Boundedness of solutions for the reversible system with low regularity in time

TL;DR

This paper proves that all solutions of a class of reversible differential systems with low regularity in time are bounded, extending the understanding of solution behavior under less smooth conditions.

Contribution

It establishes boundedness of solutions for a reversible system with coefficients of low regularity, broadening previous results that required higher smoothness.

Findings

All solutions are bounded under given conditions.

Boundedness holds despite low regularity of coefficients.

Results apply to systems with coefficients in C^1 and L^1 classes.

Abstract

In the present paper, it is proved that all solutions are bounded for the reversible system \ddot{x}+\sum_{i=0}^{l}b_{i}(t)x^{2i+1}\dot{x}+x^{2n+1}+\sum_{i=0}^{n-1}a_{i}(t)x^{2i+1}=0, 0\leq l\leq [\frac{n}{2}]-1,t\in\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}, where a_{i}(t)\in C^{1}(\mathbb{T}^{1})\;([\frac{n-1}{2}]+1\leq i\leq n-1), a_{j}(t)\in L^{1}(\mathbb{T}^{1})\;(0\leq j\leq [\frac{n-1}{2}]) and b_{k}(t)\in C^{1}(\mathbb{T}^{1})\;(0\leq k\leq l).