Generalization of a retarded Gronwall-like inequality and its applications

This paper generalizes a Lipovan's result of Gronwall-like inequalities [J. Math. Anal. Appl. 252 (2000) 389] to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. From our result Bihari's result and Pinto's result can be deduced as some special cases. Our result can be applied to give the global existence and estimate solutions. We also apply it to improve an estimation for almost periodicity of an invariant manifold.