Minimal Periods for Ordinary Differential Equations in Strictly Convex Banach Spaces and Explicit Bounds for some l^p-Spaces

TL;DR

This paper investigates the minimal periods of solutions to differential equations in Banach spaces, establishing strict inequalities in strictly convex spaces and providing improved bounds for l^p spaces near p=2.

Contribution

It proves the strictness of the period bound in strictly convex Banach spaces and refines bounds for l^p spaces close to p=2 using Wirtinger's inequality.

Findings

The inequality T L >= c is strict in strictly convex Banach spaces.

Improved lower bounds for l^p spaces near p=2.

Sharp bounds known for Hilbert spaces are extended and refined.

Abstract

Let x(t) be a non-constant T-periodic solution to the ordinary differential equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz continuous with constant L. Then there exists a constant c such that T L >= c, with c only depending on X. It is known that c >= 6 in any Banach space and that c = 2{\pi} in any Hilbert space, but whereas the bound of c = 2 pi is sharp in any Hilbert space, there exists only one known example of a Banach space such that c = 6 is optimal. In this paper, we show that the inequality is in fact strict in any strictly convex Banach space. Moreover, we improve the lower bound for l^p(R^n) and L^p(M, {\mu}) for a range of p close to p = 2 by using a form of Wirtinger's inequality for functions in W^{1,p}([0, T ], L^p(M, {\mu})).