Brake subharmonic solutions of first order Hamiltonian systems

TL;DR

This paper investigates brake subharmonic solutions in first order non-autonomous Hamiltonian systems using Galerkin approximation and $L$-Maslov index theory, establishing existence and distinctness of certain periodic solutions.

Contribution

It introduces a novel application of Galerkin approximation and $L$-Maslov index theory to prove the existence of distinct brake subharmonic solutions under specific conditions.

Findings

Existence of $jT$-periodic brake solutions.

Distinctness of solutions for different periods.

Application of $L$-Maslov index theory in this context.

Abstract

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the $L$-Maslov type index theory to study the properties of brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that when the positive integers $j$ and $k$ satisfies the certain conditions, there exists a $jT$-periodic nonconstant brake solution $z_{j}$ such that $z_{j}$ and $z_{kj}$ are distinct.