On some new integral inequalities of Growall± Bellman type H. El-Owaidy, A. Ragab, A. Abdeldaim * Department of Mathematics, Faculty of Science, AL-Azhar University, Nasr-City, Cairo 11884, Egypt Abstract In this paper, we establish some new integral inequalities of the Gronwall±Bellman type that have a wide range of the applications in the theory of ordinary dierential equations. The purpose of this paper is to extend the results proved in G.B. Pachpatte [J. Math. Phys. Sci. 8 (1974) 309±318; Bull. Soc. Grece 15 (1974) 7±12; J. Math. Phys. Sci. 9 (1975) 405±416; J. Math. An a. Appl. 49 (1975) 794±802; 5 (1975) 141±150; 189 (1995) 128±144; 69 (1965) 362±367], for the continuous inequalities of Gronwall±Bellman and Reid type. Ó 1999 Published by Elsevier Science Inc. All rights reserved. 1. Introduction The dierential and integral inequalities occupy a very privileged position in the theory of dierential and integral equations. In recent years, these in- equalities have been greatly enriched by the recognition of their potential and intrinsic worth in many applications of the applied sciences. The celebrated Gronwall inequality known now as Gronwall±Bellman±Raid inequality [2,3,6], provided explicit bounds on solutions of a class of linear integral inequalities. On the basis of various motivations, this inequality has been extended and used in various contexts. The ®rst non-linear version of this inequality is due to Bihari [4,5], which has been further generalized in several dierent directions. In the literature there are many papers which make use of this inequality very frequently, to obtain global existence, uniqueness, stability, and other Applied Mathematics and Computation 106 (1999) 289±303 www.elsevier.nl/locate/amc * Corresponding author. Fax: +20-2-2629356 0096-3003/99/$ - see front matter Ó 1999 Published by Elsevier Science Inc. All rights reserved. PII:S0096-3003(98)10131-5