A variation of Gronwall's lemma

We prove a variation of Gronwall’s lemma. The formulation and proof of the classical Gronwall’s lemma can be found in [1]. We prove here a variation of this lemma, which we were not able to find in the literature. The main difference from usual versions of Gronwall’s lemma is that −λ is negative. Lemma 1 Let g : [0,∞[→ R be a continuous function, C a real number and λ a positive real number. Assume that ∀u, t 0 ≤ u ≤ t g(t) − g(u) ≤ ∫ t