Classical theorems on differential inequalities [Coddington and Levinson, Theory of ordinary differential equations, Mc Graw-Hill Book Company, Inc., 1995, Hartman, Ordinary differential equations, John Wiley & Sons, 1964, Walter, Differential and integral inequalities, Springer-Verlag, 1970] are generalized for initial value problems of the kind and where ƒ : C([a, b]; Rn ) → Lloc (]a, b]; Rn ) is a singular Volterra operator, c 0 ∈ Rn , h : [a, b] → [0, +∞[ is continuous and positive on ]a, b], ‖ · ‖ is a norm in Rn , and [u]+ and [u]– are respectively the positive and the negative part of the vector u ∈ Rn .