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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
\[ | F(t,x)|\leq a(| x|)b(t),\;|\nabla F(t,x)|\leq a(| x|)b(t)\;\forall x\in\mathbb{R}^N\,\text{ and a.e. }\,t\in[0,T]. \]
According to whether the gradient of \(F\) is sublinearly bounded, i.e., there exist \(f,g\in L^1([0,T],\mathbb{R}^+)\) and \(0\leq\alpha<1\) such that
\[ |\nabla F(t,x)|\leq f(t)| x|^\alpha+g(t), \]
or linearly bounded, i.e., there exist \(f,g\in L^1([0,T],\mathbb{R}^+)\) such that
\[ |\nabla F(t,x)|\leq f(t)| x|+g(t), \]
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
\[ \varphi(u)=\tfrac{1}{2}\int_0^T|\dot{u}(t)|^2\,dt+\int_0^TF(t,u(t))\,dt. \]
The saddle point theorem both and the least action principle are used. The paper ends with four examples.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces
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References:

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