The functional differential equation x′(t) = x(x(t))

AbstractA classification of the solutions of the functional differential equation x′(t) = x(x(t)) is given and it is proved that every solution either vanishes identically or is strictly monotonie. For monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x(t0) = x0 where (t0, x0) is any given pair of reals in some specified subset of R2. Every monotonically increasing solution is thus obtained. It is analytic and depends analytically on t0 and x0. Only for t0 = x0 = 1 is the question of analyticity still open