Minimal periods for ordinary differential equations in strictly convex Banach spaces and explicit bounds for some [formula omitted]-spaces

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 TL⩾c, with c only depending on X. It is known that c⩾6 in any Banach space and that c=2π in any Hilbert space, but whereas the bound of c=2π 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 ℓp(Rn) and Lp(M,μ) for a range of p close to p=2 by using a form of Wirtinger's inequality for functions in Wper1,p([0,T],Lp(M,μ)).