Minimal periods for ordinary differential equations in strictly convex Banach spaces and explicit bounds for some LpLp-spaces

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