Existence of solution for perturbed fractional Hamiltonian systems

Abstract

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 $\alpha \in (1/2,1)$, $t\in \mathbb{R}$, $u\in {\mathbb{R}}^n$, $L\in C(\mathbb{R},{\mathbb{R}}^{n^2})$ 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.

arXiv admin note: substantial text overlap with arXiv:1212.5811

References in corpus (1)

• Ground state solution for a class of differential equations with left and right fractional derivatives