The letters 5, G and H denote a linearly ordered set, a normed complete Abelian group with zero element 0, and the set of functions from G to G that map O into 0, respectively. In addition, if Z E H and there exists an additive function a from S x S to the nonnegative numbers such that 11 Y(x, y)P-V(x, y)Q 11 < a(x, Y)llP-Q 12 for each {x, y, P, Q } in S x S x C x G, then V E 6S onlyif J*YYPexistsforeach {x,y,P} in S x S x G,and V E OMP onlyifxIIY(l + V)Pexistsforeach {x,y,P} in S x S x G. It is established that Z E e fi if, and only if, V E e @. Then, this relationship is used in the solution of integral equations of the form f(x) = h(x) + I c t U(u, v)J(u) + Y(u, v)f(v)], where U and V are in e s . This research extends known results in that requirements pertaining to the additivity of U and V are weakened.