Nonlinear Analysis 67 (2007) 1550–1559 www.elsevier.com/locate/na About new analogies of Gronwall–Bellman–Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems A. Gallo a , A.M. Piccirillo b,∗ a Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Claudio, 21 - 80125 Napoli, Italy b Department of Civil Engineering, Second University of Naples, Via Roma, 29 - 81031 Aversa (CE), Italy Received 12 July 2006; accepted 12 July 2006 Abstract In this article we present new integral Gronwall–Bellman–Bihari type inequalities for discontinuous functions (integro-sum inequalities). As applications, we investigate estimated solutions for impulsive differential systems, conditions of boundedness, stability, practical stability. c  2006 Elsevier Ltd. All rights reserved. MSC: 34B15; 26D15; 26D20 Keywords: Integral inequalities; Discontinuous functions; Estimates; Impulsive differential systems 1. Introduction The celebrated results by J. Gronwall, R. Bellman, I. Bihari and their various linear and nonlinear generalizations in the case of continuous functions, which satisfy integral (linear, nonlinear, one-dimensional, multi-dimensional, functional, etc.) inequalities are well known. The first analogous results (Gronwall–Bellman lemma, Bihari lemma) for the case of discontinuous functions were obtained only during the eighties of the past century [11,27,40]. As for the continuous case, these results gave a strong impulse to new investigations in the development of the Bellman–Bihari method for integral inequalities solvability in discontinuous functions (integro-sum inequalities [4, 8–18,27,32,37–40]). The main role of this method is its effective application in investigating qualitative characteristics of solutions for systems of differential, integro-differential, functional-differential equations with impulsive perturbations such as: boundedness, attraction, stability by Lyapunov, (practical) stability by Chetaev, etc. Moreover, we have the possibility to write analytical representations for estimates of solutions for systems with different kinds of (continuous, discrete, also parametric) perturbations. ∗ Corresponding author. Tel.: +39 081 7683548; fax: +39 081 7683643. E-mail addresses: angallo@unina.it (A. Gallo), annamaria.piccirillo@unina2.it (A.M. Piccirillo). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.07.038