A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: 0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from 0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.