3 The delay equation
In this section, we make a first step towards the solution of the delay equation d y
t
= σ y
t
, y
t −r
1
, . . . , y
t −r
k
d x
t
, t
∈ [0, T ], y
t
= ξ
t
, t
∈ [−r
k
, 0], 14
where x is a R
d
-valued γ-Hölder continuous function with γ
1 3
, the function σ ∈ C
3
R
nk+ 1
; R
n ,d
is bounded together with its derivatives, ξ is a R
n
-valued weakly controlled path based on x, and r
1
. . . r
k
∞. For convenience, we set r = 0 and, moreover, we will use the notation
s y
t
= y
t −r
1
, . . . , y
t −r
k
, t
∈ [0, T ]. 15
3.1 Delayed controlled paths
As in the previous section, we will first make some heuristic considerations about the properties of a solution: set ˆ
σ
t
= σ y
t
, s y
t
and suppose that y is a solution of 14 with y ∈ C
κ 1
for a given
1 3
κ γ. Then we can write the integral form of our equation as δ y
st
= Z
t s
ˆ σ
u
d x
u
= ˆ σ
s
δx
st
+ ρ
st
with ρ
st
= Z
t s
ˆ σ
u
− ˆ σ
s
d x
u
. Thus, we have again obtained a decomposition of y of the form
δ y = ˆ σδx + ρ. Moreover, it follows
still at a heuristic level that ˆ σ is bounded and satisfies
| ˆ σ
t
− ˆ σ
s
| ≤ kσ
′
k
∞ k
X
i=
| y
t −r
i
− y
s −r
i
| ≤ k + 1kσ
′
k
∞
k yk
κ
|t − s|
κ
. Thus, with the notation of Section 2.1, we have that ˆ
σ belongs to C
κ 1
and is bounded. The term ρ should again inherit both the regularities of δ ˆ
σ and x. Thus, one should have that ρ ∈ C
2 κ
2
. In conclusion, the increment
δ y should be decomposable into δ y = ˆ
σδx + ρ with
ˆ σ ∈ C
κ 1
bounded and ρ ∈ C
2 κ
2
. 16
This is again the structure we will ask for a possible solution to 14. However, this decomposition does not take into account that equation 14 is actually a delay equation. To define the integral
R
t s
ˆ σ
u
d x
u
, we have to enlarge the class of functions we will work with, and hence we will define a delayed controlled path
hereafter
DCP
in short.
Definition 3.1. Let 0 ≤ a ≤ b ≤ T and z ∈ C
κ 1
[a, b]; R
n
with
1 3
κ ≤ γ. We say that z is a delayed controlled path based on x, if z
a
= α belongs to R
n
and if δz ∈ C
κ 2
[a, b]; R
n
can be decomposed into δz
st
=
k
X
i=
ζ
i s
δx
s −r
i
,t −r
i
+ ρ
st
for s
, t ∈ [a, b],
17 where
ρ ∈ C
2 κ
2
[a, b]; R
n
and ζ
i
∈ C
κ 1
[a, b]; R
n ,d
for i = 0, . . . , k. The space of delayed controlled paths on [a
, b] will be denoted by D
κ,α
[a, b]; R
n
, and a path z ∈ 2039
D
κ,α
[a, b]; R
n
should be considered in fact as a k + 2-tuple z, ζ , . . . ,
ζ
k
. The norm on
D
κ,α
[a, b]; R
n
is given by N [z; D
κ,α
[a, b]; R
n
] = kδzk
κ
+ kρk
2 κ
+
k
X
i=
kζ
i
k
∞
+
k
X
i=
kδζ
i
k
κ
.
Now we can sketch our strategy to solve the delay equation: 1. Consider the map T
σ
defined on Q
κ,α
[a, b]; R
n
× Q
κ, ˜ α
[a − r
k
, b − r
1
]; R
n
by T
σ
z, ˜ z
t
= σz
t
, s˜ z
t
, t
∈ [a, b], 18
where we recall that the notation s˜ z
has been introduced at 15. We will show that T
σ
maps Q
κ,α
[a, b]; R
n
× Q
κ, ˜ α
[a − r
k
, b − r
1
]; R
n
smoothly onto a space of the form D
κ, ˆ α
[a, b]; R
n ,d
. 2. Define rigorously the integral
R z
u
d x
u
= J zd x for a delayed controlled path z ∈ D
κ, ˆ α
[a, b]; R
n ,d
, show that J zd x belongs to Q
κ,α
[a, b]; R
d
, and compute its decom- position 13. Let us point out the following important fact: T
σ
creates “delay”, that is T
σ
z, ˜ z
∈ D
κ, ˆ α
[a, b]; R
n ,d
, while J creates “advance”, that is J zd x ∈ Q
κ,α
[a, b]; R
n
. 3. By combining the first two points, we will solve equation 14 by a fixed point argument on
the intervals [0, r
1
], [r
1
, 2r
1
], . . . .
3.2 Action of the map T on controlled paths
