We solve the analytic Cauchy problem for the generalized two-component Camassa-Holm system introduced by R. M. Chen and Y. Liu. We show the existence of a unique local/global-in-time analytic solution under certain conditions. This is the first result about global analyticity for a Camassa-Holm-like system. The method of proof is basically that developed by Barostichi, Himonas and Petronilho. The main differences between their proof and ours are twofold: (i) the system of Chen and Liu is not symmetric in the two unknowns and our estimates are not trivial generalization of those in their articles, (ii) we have simplified their argument by using fewer function spaces and the main result is stated in a simple and natural way.

We are devoted with a singular integro-differential problem. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. We make use of Greenlee's idea [8] to rewrite the singular integro-differential problem in a regular form. A concrete example illustrating our abstract results is given.

In this paper, we study the behavior of solutions of the nonlinear first order delay differential equations and have a new sufficient condition under which every solution tends to zero asymptotically. Under certain restrictions, this condition is weaker than 3/2-condition which was showed by So et al. [7], Yoneyama [8].