The author studies the integral equation \[ x(t) = \left( g_1(t)+\int_0^t a_1(t-s)F_1(s,x(s))\, ds\right) \ldots \left( g_p(t)+\int_0^t a_p(t-s)F_p(s,x(s))\, ds\right) \] and shows, using the Banach fixed-point theorem, that the equation under certain assumptions involving the growth of the functions \(x\mapsto F_i(t,x)\) has a continuous solution. Then the author shows that under certain, quite complicated but explicit assumptions, the solution converges exponentially to \(0\).