Integral inequalities for retarded Volterra equations

Integral inequalities are very useful in the qualitative analysis of differential and integral equations. Starting with [O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl. 252 (2000) 389–401], several recent investigations, see [O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443; B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61; B.G. Pachpatte, On some retarded integral inequalities and applications, J. Inequal. Pure Appl. Math. 3 (2002), Article 18; B.G. Pachpatte, On a certain retarded integral inequality and its applications, J. Inequal. Pure Appl. Math. 5 (2004), Article 19; B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004), Article 80], were devoted to retarded integral inequalities. In this paper we consider the case of retarded Volterra integral equations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the asymptotic behaviour of the solutions.