On Generalized Cauchy and Pexider Functional Equations over a Field

Let lK be a commutative field and (P, +) be a uniquely 2-divisible group (not necessarily abelian). We characterize all functions T: IK -+ P such that the Cauchy difference T(s+ t) T(t) T(s) depends only on the product st for all s, t E ~{. Further, we apply this result to describe solutions of the functional equation F(s + t) = K(st) 0 H(s) 0 G(t), where the unknown functions F, K, H, G map the field IK into some function spaces arranged so that the compositions make sense. Conditions are established under which the equation can be reduced to a corresponding generalized Cauchy equation, and the general solution is given. Finally, we solve the equation F(s + t) = K(st) + H(s) + G(t) for functions F, K, H, G mapping IKinto P. The paper generalizes some results from [ll], [13] and, up to some extent, from [2].