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A continuous semiflow on a space of Lipschitz functions
for a differential equation with state-dependent delay
from cell biology
Istv´an Bal´azsa,1, Philipp Gettob,2,∗, Gergely R¨ostc,3
aMTA-SZTE Analysis and Stochastics Research Group,
Bolyai Institute, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary and
Department of Mathematics, University of Klagenfurt, Universit¨atsstraße 65–67,
Klagenfurt am W¨orthersee, 9020 Austria
bDepartment of Mathematics, UCLM (Universidad de Castilla y La Mancha),
post adress: Facultad de Ciencias Sociales, Av. Real F´abrica de Sedas, s/n, 45600
Talavera de la Reina, Toledo, Spain
cBolyai Institute, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary
Abstract
We analyze a system of differential equations with state-dependent delay
(SD-DDE) from cell biology, in which the delay is implicitly defined as the
time when the solution of an ODE, parametrized by the SD-DDE state,
meets a threshold. We show that the system is well-posed and that the so-
lutions define a continuous semiflow on a state space of Lipschitz functions.
Moreover we establish for an associated system a convex and compact set
that is invariant under the time-t-map for a finite time. It is known that,
due to the state dependence of the delay, necessary and sufficient conditions
for well-posedness can be related to functionals being almost locally Lips-
∗corresponding author
Email addresses: balazsi@math.u-szeged.hu (Istv´an Bal´azs), phgetto@yahoo.com
(Philipp Getto), rost@math.u-szeged.hu (Gergely R¨ost)
1The research of the author was funded by the Hungarian Scientific Research Fund,
Grant No. K 129322.
2The research of the author was funded by the DFG (Deutsche Forschungsgemein-
schaft), project number 214819831, and by the ERC starting grant EPIDELAY (658, No.
259559).
3The research of the author was funded by the ERC starting grant EPIDELAY
(658, No. 259559), by the Marie Sklodowska-Curie Grant No. 748193, and by NKFIH
FK124016.
Preprint submitted to Journal of Differential Equations July 16, 2021
chitz, which roughly means locally Lipschitz on the restriction of the domain
to Lipschitz functions, and our methodology involves such conditions. To
achieve transparency and wider applicability, we elaborate a general class
of two component functional differential equation systems, that contains the
SD-DDE from cell biology and formulate our results also for this class.
Keywords:
Delay differential equation, State-dependent delay, Well-posedness,
Invariant compact set, Almost locally Lipschitz, Stem cell model
Contents
1 Introduction 2
2 A continuous semiflow for DDE on a space of Lipschitz func-
tions 7
2.1 Functionals defined on C..................... 7
2.2 Retractions and functionals with specific domains . . . . . . . 9
3 Semiflow and invariant compact sets for a certain class of
two-dimensional DDE 16
4 Semiflow and invariant compact sets for the cell SD-DDE 26
5 Discussion and outlook 34
1. Introduction
In this paper we analyze a system of delay differential equations (DDE)
from cell biology of the form
w0(t) = q(v(t))w(t),(1.1)
v0(t) = γ(v(t−τ(vt)))
g(x1, v(t−τ(vt))) g(x2, v(t))w(t−τ(vt))eRτ(vt)
0[d−D1g](y(s,vt),v(t−s))ds
−µv(t),(1.2)
with y=y(·, ψ) and τ=τ(ψ) defined via
y0(s) = −g(y(s), ψ(−s)), s > 0, y(0) = x2and (1.3)
y(τ, ψ) = x1,(1.4)
2
where x1< x2are given parameters. As common in DDE, we introduce
xt(s) := x(t+s), s < 0, for functions xdefined in t+s∈R. Next, D1gdenotes
the derivative of gwith respect to the first argument and all values of γ,g
and τare nonnegative. Precise conditions for all functions and parameters
will be given in Section 4.
The following mathematical difficulties make the analysis challenging. If
we fix a time t, in (1.1–1.2) appear both, a point delay τand a distributed
delay s, which is an integration variable. The delay τis implicitly defined
via (1.4) as the time, when the solution yof the external nonlinear ordinary
differential equation (ODE) (1.3) meets a lower threshold x1. This ODE is
parametrized by a function ψ, which in (1.1–1.2) is specified as the second
state variable vtof the DDE system, hence also the solution of the ODE y
and the delay τhave a functional or history dependence on ψ. In summary
we have a system of DDE with a state-dependent delay (SD-DDE), which
is implicitly defined via a threshold condition, and an additional distributed
delay.
A difficulty we will encounter when analyzing the invariance of bounded
sets under the solution operator is the special type of coupling. Whereas we
can often assume that the functions in (1.1–1.2) decrease in v, the right hand
sides increase in wand the equation (1.1) has no self-regulatory mechanism.
The system describes the dynamics of a stem cell population, whose size at
time tis denoted by w(t), regulated by the mature cell population, similarly
denoted by v(t). The equations have been deduced via integration along the
characteristics from a partial differential equation describing the “transport”
of a density n(t, x) over the progenitor cell maturity x∈[x1, x2]. See [7]
and references therein for the latter and modeling background and [6] for
biological background. We will refer to (1.1–1.4) as the cell SD-DDE.
Let us introduce the space C:= C([−h, 0],Rn), where h∈(0,∞) will be
related to the maximum delay, endowed with the usual sup-norm denoted by
k · k. For discussion of the results we will also refer to C1:= C1([−h, 0],Rn),
endowed with its standard norm defined by kφk1:= kφk+kφ0k. In [7] the
authors have elaborated conditions to guarantee, via application of results
of [13, 21], that the cell SD-DDE is well-posed and the solutions define, for
n= 2, a semiflow on the solution manifold, a continuously differentiable sub-
manifold of C1, and that the semiflow is differentiable in the C1-topology. For
general SD-DDE differentiability of the semiflow in the C1-topology implies a
linearized stability theorem, see [13] and [19] for a criterion for, respectively,
stability and instability of a supposed equilibrium.
3
The cell SD-DDE (1.1–1.4) may feature a unique positive equilibrium
emerging from the zero equilibrium in a transcritical bifurcation: qmay
decrease to a negative value and q(0) should increase from negative to positive
upon variation of the bifurcation parameter, see [6, 8]. A combination of the
discussed results of [13, 19, 7] facilitated a local stability analysis of equilibria
for the cell SD-DDE in [8].
The paper [8] contains numerical and analytical evidence that the interior
equilibrium is stable upon emergence and destabilizes in a Hopf bifurcation.
The latter motivates also analytical research for periodic solutions for the
cell SD-DDE. One way to prove the existence of periodic solutions in general
is to establish their correspondence with fixed points of an operator and to
apply fixed point theory. This is done for a certain class of SD-DDE in [14].
As in many fixed point arguments, also in [14] convexity and compactness
of the domain is used, properties the solution manifold in general does not
have.
In [11] the existence of non-continuable and global solutions is established
for systems of DDE defined by functionals that are continuous on domains
that are open in the C-topology (C-open). Continuous dependence on initial
values is shown under the precondition that the solution is unique. Unique-
ness of solutions is shown if the functional is Lipschitz on compact subsets
of a C-open domain. A known problem is that, in general, for SD-DDE the
functional is not locally Lipschitz, if its domain includes functions that are
not locally Lipschitz, see e.g. [15, Section 1].
In [15] the above mentioned uniqueness problem is overcome by restrict-
ing initial conditions to Lipschitz functions and using that for SD-DDE the
functional often is almost locally Lipschitz, which roughly means locally Lip-
schitz on a domain of Lipschitz functions. Then, a combination with the
discussed results in [11] yields existence and uniqueness on a domain of Lip-
schitz functions. The equation in [15] features two feedback conditions, one
from above, at A, and one from below, at −B, that guarantee that solutions
remain in a bounded domain of the form C([−h, 0],[−B, A]). The feedback
conditions for the bounded domain firstly facilitate the proof of global ex-
istence. Moreover, they lead to the intersection, of this set with a set of
functions that share a finite Lipschitz constant, being mapped into itself by
the time-t-map. By the Arzel`a-Ascoli theorem, a set of functions that share
the same finite bound and finite Lipschitz constant is compact in C. In this
way in [15] a convex and compact set that is invariant under the time-t-map
could be established.
4
Whereas in [15] a scalar equation is treated, the cell SD-DDE is a two-
dimensional system (here and below dimension often refers to the range space
of the functional defining the equation). We will show that it preserves non-
negativity, such that feedback from below, at zero, is granted. Regarding
feedback from above, however, the following problem arises. If a solution
(w, v)(t) of the cell SD-DDE through (ϕ, ψ) exists, then
w(t) = (ϕ(t), t ∈[−h, 0],
ϕ(0)eRt
0q(v(s))ds, t > 0.(1.5)
Now, qmay take positive values for small arguments, see [6], and (1.5) shows
that for such qthe set of all Lipschitz functions of C([−h, 0],[0, A1]×[0, A2])
is not mapped into itself. This leads to the fact that an invariant compact set
cannot be established by a straightforward generalization of the approach of
[15]. For similar reasons (see the proof of Theorem 3.8 (a) below) we cannot
expect this for the set of R-Lipschitz functions, where R > 0 is a fixed given
number, either. Moreover, due to the missing of the feedback from above it
seems necessary to use a more general criterion for global existence than the
one used for the equation in [15].
The first main result of this paper is the proof that the cell SD-DDE is
well-posed and the solutions define a continuous semiflow on a state-space
of Lipschitz functions. In comparison to the solution manifold established in
[7], the new set of admissible initial conditions is much larger and, other than
the former, convex. Moreover, whereas the C1-differentiability established in
[7] requires convergence of a sequence of initial functions in C1for concluding
convergence of the sequence of solutions (in C1) the C-continuity shown here
requires only C-convergence of the initial functions. An application that we
will discuss is that convergence of a solution to a constant in Ris sufficient
to conclude that the constant is an equilibrium solution. This conclusion
can not be drawn from the differentiability result in [7]. We also show that
a criterion for global existence similar to the one in [11], and more general
than the one used in [15], can be formulated for a general class of SD-DDE.
To establish the above results for the cell SD-DDE, we first show that
solutions of a general class of DDE define a semiflow on a state space defined
by all functions that have a finite (but not necessarily the same) Lipschitz
constant and belong to C+:= C([−h, 0],Rn
+), where R+:= [0,∞). This is
essentially a specific combination of (variants of) arguments of [15], [11] and
[4].
5
We then elaborate conditions for the above results to be transferable to
the cell SD-DDE. Conditions for local, global and unique existence may dif-
fer and may be equation specific. To reach more transparence, we here try
to separate the corresponding assumptions. Moreover, we head for assump-
tions that are both general and easy to check. It turns out that continuity
of the functions defining the cell SD-DDE guarantees continuity and bound-
edness properties of the functional, the first of which yields local existence.
Added boundedness of functions allows to add boundedness of orbits on fi-
nite times, which together with the boundedness properties of the functional
yields global existence. Local Lipschitzian functions lead to almost local
Lipschitzian functionals and these to uniqueness. To prove the above we de-
velop some techniques regarding Lipschitz properties of integral-, evaluation-
and implicitly defined operators acting on Lipschitz subsets of continuous
functions.
The discussion on (1.5) motivates that in order to establish invariance for
the cell SD-DDE, it seems helpful to first consider the v-component: One
can fill (1.5) into (1.2) to obtain an equation in vparametrized by the initial
condition ϕfor wand analyze invariance for the latter equation. It will turn
out that this approach can be adapted to a more general system of the form
w0(t) = m(wt, vt), v0(t) = j(wt, vt)−µv(t) (1.6)
in combination with linear boundedness conditions for mand j.
Our second main result is then the establishment of a compact and convex
set that is invariant for finite times under the v-component of the time t-map.
The result is elaborated for both, (1.6) and the cell SD-DDE. We are not
aware of previous results in this direction that are applicable in the absence
of component-wise feedback conditions from above.
To establish the invariant set for (1.6) we first combine a linear bound-
edness condition with a monotonicity argument to derive an exponential
estimate for wthat, for the specific case m(ϕ, ψ) := q(ψ(0))ϕ(0), one could
also derive from (1.5). Then we will apply the variation of constants for-
mula to the second equation of (1.6) along with the exponential estimate
for w. The resulting bound for vwill be estimated further using two alter-
native linear boundedness conditions for j. Whereas the first will be more
general, the second is a translation of the fact, that in the cell SD-DDE the
delay functional has a lower bound τwhich is positive, to a more specific
linear boundedness condition for j. This condition then yields a minimum
6
invariance time τ > 0 that in a sense is uniform with respect to initial condi-
tions. Somewhat tedious though perhaps elementary estimation techniques
employing monotonicity arguments complete this analysis.
The remainder of the paper is structured top down: In Section 2 we
consider general DDE, but with an approach tailored to SD-DDE. Section 3
contains results for the class of two-dimensional DDE (1.6) and in Section 4
we analyze the cell SD-DDE. Finally, Section 5 contains a discussion of our
results and an outlook on future research.
2. A continuous semiflow for DDE on a space of Lipschitz functions
We start with a well-known definition of solutions for DDE.
Definition 2.1. Let Dbe an arbitrary subset of Cand suppose that φ∈ D
and f:D −→ Rn. By a solution of
x0(t) = f(xt), t ≥0,(2.1)
x0=φ, (2.2)
or a solution of (2.1) through φ, we mean a continuous function
xφ: [−h, α]−→ Rn
for some α > 0, that is such that for t∈[0, α] one has that xφ
t∈ D and on
[0, α] the function xφis differentiable and satisfies (2.1–2.2).
Continuity of a solution implies continuity of [0, α]→ D;t7→ xtand, if f
is continuous, the latter and (2.1) imply continuity of x0and thus continuous
differentiability of xφon [0, α]. Solutions on half-open intervals [−h, α) for
α∈(0,∞] are defined analogously. We shall sometimes write xinstead of
xφ.
2.1. Functionals defined on C
Non-continuable and global solutions
We refer to [11] for the standard definition of a non-continuable solution.
Theorem 2.2. Suppose that F:C −→ Rnis continuous and φ∈ C.
(a) There exists some c=c(φ)∈(0,∞]and a non-continuable solution
xφ: [−h, c)−→ Rnof
x0(t) = F(xt), t ≥0, x0=φ. (2.3)
7
If additionally F(U)is bounded whenever U⊂ C is closed and bounded then
the following hold:
(b) If c < ∞then for any closed and bounded U⊂ C there exists some
tU∈(0, c)such that xφ
t/∈Ufor all t∈[tU, c).
(c) If {xφ
t:t∈[0, α)} ⊂ C is bounded, whenever α∈(0,∞)and xφis
defined on [0, α), then c=∞, i.e., the solution is global.
The existence of a solution xφ: [−h, α]−→ Rnfor some α > 0 follows
from [11, Theorem 2.2.1] and the statement in (a) is concluded in [11, Section
2.3] from Zorn’s lemma. Note that since the non-continuable solution need
not be unique, also c(φ) need not be unique. Next, (b) follows from [11,
Theorem 2.3.2] and (c) directly from (b).
Uniqueness
To guarantee uniqueness, the notion of almost local Lipschitzian func-
tionals for n= 1 from [15] can be generalized to arbitrary finite dimensions
in a straightforward way. Define for any φ∈ C
lip φ:= sup |φ(s)−φ(t)|
|s−t|:s, t ∈[−h, 0], s 6=t∈[0,∞]
and Bδ(x0) := {x:kx−x0k< δ}, where δ > 0, | · | denotes a norm in Rn
with ndepending on context and the choice of norm k·k in Bδ(x0) should also
be clear from the context, e.g., the choice of x0. In the following, however,
we denote by k·kthe sup-norm in C. Then, a function φis Lipschitz with
Lipschitz constant k(we will write k-Lipschitz) whenever ∞> k ≥lip φ.
For each φ0∈ C,δ > 0, R > 0 define
V(φ0;δ, R) := {φ∈Bδ(φ0) : lip φ≤R}.
Definition 2.3. A functional f:D ⊂ C =C([−h, 0],Rn)−→ Rmis called
almost locally Lipschitz if fis continuous and for all φ0∈ D,R > 0 there
exist some δ=δ(φ0, R)>0 and k=k(φ0, R, δ)≥0 such that for all
ϕ, ψ ∈V(φ0;δ, R)∩ D
|f(ϕ)−f(ψ)| ≤ kkϕ−ψk.
We next prove some general facts that will be relevant in the following
sections.
8
Lemma 2.4. (a) Suppose that f, g :D ⊂ C −→ Rare arbitrary almost
locally Lipschitz functions. Then so are fg,(f, g)and f+g.
(b) Let f:D ⊂ C −→ Rbe almost locally Lipschitz and g:f(D)⊂R−→ R
be locally Lipschitz, then g◦f:D −→ Ris almost locally Lipschitz.
Proof. Since (a) is fairly standard, we only prove (b). First, clearly g◦fis
continuous. Next, let φ0∈ D,R > 0, choose ε,k1such that gis k1-Lipschitz
on Bε(f(φ0)). Choose, δ,k2such that fis k2-Lipschitz on V(φ0;δ, R) and
f(Bδ(φ0)) ⊂Bε(f(φ0)). Let ϕ, ψ ∈V(φ0;δ, R). Then the following estimate
implies the statement:
|g(f(ϕ)) −g(f(ψ))| ≤ k1|f(ϕ)−f(ψ)| ≤ k1k2kϕ−ψk.
The following theorem is proven as [15, Theorem 1.2] for the case n= 1.
The proof for general nis analogous and we omit it. For D ⊂ C, define
VD:= {φ∈ D : lip φ < ∞}. Note that if Dis convex, so is VD.
Theorem 2.5. Suppose that F:C −→ Rnis almost locally Lipschitz and let
φ∈VC. If α > 0and y, z : [−h, α]−→ Rnare both solutions of (2.3), then
y(t) = z(t)for all t∈[0, α].
Continuous dependence on initial values
The next result follows directly from [11, Theorem 2.2.2] and the above
uniqueness result.
Theorem 2.6. Suppose that F:C −→ Rnis almost locally Lipschitz, φ∈VC
and that a solution xφof (2.3) through φexists on [−h, α]for some α > 0.
Let (φk)∈(VC)Nwith φk−→ φ, as k→ ∞. Then xφis the unique solution
on [−h, α], for some k0∈Nthere exist unique solutions xkthrough φkon
[−h, α]for all k≥k0,k∈Nand xk−→ xφuniformly on [−h, α].
2.2. Retractions and functionals with specific domains
Recall that a retraction is a continuous map of a topological space into a
subset that on the subset equals the identity. It is remarked in [15] (without
proof) that the following result holds in case n= 1 and the map g, addition-
ally to satisfying the properties stated below, is a retraction. The proof of
our result is analogous and we present it for completeness.
9
Lemma 2.7. Let D ⊂ C,g:C −→ D be locally Lipschitz. Suppose that for
all φ0∈ C,δ > 0,R > 0
sup{lip g(φ) : φ∈V(φ0;δ, R)}<∞.
Then, if f:D −→ Rnis almost locally Lipschitz, so is F:C −→ Rn;F:=
f◦g.
Proof. First, Fis continuous as a composition of continuous functions. Next,
let φ0∈ C,R > 0. Define L:= sup{lip g(φ) : φ∈V(φ0; 1, R)}<∞. Choose
ε=ε(g(φ0), L)>0, k=k(g(φ0), L)≥0 such that fis k-Lipschitz on
V(g(φ0); ε, L)∩ D. Choose δ∈(0,1), l≥0 such that g(Bδ(φ0)) ⊂Bε(g(φ0))
and gis l-Lipschitz on Bδ(φ0). Then for ϕ, ψ ∈V(φ0;δ, R), one has
|F(ϕ)−F(ψ)|=|f(g(ϕ)) −f(g(ψ))| ≤ k|g(ϕ)−g(ψ)| ≤ klkϕ−ψk.
Hence, Fis kl-Lipschitz on V(φ0;δ, R) and thus almost locally Lipschitz.
A specific retraction for a specific domain
For the remainder of the section we will use the following construc-
tion (unless specified otherwise). The construction contains a modifica-
tion of the retraction in [15], the latter of which maps C([−h, 0],R) onto
C([−h, 0],[−B, A]) with −∞ <−B < A < ∞, to a retraction of the space
C([−h, 0],Rn) onto C+=C([−h, 0],Rn
+). With the result we can work with
nonnegative solutions of multi-dimensional equations. Note that C+is con-
vex and that, as discussed before, this implies convexity of VC+. We define a
map
r:R−→ [0,∞); r(u) := (u, u ∈[0,∞),
0, u < 0.(2.4)
Then ris a retraction and Lipschitz with lip r≤1. With rwe define a map
ρ:C −→ C+, ρ = (ρ1, ..., ρn)
ρi(φ)(θ) := r(φi(θ)), θ ∈[−h, 0], i = 1, ..., n. (2.5)
The following results, Lemmas 2.8–2.10, are straightforward modifications of
results in [15], from the retraction in the latter to our retraction. We omit
the corresponding proofs.
10
Lemma 2.8. ρis a retraction and maps bounded sets into bounded sets.
The next lemma follows by definition of ρfrom rbeing Lipschitz with
lip r≤1.
Lemma 2.9. One has lip ρ(φ)≤lip φ, hence if φis Lipschitz so is ρ(φ).
Moreover, ρis Lipschitz with lip ρ≤1.
The result implies that sup{lip ρ(φ) : φ∈V(φ0;δ, R)} ≤ R < ∞for all
φ0∈ C,δ > 0 and R > 0. The next lemma is a straightforward combination
of this result with Lemma 2.7.
Lemma 2.10. Suppose that f:C+−→ Rnis almost locally Lipschitz. Then
so is F:= f◦ρ:C −→ Rn. Moreover F|C+=f.
Non-continuable and global solutions and uniqueness
To guarantee that a solution remains within a domain a feedback condi-
tion can be used. The following result is a slight modification of the corre-
sponding result for one dimension, which is [15, Theorem 1.3].
Lemma 2.11. Suppose that f:C+−→ Rnsatisfies
fi(φ)≥0,if φi(0) = 0, φ = (φ1, ..., φn)∈ C+, i = 1, ..., n. (F)
Now fix φ∈ C+and assume that xis a solution of x0(t) = f(ρ(xt)) through φ
on some interval [−h, α]. Then xt∈ C+and thus ρ(xt) = xtfor all t∈[0, α]
and hence xis a solution of (2.1–2.2) on [0, α].
To prove the result one can directly apply the non-autonomous [17, Propo-
sition 1.2] to the domain Ω := R× C, the map g: Ω →Rn;g(t, φ) := f(ρ(φ))
and t0:= 0. In the next theorem we combine earlier arguments of existence,
uniqueness and invariance with the criterion for global existence of Theorem
2.2.
Theorem 2.12. Suppose that f:C+−→ Rnis continuous and satisfies (F).
Then the following hold.
(a) For every φ∈ C+there exists some c=c(φ)∈(0,∞]and a non-
continuable solution xφon [−h, c)of (2.1–2.2).
11
(b) If f(U)is bounded, whenever U⊂ C+is bounded, and if for some
φ∈ C+the set {xφ
t:t∈[0, α)}⊂C+is bounded, whenever α∈(0,∞)
and xφis a solution defined on [0, α), then c=∞, i.e., the solution is
global.
(c) If fis almost locally Lipschitz and φ∈VC+, then xφis the unique
solution (on whatever interval it is defined).
Proof. Since F:= f◦ρis continuous, by Theorem 2.2 (a) there exists a non-
continuable solution xφon [t0−h, t0+c) of (2.3) for this F. Next, suppose
that U⊂ C is bounded. Then, as remarked, ρ(U)⊂ C+is bounded and
hence by the assumption of (b) also F(U) = f(ρ(U)) is bounded. Thus by
Theorem 2.2 (c) we have shown that if {xφ
t:t∈[0, α)}⊂Cis bounded,
whenever α < ∞and xφis defined on [0, α), then c=∞. If fis almost
locally Lipschitz, then by Lemma 2.10 so is Fand thus by Theorem 2.5 we
get uniqueness. To complete the proof note that (F) guarantees via Lemma
2.11 that {xφ
t:t∈[0, α)} ⊂ C and that xφis a solution of (2.1–2.2).
Note that by definition of a solution, we have that xt∈ C+, such that all
components of xare nonnegative functions.
Remark 2.13. If fwould map only the closed and bounded sets on bounded
sets, we could not guarantee that F(U) = (f◦ρ)(U) is bounded if Uis
closed and bounded, which was required in Theorem 2.2: the above defined
retraction ρmaps bounded on bounded, but in general does not map closed
and bounded on closed sets. To see the latter, consider e.g. C:= C([0,2],R),
C+:= {x∈ C :x(t)≥0,∀t∈[0,2]}and rand ρdefined as above, but for
n= 1 and the modified Cand C+. Define U:= {xn:n≥2}⊂C, where
xn(t) :=
1
n, t < 1−1
n,
1−t, 1−1
n≤t < 1,
−n(t−1),1≤t < 1 + 1
n,
−1,1 + 1
n≤t≤2.
Then clearly Uis bounded. Too see that Uis closed, suppose that (yn)∈UN,
y∈ C,yn−→ y. Then for all n∈Nthere is some kn∈Nsuch that yn=xkn.
There can be only two cases. In the first, there exists some K∈Nsuch that
kn≤Kfor all n∈N. Then {yn:n≥2}is a finite set, hence closed.
It follows that y∈ {yn:n≥2} ⊂ Uand thus also Uis closed. In the
12
second case there exists some (nj)∈NNsuch that knj−→ ∞ as j−→ ∞.
Without loss of generality (nj) is increasing. Then for all t∈[0,1) one has
xknj(t)−→ 0 as j−→ ∞. For all t∈(1,2] one has xknj(t)−→ −1. On the
other hand, for any t∈[0,2] one has xknj(t) = ynj(t)−→ y(t). It follows
that yhas a discontinuity at t= 1, which is a contradiction. Hence only the
first case can be and thus Uis closed.
Finally, it is easy to see that
ρ(U) = {x:∃n≥2,s.th. x(t) = xn(t),∀t∈[0,1], x(t)=0,∀t∈[1,2]}.
is not closed: Indeed for (zn)∈ρ(U)N, where
zn(t) := (xn(t), t ∈[0,1],
0, t ∈(1,2],
one has zn−→ 0∈ C\ρ(U).
Continuous dependence on initial values
The negative feedback condition (F) now ensures that the results on con-
tinuous dependence can be transferred to solutions of (2.1–2.2) for a func-
tional defined on our specific domain.
Theorem 2.14. Suppose that f:C+−→ Rnis almost locally Lipschitz and
satisfies (F), let φ∈VC+and let α > 0be such that the solution xφof (2.1)
through φexists on [−h, α]. Let (φk)∈(VC+)Nwith φk−→ φ. Then for some
k0∈Nthere exist unique solutions xkthrough φkon [−h, α]for all k≥k0,
k∈N, and xk−→ xφuniformly on [−h, α].
Proof. Since xφis a solution of (2.1–2.2) we have xφ
t∈ C+for all t≥0. Thus,
for F:= f◦ρ, one has F(xφ
t) = f(xφ
t) and xφis a solution of x0(t) = F(xt)
through φ. By Theorem 2.6, xφis the unique solution of x0(t) = F(xt) and
there exists some k0, such that for all k≥k0there exist unique solutions xk
of x0(t) = F(xt) through φkon [−h, α] and xk−→ xφuniformly. By Lemma
2.11 the xksolve also (2.1–2.2).
Semiflow and asymptotic properties
If fsatisfies the assumptions for global existence and uniqueness, we can
use some standard dynamical systems theory, see e.g. [2, Sections 10, 17]. In
the following let Xdenote a metric space.
13
Definition 2.15. A map Σ : [0,∞)×X−→ Xis called a continuous
semiflow if
(i) Σ(0, x) = xfor all x∈X,
(ii) Σ(t, Σ(s, x)) = Σ(t+s, x) for all s, t ∈[0,∞), x∈X,
(iii) Σ is continuous.
For a given semiflow Σ and some x∈Xwe denote by, respectively,
γ+(x) := {Σ(t, x) : t∈[0,∞)}, ω(x) := \
t≥0
γ+(Σ(t, x))
the positive orbit and ω-limit set of xunder Σ.
Note that
ω(x) = {y∈X:∃tn−→ ∞, s.th. Σ(tn, x)−→ yas n→ ∞}.(2.6)
To show that the next theorem holds, we combine what we have compiled
so far with further results from the literature. Let us specify X:= VC+
endowed with the metric induced by the sup-norm.
Theorem 2.16. Suppose that f:C+−→ Rnis almost locally Lipschitz and
satisfies (F), f(U)is bounded whenever U⊂ C+is bounded and {xφ
t:t∈
[0, α)}is bounded whenever φ∈VC+,α∈(0,∞), and a solution xφof (2.1) is
defined on [0, α). Then for any φ∈VC+there exists a unique global solution
of (2.1) through φand for all t≥0one has xφ
t∈VC+. Hence, we can define
a map
S: [0,∞)×VC+−→ VC+;S(t, φ) := xφ
t.
This map defines a continuous semiflow on VC+with respect to the sup-norm.
Proof. First note that existence of a unique global solution for all φ∈VC+
follows from Theorem 2.12. To show invariance of the state space VC+under
the time t-map, one can use continuous differentiability of solutions. More-
over, it is established in [11] (without proof) and in [4, Proposition VII 6.1
(i)] that Definition 2.15 (i-ii) hold, when applied to any continuous initial
function. It follows that they hold also for our state space. To see continu-
ity of S, let (tk, φk)∈([0,∞)×VC+)Nand (t, φ)∈[0,∞)×VC+, such that
14
(tk, φk)→(t, φ). Now, in Theorem 2.14 let α:= t+ 1 and choose k0∈N
such that tk≤t+ 1 for all k≥k0. Then for k≥k0and θ∈[−h, 0]
|S(tk, φk)(θ)−S(t, φ)(θ)|=|xk(tk+θ)−xφ(t+θ)|
≤ |xk(tk+θ)−xφ(tk+θ)|+|xφ(tk+θ)−xφ(t+θ)|=: (I)+(I I )
in obvious notation. Let ε > 0. By Theorem 2.14 we can choose k1≥k0,
such that (I)≤ε/2 for all k≥k1,θ∈[−h, 0]. Since xφis continuous as a
solution, it is uniformly continuous on [−h, t + 1] and we can choose k2≥k1
such that (II )≤ε/2 for all k≥k2,θ∈[−h, 0]. It follows that
kS(tk, φk)−S(t, φ)k ≤ ε, ∀k≥k2,
which concludes the proof.
Suppose that for the remainder of the section fsatisfies the assumptions
of Theorem 2.16. A sufficient criterion for relative compactness of the positive
orbit for DDE with infinite delay is already established in [9, Lemma 2].
Establishing such a criterion in our setting is similar and easier, because we
have finite delay:
Corollary 2.17. Let φ∈VC+and suppose that x=xφis bounded on [0,∞).
Then γ+(φ)is compact.
To see that the statement holds, first note that by the boundedness as-
sumption on falso x0is bounded on [0,∞). Then, similarly to [9, Lemma
2], one can show sequential compactness using the theorem of Arzel`a–Ascoli.
We omit further details.
Finally, continuous dependence and Definition 2.15 (ii) can be combined
to prove the following result. Recall that for an equilibrium of a (global)
semiflow induced by a DDE the corresponding solution is necessarily a con-
stant function on [−h, ∞).
Corollary 2.18. Suppose that φ∈VC+and xφ(t)−→ x∗∈Rn
+as t→ ∞.
Define φ∗(θ) := x∗on [−h, 0]. Then φ∗∈VC+,S(t, φ)−→ φ∗and S(t, φ∗) =
φ∗, hence xφ∗(t) = x∗for all t≥ −h, i.e., xφ∗is an equilibrium solution.
Moreover ω(φ) = {φ∗}and γ+(φ)is compact.
Proof. Let (tk)∈(R+)N,tk→ ∞. Then S(tk, φ) = xφ
tk−→ φ∗uniformly
by our assumption. Fix t > 0. Then also S(t+tk, φ)−→ φ∗as k→ ∞.
But also S(t+tk, φ) = S(t, S(tk, φ)) −→ S(t, φ∗) by Theorem 2.16. Hence
S(t, φ∗) = φ∗. Thus xφ∗is an equilibrium solution. Next, ω(φ) = {φ∗}follows
trivially from S(t, φ)−→ φ∗, (2.6) and S(t, φ∗) = φ∗.
15
3. Semiflow and invariant compact sets for a certain class of two-
dimensional DDE
We incorporate the dimension into the notation by introducing C+
n:=
C([−h, 0],Rn
+) and consider continuous functionals m:C+
2−→ R,j:C+
2−→
R+and a parameter µ > 0. Next, we specify the functional f:C+
2−→ R2;
f(ϕ, ψ) := (m(ϕ, ψ), j(ϕ, ψ)−µψ(0))T,(3.1)
such that the general DDE (2.1) can be related to the class of two-dimensional
DDE (1.6).
The second component of ffeatures evaluation at zero and a further
specification of mand jbelow will involve more general evaluation operators.
We start with a result on smoothness of such operators. Let aand bbe such
that b>a≥0 and denote by Jand Ji,i= 1,2,3 arbitrary subsets of R.
We define
ev :C([−b, −a], J)×[−b, −a]−→ J;ev(ϕ, s) := ϕ(s).(3.2)
Lemma 3.1. The operator ev is continuous. For an arbitrary continuous
functional r:C([−b, −a], J1)−→ [a, b], the functionals
ev ◦(id × −r) : C([−b, −a], J1)−→ J1;ψ7→ ev(ψ, −r(ψ)) = ψ(−r(ψ)),
ev ◦(id, −r) : De−→ J2; (ϕ, ψ)7→ ev(ϕ, −r(ψ)) = ϕ(−r(ψ)),(3.3)
where De:= C([−b, −a], J2×J1), are continuous. If ris almost locally
Lipschitz, then so are ev ◦(id×−r)and ev ◦(id, −r). If ris locally Lipschitz,
then for all (ϕ, ψ)∈ De,R > 0there exist k=k((ϕ, ψ), R),δ=δ((ϕ, ψ), R),
such that ev ◦(id, −r)is k-Lipschitz on (V(ϕ;δ, R)×Bδ(ψ)) ∩ De. If ris
constant, then both functionals are Lipschitz (in fact bounded and linear).
The proof is straightforward and we omit it. The lemma is sharp in the
sense that in general neither ev nor the operators defined in the lemma are
locally Lipschitz for the given domains, even if ris Lipschitz.
To guarantee existence of solutions in a general way, we define a feedback
law and boundedness conditions via
m(ϕ, ψ)≥0,if ϕ(0) = 0 and (ϕ, ψ)∈ C+
2,(Fm)
m(U), j(U) bounded,if U⊂ C+
2bounded,(Bmj)
16
{(w, v)φ
t:t∈[0, α)}⊂C+
2bounded for any solution
(w, v)φ
tdefined on [0, α), α ∈(0,∞).(Bφ)
To establish an invariant bounded set of Lipschitz functions, motivated by
(1.1), we define the linear boundedness condition
sup
(ϕ,ψ)∈C+
2, ϕ(0)6=0
|m(ϕ, ψ)|
ϕ(0) <∞(lBm)
and a second boundedness assumption for jvia
j(B1×B2) bounded,if B1×B2⊂ C+
2, B1bounded.(sBj)
In case (lBm) holds, we define kmas the supremum and note that by con-
tinuity of mone has that |m(ϕ, ψ)| ≤ ϕ(0)kmfor all (ϕ, ψ)∈ C+
2and both,
(Fm) and the boundedness condition for min (Bmj) hold. Again, if (lBm)
holds, we introduce the nonnegative quantity
q:= max{sup
(ϕ,ψ)∈C+
2, ϕ(0)6=0
m(ϕ, ψ)
ϕ(0) ,0}
and observe that q≤km<∞. Then, if a solution (w, v) through (ϕ, ψ)∈ C+
2
exists, one has w0(t) = m(wt, vt)≤qw(t). Under this differential inequality
one obtains
w(t)≤ kϕkqe(t),∀t≥ −h, where qe(t) := (1, t ∈[−h, 0],
eqt, t > 0.(3.4)
Note that qeis continuous, nondecreasing, increasing on [0,∞) if q > 0, and
differentiable on [−h, 0) ∪(0,∞). Moreover, qe(t) is increasing in qfor any
fixed t > 0.
Obviously (sBj) implies the boundedness condition for jin (Bmj). We
will use the following variation of constants formula. If (w, v) is a solution
through (ϕ, ψ) defined on [0, t], then
v(t) = ψ(0)e−µt +e−µt Zt
0
eµsj(ws, vs)ds. (VOC)
Lemma 3.2. (a) The functional fis continuous. If (Fm) holds, then f
satisfies (F) and for any φ∈ C+
2there exists some c=c(φ)∈(0,∞]and a
17
non-continuable solution (w, v)φon [−h, c)of (1.6) through φ.
(b) If mand jsatisfy (Bmj), then f(U)is bounded, whenever U⊂ C+
2is
bounded. If moreover some φ∈ C+
2satisfies (Bφ), then any non-continuable
solution of (1.6) through φis global.
(c) If mand jare almost locally Lipschitz then so is fand a solution is
unique where it exists.
(d) If (lBm) holds, so does (Fm). If moreover (sBj) holds, then so does (Bmj)
and for all φ∈ C+
2so does (Bφ).
Proof. (a) Continuity of ffollows from continuity of the functionals m,j,
(3.3) for the case r= 0 (evaluation at zero) and appropriate projections.
Property (F) follows directly from (Fm) and non-negativity of j. Then,
application of Theorem 2.12 (a) yields (a) of the lemma.
(b) The first statement is trivial, the second follows by Theorem 2.12 (b).
(c) The Lipschitz property for fcan be deduced using Lemmas 2.4 and 3.1.
Then, application of Theorem 2.12 (c) yields the second statement.
(d) (Fm) holds by (lBm) as discussed. Regarding the second statement,
(sBj) and (lBm) obviously imply (Bmj). To show (Bφ), we show sufficient
properties for each component. Suppose that (w, v) is defined on [−h, α), α∈
(0,∞). Then boundedness of won [−h, α) follows from (3.4). Boundedness
of von [0, α) can be shown using (VOC), boundedness of the w-component
on [0, α) and (sBj).
Note that, by (a) and (b), (d) provides alternative sufficient conditions
for the existence of non-continuable and global solutions, respectively. If m
and jare almost locally Lipschitz and, either (Fm), (Bmj) and (Bφ) on VC+
2,
or (lBm) and (sBj) hold, then, by Theorem 2.16, solutions define a semiflow
on VC+
2.
Proposition 3.3. Suppose that mand jare almost locally Lipschitz, µ > 0
and that (lBm) holds. Let A,B,Rand Tdenote positive numbers and let
(w, v)denote the solution through (ϕ, ψ)∈VC+
2. Choose A,Rand Tsuch
that kmAeqT ≤R. Then, if kϕk ≤ Aand lip ϕ≤Rone has lip wt≤Rfor
all t∈[0, T ].
Proof. Since lip ϕ≤R, it remains to show that lip w|[0,T ]≤R. This follows
since on [0, T ] by (lBm) and (3.4) one has
|w0(t)|=|m(wt, vt)| ≤ kmw(t)≤kmkϕkeqt ≤kmAeqt ≤R.
18
Next, we will establish an invariant set for the v-component of the time-
t-map. For given B > 0 and R > 0 this set will be of the form
CB,R := {χ∈C([−h, 0],[0, B]),lip χ≤R}.(3.5)
Note that CB,R is convex and, by the Arzel`a–Ascoli theorem, compact. Recall
that solutions of (1.6) are nonnegative. For further invariance analysis and
motivated by the expression on the right hand side of (1.2) we introduce two
new boundedness conditions for jvia
∃kj>0, s.th. j(ϕ, ψ)≤kjkϕk,∀(ϕ, ψ)∈ C+
2,(lBj)
∃kj>0, s.th. j(ϕ, ψ)≤kjϕ(−τ(ψ)),∀(ϕ, ψ)∈ C+
2,(τBj)
where τ:C+
1−→ [τ , h] for some τ∈[0, h). Obviously (lBj) is a weaker
requirement than (τBj) and any of the two implies (sBj). On the other
hand, as we will see, (τBj) may lead to better results while still applicable
to the cell SD-DDE.
Suppose that for the remainder of the section (lBm) and either (lBj) or
(τBj) hold. A combination of (VOC) with (lBj–τBj) and then (3.4) leads to
the following lemma.
Lemma 3.4. For any given solution (w, v)through (ϕ, ψ)∈ C+
2one has
v(t)≤(e−µtψ(0) + kϕkfl(t),if (lBj) holds,
e−µtψ(0) + kϕkfτ(t),if (τBj) holds,∀t≥0,(3.6)
introducing fl, fτ:R+−→ R+;
fl(t) := kje−µt Zt
0
e(µ+q)sds, fτ(t) := e−µtkjZt
0
eµsqe(s−τ)ds. (3.7)
Under the respective assumptions, one has
fl(t) = kj
µ+q(eqt −e−µt),(3.8)
fτ(t) = (kj
µ(1 −e−µt),if t≤τ
kj
q(e−µ(t−τ)−e−µt)+µ(eq(t−τ)−e−µt )
µ(µ+q),if t > τ. (3.9)
19
Proof. If (lBj) holds, the second addend in (VOC) can be estimated with
(3.4) as
e−µt Zt
0
eµsj(ws, vs)ds ≤kje−µt Zt
0
eµskwskds ≤ kϕkkje−µt Zt
0
e(µ+q)sds.
If one uses the definition of flin (3.7), the previous estimation yields the first
estimate in (3.6) and a straightforward integration of that expression leads
to (3.8).
If (τBj) holds, then similarly
e−µt Zt
0
eµsj(ws, vs)ds ≤e−µt kjZt
0
eµsw(s−τ(vs))ds
≤e−µtkjkϕkZt
0
eµsqe(s−τ(vs))ds ≤ kϕke−µt kjZt
0
eµsqe(s−τ)ds,
which proves the second estimate in (3.6). Another straightforward integra-
tion yields (3.9).
When writing about flor fτwe agree that from now on, respectively,
(lBj) or (τBj), holds.
Lemma 3.5. Both, fland fτ, are zero in zero. If q= 0, then
fl(t) = fτ(t) = kj
µ(1 −e−µt).
If q > 0the following hold. If τ= 0, then fl=fτand if τ > 0, then for each
fixed t > 0, one has fl(t)> fτ(t), the image fl(t)is increasing in qand the
image fτ(t)is nondecreasing in qif t≤τand increasing in qif t > τ . Both,
fland fτ, tend to ∞at ∞, are increasing and continuously differentiable.
One has
lim
t→0+
fi(t)
1−e−µt =kj
µ, i ∈ {l, τ }.
The functions gland gτdefined by
gi:R+−→ R;gi(t) := fi(t)
1−e−µt , i ∈ {l, τ },
respectively, increase from kj/µ to infinity on R+, equal kj/µ on [0, τ]and
increase to infinity on [τ , ∞).
20
Proof. The statements related to being zero in zero and the cases q= 0
and τ= 0 are obvious. Let now q > 0 and τ > 0. Monotonicity in qas
stated follows from (3.7) and the fact that exp(qs) and qe(s−τ) have related
monotonicity properties. To understand that fl(t)> fτ(t) it is sufficient to
compare the two functions given in the previous sentence. We omit further
details on these statements. Now, note that by the rule of l’Hˆopital (case
“0/0”) and (3.8) we have
lim
t→0+gl(t) = lim
t→0+
fl(t)
1−e−µt = lim
t→0+
f0
l(t)
µe−µt =kj
µ.(3.10)
For fτthe limit statement follows directly from (3.9). Next, by (3.8)
sgn d
dtgl(t) = sgn g(t),where g(t) := qeq t −(q+µ)e(q−µ)t+µe−µt.
Then g(0) = 0 and g0(t) = q2eqt (1−e−µt)+µ2e−µt(eq t −1) >0. Thus g(t)>0
for all t > 0 and hence glis increasing.
To show that gτis increasing, we prove that
g(t) := q(e−µ(t−τ)−e−µt) + µ(eq(t−τ)−e−µt )
1−e−µt
is increasing for t > τ. Using positivity of the sign of both, the denominator
of g0and µ, it is straightforward to compute that sgn g0(t) = sgn h(q) where
h(q) := qeq(t−τ)(1 −e−µt )−µeq(t−τ)−µt +qe−µt(1 −eµτ ) + µe−µt.
Then h(0) = 0. Next, similarly
h0(q) = eq(t−τ){q(t−τ)(1 −e−µt)−tµe−µt + (τ µ −1)e−µt + 1}
+e−µt(1 −eµτ ),
h0(0) = j(t),where j(t) := 1 −e−µ(t−τ)−µ(t−τ)e−µt .
Then j0(t) = µe−µt[eµτ −1 + µ(t−τ)] >0, hence h0(0) = j(t)> j(τ) = 0.
Now,
h00(q) = (t−τ)eq(t−τ)k(q),defining
k(q) := q(t−τ)(1 −e−µt)−tµe−µt + (τ µ −2)e−µt + 2.
21
Then, applying ex≥1 + xto x=µ(t−τ), one gets
k(0) = 2 −[2 + µ(t−τ)]e−µt ≥1−e−µt + 1 −e−µτ >0,
k0(q) = (t−τ)(1 −e−µt)>0.
Hence, kis positive for q > 0, thus so is h00, hence so is h0, thus so is h,
hence so is sgn g0. We have shown that gτis increasing. Monotonicity of fl
follows from monotonicity of gland the same conclusion holds for fτ. Using
that (1 −e−µt)−1converges to one at infinity the remaining statements are
straightforward to deduce.
Lemma 3.6. Assume that, respectively, (lBj) or (τBj) holds and that A,B
and Tare such that Agl(T)≤Bor Agτ(T)≤B. Then, if kϕk ≤ Aand
|ψ(0)| ≤ B, one has that v(t)≤Bfor all t∈[−h, T ].
Proof. We prove the statement for (lBj) only, since the proof for (τBj) is
similar. By (3.6) one has v(t)≤Be−µt +Afl(t) for t∈(0, T ]. Hence
v(t)≤Bif Agl(t)≤Band the latter follows by assumption and Lemma 3.5.
An elaboration of the maximum in the following lemma will be carried
out further down.
Lemma 3.7. Let kϕk ≤ Aand |ψ(0)| ≤ B. Let T > 0and choose
R≥
maxt∈[T−h,T ]∩[0,∞)max{kjqe(t)A, µ(e−µtB+Afl(t))},
if (lBj) holds,
maxt∈[T−h,T ]∩[0,∞)max{kjqe(t−τ)A, µ(e−µtB+Afτ(t))},
if (τBj) holds.
Then, if lip ψ≤Rand (w, v)is a solution through (ϕ, ψ), also lip vT≤R.
Proof. We should show that lip v|[T−h,T]≤R. First,
lip v|[T−h,T ]∩[−h,0] = lip ψ|[T−h,T ]∩[−h,0] ≤R.
Next, if (lBj) holds, we get v0(t)≤j(wt, vt)≤kjkwtk ≤ kjqe(t)kϕk.If (τBj)
holds, then v0(t)≤kjw(t−τ(vt)) ≤kjqe(t−τ(vt))kϕk ≤ kjqe(t−τ)kϕk.
Moreover, in the respective cases,
v0(t)≥ −µv(t)≥(−µ(e−µt|ψ(0)|+kϕkfl(t)),
−µ(e−µt|ψ(0)|+kϕkfτ(t))
22
and thus for t > 0
|v0(t)| ≤ (max{kjqe(t)kϕk, µ(|ψ(0)|e−µt +kϕkfl(t))},
max{kjqe(t−τ)kϕk, µ(|ψ(0)|e−µt +kϕkfτ(t))}.
Hence, in either case, lip v|[T−h,T]∩[0,∞)≤maxt∈[T−h,T ]∩[0,∞)|v0(t)| ≤ R.
We can summarize our results on invariance.
Theorem 3.8. Suppose that mand jare almost locally Lipschitz, µ > 0,
(lBm) holds and so does (lBj) or (τBj). Denote by (w, v)the solution through
(ϕ, ψ)∈VC+
2and let A,B,Rand Tdenote positive numbers such that
ψ(0) ≤Band lip ψ≤R, both of which follow if ψ∈CB,R , and kϕk ≤ A.
(a) If (lBj) holds, Agl(T)≤Band R≥max{µB, kjAeqT }, then one has
vt∈CB,R for all t∈[0, T ].
(b) If (τBj) holds, Agτ(T)≤Band
R≥max{µB, kjAqe(T−τ)},
then, again vt∈CB,R for all t∈[0, T ].
Proof. (a) Let ˜
T∈[0, T ]. Then
eqT ≥max
t∈[˜
T−h, ˜
T]∩[0,∞)
qe(t).
Moreover
max
t∈[0,˜
T]
(e−µtB+Afl(t)) = B
since e−µtB+Afl(t)|t=0 =Band e−µt B+Afl(t)≤Bfor all t∈(0,˜
T] since
Agl(t)≤Bfor all t∈(0,˜
T] by the assumptions and monotonicity of gl.
Hence, by the assumptions
R≥max{µB, AkjeqT }
≥max{max
t∈[˜
T−h, ˜
T]∩[0,∞)
µ(e−µtB+Afl(t)),max
t∈[˜
T−h, ˜
T]∩[0,∞)
Akjqe(t)}
= max
t∈[˜
T−h, ˜
T]∩[0,∞)
max{µ(e−µtB+Afl(t)), Akjqe(t)}.
23
Then, by Lemma 3.7 applied to T:= ˜
Tone has lip v˜
T≤R. As ˜
Twas chosen
arbitrarily, one has lip vt≤Rfor all t∈[0, T ]. The boundedness property
follows by Lemma 3.6.
(b) By the assumption and the monotonicity of gτ, shown in Lemma 3.5, one
has that B≥Agτ(t). Then by Lemma 3.4
v(t)≤Be−µt +Afτ(t)≤B, ∀t∈[0, T ] and
R≥max{Akjqe(T−τ), µB}
≥max
t∈[0,T ]max{Akjqe(t−τ), µ(Be−µt +Afτ(t))}.
Hence the Lipschitz-property follows by Lemma 3.7.
For further discussion we state some technical results.
Lemma 3.9. For all t > 0one has kjeqt
µ≥gl(t)and kjqe(t−τ)≥µgτ(t). If
q > 0, the corresponding strict inequalities hold and, if q= 0, equalities hold.
Proof. The case q= 0 is trivial. We present the case q > 0. For t>s≥0
one has
eq(t−s)>q(e−µ(t−s)−e−µt) + µ(eq(t−s)−e−µt )
(µ+q)(1 −e−µt)(3.11)
⇔eq(t−s)f(t)>0,where
f(t) := q+ (µ+q)e−q(t−s)−µt −(µ+q)e−µt −qe−(q+µ)(t−s).
Then limt↓sf(t) = 0 and
f0(t) = (µ+q)e−µt[qe−q(t−s)(eµs −1) + µ(1 −e−q(t−s))] >0.
Hence f(t)>0 for all t > s and (3.11) holds. Setting s= 0 shows the first
inequality and setting s=τshows the second inequality for t≥τ. The
second inequality for t<τ follows directly by definition of qeand fτ.
By the lemma, to guarantee the preconditions in Theorem 3.8 (a) and
(b), it would be sufficient to have
R≥µB ≥(kjAeqT ,respectively,
kjAqe(T−τ),
24
which is stronger but easier to check than the present assumptions. In par-
ticular, if q= 0, the preconditions in (a) and (b) are satisfied for any A,B,
and Rwith R≥µB ≥Akjand arbitrary T.
Recall that if τ= 0 then fτ=fl. Then it becomes obvious that also in
Theorem 3.8 and Lemma 3.9 the cases (lBj) and (τBj) coincide.
If τ > 0 and q > 0, through (b) a lower bound, a lower Lipschitz constant
and a larger invariance time than through (a) can be achieved. We formulate
this more precisely in obvious notation without proof:
Corollary 3.10. Fix positive numbers Aand T. Then
Ba:= Agl(T)> Agτ(T) =: Bb,
Ra:= max{µBa, kjAeqT }>max{µBb, kjAqe(T−τ)}=: Rb.
Now, fix A,Band Rsuch that R≥µB > Akj>0and define ta:=
min{ta1, ta2}and tb:= min{tb1, tb2}, with tij defined via
Agl(ta1) = B, Agτ(tb1) = B,
R= max{µB, Akjeqta2}, R = max{µB, Akjqe(tb2−τ)}.
Then taj < tbj ,j= 1,2, hence ta< tb.
Theorem 3.11. Suppose that mand jare almost locally Lipschitz, µ > 0,
(lBm) holds, so does (τBj) and denote by (w, v)=(w, v)ϕ,ψ the solution
through (ϕ, ψ)∈VC+
2. Let A,Band Rdenote positive numbers, such that
Akj< Bµ ≤R. If q > 0, choose δ > 0such that Akjeqδ =µB, and if q= 0,
choose any δ > 0. Then, if |ψ(0)| ≤ Band lip ψ≤R, so in particular if
ψ∈CB,R , and kϕk ≤ Aone has vt∈CB,R for all t∈[0, τ +δ].
The result can be concluded from Theorem 3.8 (b): To prove this for
q > 0, define T=τ+δin (b), and apply the second estimate of Lemma 3.9
with t=T. We omit further details.
Note that, if (τBj) holds and additionally τ > 0, then by Theorem 3.11
the positive time τfor which invariance holds is uniform for all A,Bsatisfying
Akj/µ < B. If merely (lBj) holds and q > 0, we cannot get such a lower
bound on invariance time through Theorem 3.8 (a).
Remark 3.12. Recall that q≤km, and that qe(t), fl(t) and fτ(t) are either
nondecreasing or increasing in q. Then it is easy to see that our statements
essentially remain true, if one replaces qby km, in particular in these func-
tions, but become weaker. Hence qis the more suitable quantity here.
25
Remark 3.13. Define
˜q:= sup
(ϕ,ψ)∈C+
2, ϕ(0)6=0
m(ϕ, ψ)
ϕ(0) .
If ˜q≤0, then q= 0 and Aand Bcan be chosen such that we obtain invariant
sets for arbitrary times. In this sense the analysis of this case is ahead of
the analysis of the case ˜q > 0. If ˜q < 0, the estimates could probably still
be improved if one would use functions defined in terms of ˜qinstead of q.
However, this would be at the expense of a more involved notation. On
the other hand the case ˜q < 0 can be related to extinction and the absence
of both, positive equilibria and oscillations, which makes it mathematically
simpler in many senses.
4. Semiflow and invariant compact sets for the cell SD-DDE
We specify the functionals m,jand τintroduced in Section 3 such that
the two-dimensional DDE (1.6) becomes the cell SD-DDE (1.1–1.4). We will
guarantee that the functionals mand jhave smoothness properties that are
such that the results of Section 3 can be applied.
Consider a function g:Dg−→ [0, K ], where Dg:= Bb(x2)×R+and
b, x2, K ∈Rare given parameters with b > 0, K > 0. Define h:= b/K > 0.
The following result contains an application of the Picard–Lindel¨of theorem.
Lemma 4.1. Suppose that gis continuous on Dgand Lipschitz in the first
argument, uniformly on compact intervals of R+with respect to the second.
Then for ψ∈ C+
1there exists a unique solution y=y(·, ψ)on [0, h]of (1.3).
Proof. Fix ψ. Define fψ: [0, h]×Bb(x2)−→ R;fψ(s, y) := −g(y, ψ(−s))
and with fψa non-autonomous ODE y0(s) = fψ(s, y(s)). Then fψsatisfies
the conditions of the Picard–Lindel¨of Theorem, e.g. [12, Theorem II.1.1],
which guarantees that there exists a unique solution yon [0, h], since we
defined h:= b/K.
Next, we show that the Gronwall inequality leads to the following result.
In the proof of part (b) of Lemma 4.2 below we will need, additionally to the
assumptions of Lemma 4.1, that gis locally Lipschitz in the second argument,
uniformly with respect to the first. Note that the two together are equivalent
to gbeing locally Lipschitz.
26
Lemma 4.2. (a) Under the assumptions of Lemma 4.1, the map Y:C+
1−→
C([0, h], Bb(x2)); Y(ψ)(t) := y(t, ψ)is continuous.
(b) If gis locally Lipschitz, then Yis locally Lipschitz.
Proof. Fix ψ∈ C+
1and define A:= {ψ(−s) : s∈[0, h], ψ ∈ C+
1∩B1(ψ)}.
Note that
A= [max{min
s∈[0,h]ψ(−s)−1,0},max
s∈[0,h]ψ(−s) + 1].
Hence, Ais a compact subinterval of R+. Choose L=L(ψ), such that gis
L-Lipschitz in the first argument, uniformly with respect to the second on
A. Let ψ, χ ∈ C+
1∩B1(ψ) and t∈[0, h]. Then
|y(t, ψ)−y(t, χ)| ≤ Zt
0
|g(y(s, ψ), ψ(−s)) −g(y(s, χ), χ(−s))|ds
≤Zt
0
|g(y(s, ψ), ψ(−s)) −g(y(s, χ), ψ(−s))|ds
+Zt
0
|g(y(s, χ), ψ(−s)) −g(y(s, χ), χ(−s))|ds
≤LZt
0
|y(s, ψ)−y(s, χ)|ds
+Zh
0
|g(y(s, χ), ψ(−s)) −g(y(s, χ), χ(−s))|ds.
Then, by Gronwall’s inequality, see e.g. [10, Corollary I 6.6] one has
|y(t, ψ)−y(t, χ)| ≤ eLh Zh
0
|g(y(s, χ), ψ(−s)) −g(y(s, χ), χ(−s))|ds (4.1)
for all ψ, χ ∈ C+
1∩B1(ψ) and t∈[0, h].
To prove (a) first note that (4.1) holds for χ:= ψ. Then continuity of Y
at ψcan be deduced using (4.1) and uniform continuity of gon the compact
set Bb(x2)×A. To prove (b), note that, since gis locally Lipschitz in the
second argument, uniformly with respect to the first, by compactness of A,
gis Lipschitz on Ain the second argument, uniformly with respect to the
first argument, i.e., there exists some KA, such that
|g(y, z1)−g(y, z2)| ≤ KA|z1−z2|,∀y∈Bb(x2), z1, z2∈A. (4.2)
The proof can be completed, using (4.1) and (4.2).
27
Remark 4.3. Note that the smoothness formulated in the previous lemma
can be formulated as smoothness of solutions with respect to variation of a
parameter in a Banach space in a non-autonomous ODE. We refer to the
discussion section in [7] for details and a discussion of literature. Somewhat
similar, the conclusion of Lemma 4.2 (a) is a special case of the conclusion of
[11, Theorem 2.2.2], if one considers in the latter only continuous dependence
with respect to the functional for a non-autonomous functional differential
equation. The preconditions in [11, Theorem 2.2.2], however, are somewhat
different. Openness of both components of the domain of the functional is
required and existence and uniqueness of solutions are assumed. On the other
hand rather than Lipschitz assumptions, mere continuity of the functional is
required. Here we proved (a) at little extra cost with respect to the proof of
(b).
Lemma 4.4. Suppose that additionally to the assumptions of Lemma 4.1
there are parameters x1, ε ∈R, such that 0< ε < K and x2−x1∈(0, bε/K)
and suppose that g(y, z)≥εfor all (y, z)∈ Dg. Then x1∈Bb(x2)and there
exists a unique
τ=τ(ψ)∈[x2−x1
K,x2−x1
ε]⊂(0, h]
solving (1.4).
Proof. Fix ψand denote by y=y(·, ψ) the solution of (1.3) and by τ=τ(·, ψ)
possible solutions of (1.4). Then yis decreasing by (1.3) since g > 0 which
shows that there can be at most one τsolving (1.4). By the fundamental
theorem of calculus applied to (1.3) and using that ε≤g(y, z)≤Kfor all
(y, z)∈ Dg, one can show that y((x2−x1)/K)≥x1≥y((x2−x1)/ε). By the
intermediate value theorem there exists some τin the stated interval solving
(1.4). The remaining statements follow by our assumptions on parameters.
In the setting of the previous lemma, we can now define a functional
τ:C+
1−→ [0, h],with τ(C+
1)⊂[τ , h),where τ:= (x2−x1)/K > 0.
Smoothness of Yis inherited by τin the following sense.
Lemma 4.5. Suppose that the assumptions of Lemma 4.4 hold. Then the
functional τ:C+
1−→ [0, h]is continuous. If additionally to the stated as-
sumptions gis locally Lipschitz, then so is τ.
28
Proof. Let ψ, ψ ∈ C+
1. By definition of τ(ψ) and τ(ψ) one has
y(τ(ψ), ψ) = y(τ(ψ), ψ) (= x1).
Hence,
|y(τ(ψ), ψ)−y(τ(ψ), ψ)|=|y(τ(ψ), ψ)−y(τ(ψ), ψ)|.
The left hand side is dominated by kY(ψ)−Y(ψ)k. By the mean value
theorem, there exists some t∈[0, h], such that the right hand side equals
|D1y(t, ψ)||τ(ψ)−τ(ψ)|
=|g(y(t, ψ), ψ(−t))||τ(ψ)−τ(ψ)| ≥ ε|τ(ψ)−τ(ψ)|.
Thus |τ(ψ)−τ(ψ)| ≤ 1
ε|Y(ψ)−Y(ψ)|and the proof is completed by applying
Lemma 4.2.
Next, we introduce functions q:R+−→ R,γ:R+−→ R+and d:
Dg−→ R. Define
m:C+
2−→ R;m(ϕ, ψ) := q(ψ(0))ϕ(0).
If the assumptions of Lemma 4.4 (and Lemma 4.1) hold and moreover gis
partially differentiable in the first argument and D1gand dare continuous,
we can define j:C+
2−→ R+via
j(ϕ, ψ) := γ(ψ(−τ(ψ)))
g(x1, ψ(−τ(ψ))) g(x2, ψ(0))ϕ(−τ(ψ))eRτ(ψ)
0[d−D1g](y(s,ψ),ψ(−s))ds .
(4.3)
Note that continuity of the partial derivative of gimplies the Lipschitz prop-
erty required in Lemma 4.1. To guarantee smoothness of j, it is useful to
introduce a notation that summarizes ingredients with the same type of de-
lay. We will use below that jis a special case of the functional defined in the
following corollary.
Corollary 4.6. Let β:R+−→ R+,r:C+
1−→ [0, h]and G:C+
1−→ Rbe
continuous maps and suppose that g(x2,·)is continuous. Then the functional
C+
2−→ R+;
(ϕ, ψ)7−→ β(ψ(−r(ψ)))g(x2, ψ(0))ϕ(−r(ψ))eG(ψ)(4.4)
is continuous. If βand g(x2,·)are locally Lipschitz and rand Gare almost
locally Lipschitz, it is almost locally Lipschitz.
29
The result is a straightforward application of Lemma 3.1 and discussed
or straightforward rules. We omit further details. Below we will apply the
corollary to even locally Lipschitz rand G. However, Lemma 3.1 and the
discussion below it should make clear that assuming locally Lipschitz rand
Gdoes not allow to sharpen the second statement of the corollary to “locally
Lipschitz”.
Lemma 4.7. Consider arbitrary continuous operators
Q:DQ−→ RQ, r :DQ−→ [a, b], G :DG:= DQ× RQ−→ RG,where
DQ:= C([−b, −a], J1),RQ:= C([a, b], J2),RG:= C([a, b], J3),
a, b ∈Rwith a < b and J1,J2and J3are nonempty subsets of R. Define
G:DQ−→ J3;G(ψ) := G(ψ, Q(ψ))(r(ψ)).
Then Gis continuous. If G,Qand rare locally Lipschitz and moreover
∀x0∈ DG∃k=k(x0), δ =δ(x0), s.th. sup
x∈DG∩Bδ(x0)
lip G(x)≤k, (4.5)
then Gis locally Lipschitz.
Proof. First, we decompose Gas
ψ(id×Q)×r
7−−−−−→ ((ψ, Q(ψ)), r(ψ)) (G,id)
7−−−→ (G(ψ, Q(ψ)), r(ψ))
ev
7−→ G(ψ, Q(ψ))(r(ψ)).
As the involved maps are continuous, the decomposition shows that so is G.
To prove the second statement, first, by similar arguments, ψ7→ G(ψ, Q(ψ))
is locally Lipschitz as a composition. Next, in notation similar to the one
in Lemma 3.1, we can write G(ψ) = ev ◦(id, r)(G(ψ, Q(ψ)), ψ). Then the
remainder of the proof is similar to showing the last statement of Lemma
3.1, if one uses that functions in the first argument of ev ◦(id, r) lie in RG
and thus are Lipschitz by (4.5).
The following lemma will be helpful to apply Gof the previous lemma to
the integral term appearing in the exponent of j.
30
Lemma 4.8. Let J1and J2be nonempty subsets of Rand let a, b ∈R,a<b.
Suppose that l:J2×J1−→ Ris continuous. Define
H:DH:= C([a, b], J2×J1)−→ C([a, b],R); H(χ)(t) := Zt
a
l(χ(s))ds.
Then His continuous. If lis locally Lipschitz, then so is H.
Proof. First note that for any ξ, χ ∈ DHa standard estimation yields
kH(ξ)−H(χ)k ≤ (b−a) max
s∈[a,b]|l(ξ(s)) −l(χ(s))|.(4.6)
To prove continuity in an arbitrary χ∈ DH, fix ε > 0. First note that
χ([a, b]) ⊂J2×J1is compact. Hence we can choose a neighborhood Uof
χ([a, b]) in R2such that K:= U∩(J2×J1) is a compact subset of J2×J1.
Next, we can choose δ > 0 such that both, ξ∈ DHand kξ−χk< δ implies
ξ(s)∈Kfor all s∈[a, b] and, by uniform continuity of lon K, one has
|l(y)−l(z)| ≤ ε
(b−a),if |y−z| ≤ δ, y, z ∈K. (4.7)
Combining the last argument with (4.6) yields that
kH(ξ)−H(χ)k ≤ εif ξ∈ DH,kξ−χk< δ.
This proves the continuity statement.
In the following we show that His locally Lipschitz in φ∈ DH. Similar
as above, we can choose a neighborhood Uof φ([a, b]) in R2such that K:=
U∩(J2×J1) is a compact subset of J2×J1. Since lis locally Lipschitz, it is
L-Lipschitz on Kfor some constant L. Next, we can choose δ > 0 such that
ξ, χ ∈ DH∩Bδ(φ) implies ξ(s), χ(s)∈Kfor all s∈[a, b]. Hence we can use
(4.6) and the Lipschitz property for lso see that His L(b−a)-Lipschitz on
Bδ(φ), which completes the proof.
See also [3] for details on smoothness properties of related Nemytskii-
operators. Note that in Lemmas 4.7 and 4.8 we obtain locally Lipschitz
operators, whereas in Corollary 4.6 only almost locally Lipschitz operators
are required. However, in Lemma 4.8 it is fairly natural to assume that
the kernel lis locally Lipschitz and, if one would state on a merely almost
locally Lipschitz integral operator, it is not clear how to formulate a weaker
natural precondition for the kernel. It should become clear below that similar
considerations apply to Qand r. We are now ready to establish smoothness
of mand j.
31
Lemma 4.9. (a) If qis continuous, then the functional mis continuous and
m(U)is bounded, whenever U⊂ C+
2is bounded. If qis locally Lipschitz, then
so is m.
(b) If the assumptions of Lemma 4.4 (and Lemma 4.1) hold, gis partially
differentiable in the first argument and D1g,dand γare continuous, then
the functional jis continuous and j(U)is bounded, whenever U⊂ C+
2is
bounded. If additionally gis locally Lipschitz and D1g,γand dare locally
Lipschitz, then jis almost locally Lipschitz.
Proof. We prove only (b), since (a) does not involve additional arguments.
In the notation of Lemma 4.7 define a:= 0, b:= h,J1:= R+,J2:= Bb(x2),
J3:= R,β:= γ(·)/g(x1,·), Q:= Y,r:= τ,
G:C+
1×C([0, h], Bb(x2)) −→ C([0, h],R);
G(ψ, z)(t) := Zt
0
[d−D1g](z(s), ψ(−s))ds
and with these ingredients Gand (4.4). To see the first statement, by Corol-
lary 4.6 it is sufficient to show that β,rand Gare continuous. Continuity
of βfollows from continuity of γand g. Continuity of r=τwas shown in
Lemma 4.5. Continuity of Gfollows from Lemma 4.7, provided we show that
Gis continuous. To see this, first apply Lemma 4.8 to the case l:= d−D1g
and then observe that G=H◦Φ, introducing the Lipschitz map
Φ : C+
1×C([0, h], Bb(x2)) −→ C([0, h],Dg); Φ(ψ, z)(s) := (z(s), ψ(−s)).
To see that the statement on boundedness holds, let B1×B2⊂ C+
2be
bounded. Then {ψ(−τ(ψ)) : ψ∈B2},{ϕ(−τ(ψ)) : (ϕ, ψ)∈B1×B2}and
{(y(s, ψ), ψ(−s)) : ψ∈B2s∈[0, h]}are bounded. Each of the functions γ,
g(x2,·), id|R+,dand D1ghas closed domain that contains the corresponding
(see definition of j) of the above sets. Hence for each function the domain
contains also the closure of the set. Since each function is continuous and
any such closure compact, the functions are bounded on the closures of the
sets, thus also on the sets themselves. Finally note that g(x1,·)≥εon its
domain. Then boundedness of jon B1×B2should be obvious. To see that
jis almost locally Lipschitz, similarly as above, it is sufficient to show that
β,rand Gare locally Lipschitz. The map βis locally Lipschitz since γand
g(x1,·) are. The functional r=τis locally Lipschitz by Lemma 4.5. Next,
similarly as we showed continuity of Gabove, it follows by Lemma 4.8 that
32
under the corresponding assumptions Gis also locally Lipschitz. To see that
(4.5) holds, one can use boundedness of lon a suitable compact set, which
follows from continuity of lwith an argument similarly as above. Then, by
Lemma 4.7, also Gis locally Lipschitz.
Lemma 4.9 refers to conditions under which (1.3–1.4) can be solved and
mand jcan be defined. Under these conditions (1.6) is equivalent to the
cell SD-DDE.
Recall that we can achieve global existence via the results of Section 3 if
we guarantee (lBm) and (sBj). If we add to the assumptions that γ,dand
D1gare bounded it is easy to see that (τBj) and thus in particular (sBj)
holds. For our specification of m, (lBm) holds if and only if qis bounded. In
this case, in the notation of Section 3
q= max{sup
z∈R+
q(z),0}<∞.
With the results established in this subsection for mand jwe can now apply
the results of Section 3 on (1.6) to conclude well-posedness and invariance
properties for the cell SD-DDE. For better overview, we repeat the main
assumptions.
Theorem 4.10. Consider parameters x1, x2, b, ε, K with b > 0,0< ε < K
and x2−x1∈(0, bε/K). Suppose that for Dg:= Bb(x2)×R+and g:Dg−→
[ε, K]one has that gis partially differentiable in the first argument and g
and D1gare continuous and define h:= b/K > 0. Suppose that moreover
γ:R+−→ R+,d:Dg−→ Rand q:R+−→ R, are continuous. Then the
following hold.
(a) For all φ∈ C+
2there exists some c=c(φ)and a non-continuable solution
(w, v)φof the cell SD-DDE (1.1–1.4) on [−h, c)through φ.
(b) If g,D1g,γ,dand qare locally Lipschitz, then for any φ∈VC+
2there
exists a unique c=c(φ)and a unique non-continuable solution on [−h, c)
through φ.
(c) If D1g,γ,dand qare bounded, then for any φ∈ C+
2there exists a global
solution through φ.
(d) If the preconditions of both, (b) and (c), hold then for any φ∈VC+
2there
exists a unique global solution through φ. The solutions define a continuous
semiflow in the sense of Theorem 2.16 and with gτas in Section 3 satisfy the
invariance properties in Proposition 3.3, Theorem 3.8 (b) and Theorem 3.11.
33
Finally, τ > 0, i.e., there exists a positive invariance time which is uniform
in the sense discussed in Section 3.
Proof. (a) It is trivial that (Fm) holds. By Lemma 4.9, mand jare contin-
uous. Then the statement follows by Lemma 3.2 (a).
(b) By Lemma 4.9, mis locally Lipschitz and jis almost locally Lipschitz
and the statement follows by Lemma 3.2 (c)
(c) The statement follows by Lemma 3.2 (d) and by what we have discussed.
(d) Existence of a unique global solution follows trivially from (b) and (c).
Next, by the conditions of (b) and by Lemma 4.9 the functionals mand j
are almost locally Lipschitz. Moreover, we have discussed, that (lBm) and
(sBj) hold. Then by Lemma 3.2 (e) solutions define a continuous semiflow.
The invariance properties follow by Theorem 3.8 (d).
5. Discussion and outlook
In Section 4 we have elaborated conditions on the functions q,γ,gand d
and the positive parameter µthat guarantee well-posedness of the cell SD-
DDE and invariance properties. For a further specification of these functions
we refer to [6, 5, 16]. The functions introduced in this paper are essentially
generalizations of these specifications. We also remark that the exact nature
of the cellular and sub-cellular processes related to these ingredients is subject
to current research, see e.g. [20].
To establish existence of periodic solutions for a certain class of DDE with
state-dependent delay, in [14] the authors include the assumption that the
initial function should be at equilibrium value at time zero in their definition
of the invariant set. In future analysis of the cell SD-DDE one could include
such assumptions and try to investigate convex and compact sets that are
invariant under the original untransformed system (1.1–1.4), i.e., sets that
are invariant for both components of the state. Motivated by the fact that
(global) existence of periodic solutions often can be concluded from behavior
in a finite time interval, we also have some hope that the invariance for finite
time, as established here, may be sufficient to obtain results on the existence
of periodic solutions.
In population dynamics, ultimate boundedness and dissipativity, apart
from being interesting on their own, often can be used to conclude popula-
tion persistence [18] and these topics are essentially open problems for the cell
34
SD-DDE. In relation to the absence of feedback from above (see the Intro-
duction), in ongoing research on these problems, the authors reencountered
some of the discussed challenges they found in the invariance analysis of this
manuscript. In this sense there is hope that future research may benefit from
the research in this manuscript. The authors are also involved in ongoing
research on global stability of equilibria for the cell SD-DDE. Should results
be achieved in any of these areas, they may be formulated for the enlarged
set of initial conditions established in this manuscript.
Corollary 2.18 is an example for how asymptotic behavior in terms of con-
vergence in Rcan be concluded from continuous dependence of the solution
on the initial value in the C-topology. Using continuous dependence of the
solution on the initial value in C1, as established in [7], one could possibly
prove similarly that the limit is an equilibrium, if the convergence of the so-
lution to the constant is in C1. The latter, however, is a stronger prerequisite
and may be too strong in applications.
Section 4 shows that it is feasible to guarantee that a functional appearing
in a real world application is almost locally Lipschitz, but that this should be
taken care of by the mathematical rather than by the modelling community
and we hope to provide some more generally applicable ideas on how this
can be done.
Acknowledgements: The manuscript was inspired by discussions with Ti-
bor Krisztin during a postdoctoral stay of Ph.G. at the University of Szeged.
Ph.G. thanks Stefan Siegmund und Reinhard Stahn at Technische Univer-
sit¨at Dresden for help with the manuscript.
[1] T. Alarc´on, Ph. Getto and Y. Nakata, Stability analysis of a renewal
equation for cell population dynamics with quiescence, SIAM J. Appl.
Math. 74 (4), 1266–1297 (2014)
[2] H. Amann, Ordinary Differential Equations, An Introduction to Non-
linear Analysis, Walter de Gruyter, Berlin, New York, 1990.
[3] J. Appell, M. V¨ath, Elemente der Funktionalanalysis. Vieweg, 2005.
[4] O. Diekmann, S. van Gils, S.M. Verduyn Lunel, H.-O. Walther, De-
lay Equations, Functional-, Complex-, and Nonlinear Analysis, Springer
Verlag, New York, 1995.
35
[5] M. Doumic, A. Marciniak-Czochra, B. Perthame, J. P. Zubelli, A struc-
tured population model of cell differentiation, SIAM J. Appl. Math. 71,
1918–1940 (2011)
[6] Ph. Getto, A. Marciniak-Czochra, Mathematical modelling as a tool
to understand cell self-renewal and differentiation, in M dM. Vivanco
(Ed.), Mammary stem cells - Methods in Molecular Biology, Springer
protocols, Humana press 247–266.
[7] Ph. Getto, M. Waurick, A differential equation with state-dependent
delay from cell population biology, J. Differential Equations 260, 6176–
6200 (2016)
[8] Ph. Getto, M. Gyllenberg, Y. Nakata, F. Scarabel, Stability analysis of
a state-dependent delay differential equation for cell maturation: ana-
lytical and numerical methods, J. Math. Biol., 79 (1), 281–328 (2019)
[9] J.K. Hale, Dynamical Systems and Stability, J. Math. Anal. Appl., 26,
39–59 (1969)
[10] J.K. Hale, Ordinary Differential Equations, John Wiley and sons inc.,
1969.
[11] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential
Equations, Springer Verlag, New York, 1991.
[12] Ph. Hartman, Ordinary Differential Equations, John Wiley & Sons, New
York, London, Sydney, 1964.
[13] F. Hartung, T. Krisztin, H.-O. Walther, J. Wu, Functional Differen-
tial Equations with state dependent delays: Theory and Applications,
Chapter V in Handbook of Differential Equations: Ordinary Differential
Equations, Volume 4, Elsevier.
[14] J. Mallet-Paret, R. D. Nussbaum, Boundary Layer Phenomena for
Differential-Delay Equations with State-Dependent Time Lags I, Arch.
Rational Mech. Anal. 120, 99–146 (1992)
[15] J. Mallet-Paret, R.D. Nussbaum, P. Paraskevopoulos, Periodic Solutions
for Functional Differential Equations with Multiple State-Dependent
Time Lags, Topol. Meth. Nonl. Anal. 3, 101–162 (1994)
36
[16] A. Marciniak-Czochra, T. Stiehl, A. D. Ho, W. Jaeger, W. Wagner,
Modeling of asymmetric cell division in hematopoietic stem cells: Reg-
ulation of self-renewal is essential for efficient repopulation, Stem Cells
Dev. 17, 1–10 (2008)
[17] H.L. Smith, Monotone Semiflows Generated by Functional Differential
Equations, J. Differential Equations 66, 420–442 (1987).
[18] H.L. Smith, H.R. Thieme, Dynamical Systems and Population Persis-
tence, Graduate Studies in Mathematics Vol. 118, American Mathemat-
ical Society, Providence, Rhode Island, 2010.
[19] E. Stumpf, Local stability analysis of differential equations with state-
dependent delay, Discr. Cont. Dyn. Sys. a 6, 3445–3461 (2016)
[20] M dM. Vivanco (Ed.), Mammary stem cells - Methods in Molecular
Biology, Springer protocols, Humana press.
[21] H.-O. Walther, The solution manifold and C1-smoothness for differential
equations with state-dependent delay, J. Differential Equations 195, 46–
65 (2003)
37
