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

### ON GENERALIZED CAUCHY AND PEXIDER FUNCTIONAL
EQUATIONS OVER A FIELD

### Mariusz Bajger

Department of Matematics, The University of Queensland,
Brisbane 4072, Australia

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
*s**t* for all *s*, *t*
∈ **K**. Further, we apply this result to describe
solutions of the functional equation
*F*(*s*+*t*) = *K*(*s**t*) ◦
*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*(*s**t*)
+ *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.

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