The equations
k
(
s
+
t
)
=
ℓ
(
s
)
+
n
(
t
)
k(s+t)=\ell (s)+n(t)
and
k
(
s
+
t
)
=
m
(
s
)
n
(
t
)
k(s+t)=m(s)n(t)
, called Pexider equations, have been completely solved on
R
2
.
\mathbb {R}^2.
If they are assumed to hold only on an open region, they can be extended to
R
2
\mathbb {R}^2
(the second when
k
k
is nowhere 0) and solved that way. In this paper their common generalization
k
(
s
+
t
)
=
ℓ
(
s
)
+
m
(
s
)
n
(
t
)
k(s+t)=\ell (s)+m(s)n(t)
is extended from an open region to
R
2
\mathbb {R}^2
and then completely solved if
k
k
is not constant on any proper interval. This equation has further interesting particular cases, such as
k
(
s
+
t
)
=
ℓ
(
s
)
+
m
(
s
)
k
(
t
)
k(s+t)=\ell (s)+m(s)k(t)
and
k
(
s
+
t
)
=
k
(
s
)
+
m
(
s
)
n
(
t
)
,
k(s+t)=k(s)+m(s)n(t),
that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in
R
2
.
\mathbb {R}^2.
Thus these problems are solved too.