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arXiv 1402.6919math.AP
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扰动分数阶哈密顿系统解的存在性

Existence of solution for perturbed fractional Hamiltonian systems

César Torres

备注：
arXiv admin note: substantial text overlap with arXiv:1212.5811
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摘要：

本文证明了由\begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t), \end{eqnarray}给出的一类扰动分数阶哈密顿系统解的存在性，其中$α\in (1/2, 1)$，$t\in \mathbb{R}$，$u\in \mathbb{R}^{n}$，$L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$是对称正定矩阵，对所有的$t\in \mathbb{R}$，$W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$，$\nabla W$是$W$在$u$的梯度.本文的新颖之处在于，假设$L$在无穷远处是强制的，我们证明了（\ref{eq00}）至少有一个非平凡解。

英文摘要：

In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t), \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$ and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming $L$ is coercive at infinity we show that (\ref{eq00}) at least has one nontrivial solution.

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