Cross-sections of solution funnels

Let X   be a separable infinite dimensional real Banach space. We denote by F(X)F(X) the class of continuous functions f:R×X→Xf:R×X→X such that the ODE u′=f(t,u),u(t0)=x,t0∈R,x∈X, has a global solution for any initial condition. Our main result states that A⊂XA⊂X is the cross-section of a solution funnel of the ODE u′=f(t,u),u(0)=0u′=f(t,u),u(0)=0, for some f∈F(X)f∈F(X), if and only if A is an analytic set.