Existence of solution for perturbed fractional Hamiltonian systems

TL;DR

This paper proves the existence of at least one nontrivial solution for a class of perturbed fractional Hamiltonian systems with variable coefficients, extending the understanding of such systems under coercivity conditions.

Contribution

It demonstrates the existence of solutions for perturbed fractional Hamiltonian systems with variable coefficients, assuming coercivity at infinity, which is a novel result in this context.

Findings

Existence of at least one nontrivial solution established.

Solution existence proven under coercivity assumptions.

Extends fractional Hamiltonian system theory to variable coefficient cases.

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}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(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.