Bounded solutions for a class of Hamiltonian systems

TL;DR

This paper develops a method to find bounded solutions for Hamiltonian systems by approximating solutions of Dirichlet problems on expanding intervals, with implications for both ODEs and PDEs.

Contribution

It introduces a novel approach to obtain bounded solutions as limits of Dirichlet problem solutions for Hamiltonian systems and extends this method to PDEs.

Findings

Bounded solutions are obtained as limits of Dirichlet problem solutions.

A priori estimates enable passage to the limit as interval size tends to infinity.

Method extends from ODEs to PDEs, broadening applicability.

Abstract

We obtain bounded for all $t$ solutions of ordinary differential equations as limits of the solutions of the corresponding Dirichlet problems on $(-L,L)$, with $L \rightarrow \infty$. We derive a priori estimates for the Dirichlet problems, allowing passage to the limit, via a diagonal sequence. This approach carries over to the PDE case.