For each of m and n a positive integer denote by
S
(
m
,
i
)
S(m,i)
the space of all real-valued symmetric i-linear functions on
E
m
,
i
=
1
,
2
,
…
,
n
{E_m},i = 1,2, \ldots ,n
. Denote by L a nonzero linear functional on
S
(
m
,
n
)
S(m,n)
, denote by f a real-valued analytic function on
E
m
×
R
×
S
(
m
,
1
)
×
⋯
×
S
(
m
,
[
n
/
2
]
)
{E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])
and denote by
α
\alpha
a member of
D
(
f
)
D(f)
. Denote by H the space of all real-valued functions U, analytic at the origin of
E
m
{E_m}
, so that
α
=
(
0
,
U
(
0
)
,
U
′
(
0
)
,
…
,
U
(
[
n
/
2
]
)
(
0
)
)
\alpha = (0,U(0),U’(0), \ldots ,{U^{([n/2])}}(0))
. For
U
∈
H
,
f
U
(
x
)
≡
f
(
x
,
U
(
x
)
,
U
′
(
x
)
,
…
,
U
(
[
n
/
2
]
)
(
x
)
)
U \in H,{f_U}(x) \equiv f(x,U(x),U’(x), \ldots ,{U^{([n/2])}}(x))
for all x for which this is defined. A one-parameter semigroup (nonlinear if
f
≠
0
f \ne 0
) K on H is constructed so that if
U
∈
K
U \in K
, then
K
(
λ
)
U
K(\lambda )U
converges, as
λ
→
∞
\lambda \to \infty
, to a solution Y to the partial differential equation
L
Y
(
n
)
=
f
Y
L{Y^{(n)}} = {f_Y}
. A resolvent j for this semigroup is determined so that
J
(
λ
)
U
J(\lambda )U
also converges to y as
λ
→
∞
\lambda \to \infty
and so that
J
(
λ
/
n
)
n
U
J{(\lambda /n)^n}U
converges to
K
(
λ
)
U
K(\lambda )U
as
n
→
∞
n \to \infty
. The solutions
Y
∈
H
Y \in H
of
L
Y
(
n
)
=
f
Y
L{Y^{(n)}} = {f_Y}
are precisely the fixed points of the semigroup K.