KYUNGPOOK Math. J. 54(2014), 555-575 http://dx.doi.org/10.5666/KMJ.2014.54.4.555 On Some New Nonlinear Integral Inequalities of Gronwall- Bellman Type Hassan mostafa El-Owaidy, Abdelwahab Abbas Ragab and Ahmed Abdel-moneim Eldeeb * Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt e-mail : elowaidy@yahoo.com, awaragab158@yahoo.com and ahmed_na707@yahoo.com(ahmedeldeeb@azhar.edu.eg) Waleed mostafa kamal Abuelela Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt Deanery of Academic Services, Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia e-mail : w_abuelela@yahoo.com Abstract. In this paper, we establish some new nonlinear integral inequalities of Gronwall-Bellman type. These inequalities generalize some famous inequalities which can be used in applications as handy tools to study the qualitative as well as quantitative properties of solutions of some nonlinear ordinary differential and integral equations. More accurately we extend certain results which have been proved in A. Abdeldaim and M. Yak- out [1] and H. El-Owaidy, A. A. Ragab, A. Abdeldaim [7] too. 1. Introduction It is well known that, the differential and integral inequalities of one variable which provide explicit bounds on unknown functions, occupy a very privileged po- sition in the development of the theory of linear and nonlinear ordinary differential and integral equations see for example [2, 8, 9, 10, 11]. In the qualitative theory of differential and Volterra integral equations, Gronwall type inequalities of one vari- able for the real functions play a very important role. In the recent years, these inequalities have been greatly enriched by the recognition of their potential and * Corresponding Author. Received December 28, 2012; accepted April 15, 2013. 2010 Mathematics Subject Classification: 54C60, 54H25. Key words and phrases: Gronwall-Bellman inequality, integral inequalities, ordinary dif- ferential equations, integral equations. 555