Glasnik Matematicki, Vol. 33, No.2 (1998), 239-249.
ON GENERALIZED CAUCHY AND PEXIDER FUNCTIONAL
EQUATIONS OVER A FIELD
Department of Matematics, The University of Queensland,
Brisbane 4072, Australia
Department of Applied Mathematics, The University of Adelaide,
Adelaide 5005, Australia
K be a commutative
field and (P,+) be a uniquely 2-divisible group
(not necessarily abelian). We characterize all functions T :
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.
Key words and phrases. Cauchy equation, Pexider
equation, Cauchy difference.
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