NONLINEAR STUDIES - www.nonlinearstudies.com Vol. 19, No. 1, pp. 13-24, 2012 c ⃝ CSP - Cambridge, UK; I&S - Florida, USA, 2012 Multidimensional impulse inequalities and general Bihari type inequalities for discontinuous functions with delay A. Gallo 1 , A.M. Piccirillo 2 ⋆ 1 Department of Mathematics and Applications “R.Caccioppoli” University of Naples “Federico II” Via Claudio 21 - 80125 Napoli, Italy E-mail: angallo@unina.it 2 Department of Civil Engineering Second University of Naples Via Roma, 29 - 81031 Aversa (CE), Italy Phone +39 081 7683548 - fax +39 081 7683643 E-mail: annamaria.piccirillo@unina2.it ⋆ Corresponding Author. E-mail address: angallo@unina.it Abstract. In this article we investigate some impulsive integro-functional inequalities for functions of n independent variables. The problem of reducing multidimensional integro-sum functional in- equalities to one-dimensional inequalities is also considered (using conditions of Chaplygin problem solvability for impulsive integral inequalities). Some new analogies of Wendroff–type inequalities for discontinuous functions with finite jumps are obtained. 1 Introduction Over the last 20 years, the theory of ordinary impulsive differential systems has undergone extensive development, and this explains the appearance of new problems in several fields of the investigation such as: physics, biology, chemistry, electronics and many others. The integral representations of the solutions of ordinary impulsive differential systems usually have both a continuous and a discrete part and may be written in the following form: x(t )= φ(t )+ ∫ t t 0 k(t , s, x(s))ds + ∑ t 0 <t k <t ψ i (t , τ i )I i (u(τ i − 0)) (A) Mathematics Subject Classification: Primary: 34B15, 26D15, 26D20 Keywords: Inequalities, discontinuous functions, delay, impulse integral inequalities.