In this paper we prove some fixed point theorems of the Banach and Krasnosel’skii type for mappings on the $m$-tuple Cartesian product of a Banach algebra $X$ over $ℝ$. Using these theorems existence results for a system of integral equations of the Gripenberg’s type are proved. A sufficient condition for the nonexistence of blowing-up solutions of this system of integral equations is also proved.