AbstractWe construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$
C
r
1
=
C
1
(
[
-
r
,
0
]
,
R
)
and find $$h\in (0,r)$$
h
∈
(
0
,
r
)
so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$
x
′
(
t
)
=
-
x
(
t
-
d
(
x
t
)
)
defines a continuous semiflow of continuously differentiable solution operators on the solution manifold $$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$
X
=
{
ϕ
∈
C
r
1
:
ϕ
′
(
0
)
=
-
ϕ
(
-
d
(
ϕ
)
)
}
,
and along each solution the delayed argument $$t-d(x_t)$$
t
-
d
(
x
t
)
is strictly increasing, and there exists a solution whose short segments$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$
x
t
,
s
h
o
r
t
=
x
(
t
+
·
)
∈
C
h
2
,
t
≥
0
,
are dense in an infinite-dimensional subset of the space $$C^2_h$$
C
h
2
. The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.
Dense Short Solution Segments from Monotonic Delayed Arguments
