Explicit bound on some retarded integral inequalities and applications

Document Type: Original Article


1 Laboratoire des telecommunications, Faculte des Sciences et de la Technologie, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria.

2 University of Badji Mokhtar Annaba, Algeria.


In this paper, we establish some new retarded integral inequalities, which can be used as a tool in the qualitative
study of certain properties of retarded integrodifferential equations.


[1] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10 (1943). 643-647.

[2] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations. Rend. Circ. Mat. Palermo (2) 7(1958), 261-269.

[3] H. El-Owaidy, A. Abdeldaim, A.A. El-Deeb, On some new nonlinear retarded integral inequalities with iterated integrals
and their applications in differential-integral equations, Math. Sci. Lett., 3 (2014), 157-164.

[4] Y. H. Kim, On some new integral inequalities for functions in one and two variables, Acta Math. Sin. (Engl. Ser.), 21
(2005), 423-434.

[5] O. Lipovan, A retarded Gronwall-like inequality, and its applications, J. Math. Anal. Appl., 252 (2000), 389-401.

[6] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. (2), 20 (1919), 292-296.

[7] B. G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl., 267 (2002), 48-61.

[8] M. H. M. Rashidi, Explicit bounds on retarded Gronwall-Bellman inequality, Tamkang J. Math., 43 (2012), 99-108.