Better bounds for periodic solutions of differential equations in Banach spaces

Abstract

Let f f be Lipschitz with constant L L in a Banach space and let x ( t ) x(t) be a P P -periodic solution of x ′ ( t ) = f ( x ( t ) ) x’(t) = f(x(t)) . We show that P ⩾ 6 / L P \geqslant 6/L . An example is given with P = 2 π / L P = 2\pi /L , so the bound is nearly strict. We also give a short proof that P ⩾ 2 π / L P \geqslant 2\pi /L in a Hilbert space.