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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.

34B15Nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE
58E30Variational principles on infinite-dimensional spaces
Full Text: DOI
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