Original scientific paper

On generalized Cauchy and Pexider functional equations over a field

Mariusz Bajger

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.

Keywords

Cauchy equation; Pexider equation; Cauchy difference

Hrčak ID:

8189

URI

https://hrcak.srce.hr/8189

Publication date:

1.12.1998.

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