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On the Cauchy problem for the differential equation f(t, x, x′, …, x(k)) = 0

In the sequel, given k, n ∈ ℕ, p ∈ [1, ∞] and a compact real interval I, we denote by Wk, p(I, ℝn) (simply by Wk,p(I if n = 1) the space of all functions u ∈ Ck−1(I, ℝn) such that uk−1 is absolutely continuous in I and u(k) ∈ Lp (I, ℝn). Very recently, in [11], J. R. L. Webb and S. C. Welsh obtained the following existence result.