ is measurable in $t$, continuously differentiable in $x$ and there exist $a\in C(\mathbb{R}^+,\mathbb{R}^+)$ and $b\in L^1([0,T],\mathbb{R}^+)$ such that $$\vert F(t,x)\vert\le a(\vert x\vert)b(t),\;\vert\nabla F(t,x)\vert\le a(\vert x\vert)b(t)\ \forall x\in\mathbb{R}^N\,\text{ and a.e. }\,t\in[0,T].$$ According to whether the gradient of $F$ is sublinearly bounded, i.e., there exist $f,g\in L^1([0,T],\mathbb{R}^+)$ and $0\le\alpha<1$ such that $$\vert\nabla F(t,x)\vert\le f(t)\vert x\vert^\alpha+g(t),$$ or linearly bounded, i.e., there exist $f,g\in L^1([0,T],\mathbb{R}^+)$ such that $$\vert\nabla F(t,x)\vert\le f(t)\vert x\vert+g(t),$$ the authors prove five existence theorems for solutions in the Hilbert space of absolutely periodic functions $u$ with $\dot{u}\in L^2([0,T],\mathbb{R}^N)$. The obtained solutions minimize the corresponding function $\varphi$ defined by $$\varphi(u)=\tfrac{1}{2}\int_0^T\vert\dot{u}(t)\vert^2\,dt+\int_0^TF(t,u(t))\,dt.$$ The saddle point theorem both and the least action principle are used. The paper ends with four examples.

MSC:
34B15Nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE
58E30Variational principles on infinite-dimensional spaces
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References:
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