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

A 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 $\text{R}$². 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.