Suppose f , h f,h and G G are functions with values in a normed complete ring. With suitable restrictions on these functions, it is established that \[ f ( x ) = h ( x ) + ∫ a x f ( u ) G ( u , v ) f(x) = h(x) + \int _a^x {f(u)G(u,v)} \] for a ≤ x ≤ b a \leq x \leq b only if ∫ a x h ( u ) G ( u , v ) v Π x ( 1 + G ) \int _a^x {h(u)G{{(u,v)}_v}{\Pi ^x}} (1 + G) exists and is f ( x ) − h ( x ) f(x) - h(x) for a ≤ x ≤ b a \leq x \leq b , and that \[ f ( x ) = h ( x ) + ∫ a x G ( u , v ) f ( u ) f(x) = h(x) + \int _a^x {G(u,v)f(u)} \] for a ≤ x ≤ b a \leq x \leq b only if ∫ a x x Π v ( 1 + G ) G ( u , v ) h ( u ) \int _a^x {_x{\Pi ^v}} (1 + \mathcal {G})G(u,v)h(u) exists and is f ( x ) − h ( x ) f(x) - h(x) for a ≤ x ≤ b a \leq x \leq b , where G ( s , r ) = G ( r , s ) \mathcal {G}(s,r) = G(r,s) .