In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

Using a canonical linear embedding of the algebra ${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of $\overline{\C}$-valued $\C$-linear maps on the space ${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.