Glasnik Matematicki, Vol. 33, No.2 (1998), 239-249.

ON GENERALIZED CAUCHY AND PEXIDER FUNCTIONAL EQUATIONS OVER A FIELD

Mariusz Bajger

Department of Applied Mathematics, The University of Adelaide, Adelaide 5005, Australia
e-mail: mbajger@maths.adelaide.edu.au

Abstract.   Let K be a commutative field and (P,+) be a uniquely 2-divisible group (not necessarily abelian). We characterize all functions T : K → P such that the Cauchy difference T(s+t) - T(t) - T(s) depends only on the product st for all s, t ∈ K. Further, we apply this result to describe solutions of the functional equation F(s+t) = K(st) ◦ H(s) ◦ G(t), where the unknown functions F, K, H, G map the field K 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 K into P.

1991 Mathematics Subject Classification.   39B52, 39B12.

Key words and phrases.   Cauchy equation, Pexider equation, Cauchy difference.