Nonlinear Analysis 66 (2007) 2140–2165 www.elsevier.com/locate/na About a generalization of Bellman–Bihari type inequalities for discontinuous functions and their applications Yu.A. Mitropolskiy a , G. Iovane b,∗ , S.D. Borysenko c a International Mathematical Center of the National Academy of Sciences of Ukraine, Tereshcenkivska, 3, Kyiv, Ukraine b Department of Engineering of Informatics and Applied Mathematics, University of Salerno, Ponte don Melillo str., 84084 Fisciano, Salerno, Italy c Department of Differential Equations, National Technical University of Ukraine, “KPI” 37 Peremohy Prosp. 03056 Kyiv, Ukraine Received 3 March 2006; accepted 3 March 2006 Abstract In the present paper we introduce the conditions of solvability for Chaplygin’s problem with discontinuous functions in two independent variables, satisfying integro-sum inequalities. The new type of nonlinear integral and Wendroff’s inequality for discontinuous functions are investigated. As applications, the conditions of boundedness solutions of partial differential equations of hyperbolic type with impulse influence on some hypersurfaces {Γ j }⊂ R 2 + are obtained. Some historical aspects of the theory of integro- sum inequalities are presented. c  2006 Elsevier Ltd. All rights reserved. MSC: 34B15; 26D15; 26D20 Keywords: Integral inequalities; Integro-sum inequalities; Impulsive integro-differential equations; Boundedness; Estimates 1. Introduction The theory of integral inequalities [1,2,4,5,26,28,45] and its numerous linear, nonlinear generalizations for continuous, discontinuous functions of one and n independent variables ∗ Corresponding author. Tel.: +39 089 964268; fax: +39 089 964191. E-mail addresses: iovane@diima.unisa.it (G. Iovane), borys@mbox.com.ua (S.D. Borysenko). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.03.006