Projected Gronwall-Bellmanʹs inequality for integrable functions

Without continuity of functions and monotonicity of some functions in Lemma 1, in this paper we discuss a projected Gronwall-Bellmanʹs inequality with only integrability. In the proof of our main result some techniques are applied to overcome difficulties caused by lack of continuity and monotonicity. As applications of our main results we also give estimates in various concrete cases.