Dense Short Solution Segments from Monotonic Delayed Arguments

We construct a delay functional d on an open subset of the space Cr1=C1([-r,0],R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1_r=C^1([-r,0],\mathbb {R})$$\end{document} and find h∈(0,r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in (0,r)$$\end{document} so that the equation x′(t)=-x(t-d(xt))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$\end{document}defines a continuous semiflow of continuously differentiable solution operators on the solution manifold X={ϕ∈Cr1:ϕ′(0)=-ϕ(-d(ϕ))},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$\end{document}and along each solution the delayed argument t-d(xt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t-d(x_t)$$\end{document} is strictly increasing, and there exists a solution whose short segmentsxt,short=x(t+·)∈Ch2,t≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$\end{document}are dense in an infinite-dimensional subset of the space Ch2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2_h$$\end{document}. The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.

with α > 0 and constant time lag r > 0. This is the simplest delay differential equation modelling negative feedback with respect to the zero solution. Let C 0 r denote the Banach space of continuous functions [−r , 0] → R with the maximum norm, |φ| 0,r = max −r ≤t≤0 |φ(t)|.
The results in [13,14] established another kind of complicated solution behaviour, namely, the existence of delay functionals d and parameters α > 0 so that for a positive number h < r there are solutions whose short solution segments are dense in open subsets of the space C 1 h . In [13] density of short segments in the whole space C 1 h was achieved for a continuous delay functional on a set Y ⊂ C 1 r which is large in some sense but not open, nor a differentiable submanifold. Because of this lack of regularity results from [8,9] on well-posedness of initial value problems and on differentiability of solutions with respect to initial data do not apply.
In [14] we constructed a continuously differentiable delay functional d : U → [0, r ], U ⊂ C 1 r open, so that the results from [8] apply, and found h ∈ (0, r ) so that the previous equation with α = 1, namely, has a solution x : [−r , ∞) → R whose short segments are dense in an open subset of the space C 1 h . The construction involves that the delayed argument function along the solution x is not monotonic, and this oscillatory behaviour seems crucial for density of short segments in an open subset of the space C 1 h . Before stating the result of the present paper let us mention that equations with nonconstant, state-dependent delay are not covered by the theory with state space C 0 r which is familiar from monographs on delay differential equations [1][2][3]. We recall what was shown in [8] for delay differential equations in the general form under hypotheses designed for applications to examples with state-dependent delay. Let C 0 r ,n and C 1 r ,n denote the analogues of the spaces C 0 r and C 1 r , for maps [−r , 0] → R n . Assume f : U → R n , U ⊂ C 1 r ,n open, is continuously differentiable so that (e) each derivative D f (φ) : C 1 r ,n → R n , φ ∈ U , has a linear extension D e f (φ) : C 0 r ,n → R n and the map U × C 0 r ,n (φ, χ) → D e f (φ)χ ∈ R n is continuous.
The extension property (e) is a variant of the notion of being almost Fréchet differentiable for maps C 0 r ,n ⊃ V → R n which was introduced in [7]. Suppose also there exists φ ∈ U with φ (0) = f (φ). Then the nonempty set is a continuously differentiable submanifold with codimension n in C 1 r ,n , and each initial value problem with the said maximal solution x = x φ , defines a continuous semiflow of continuously differentiable solution operators In the present paper we prove the following result on complicated motion caused by a delay functional so that the delayed argument functions along solutions of Eq. (1.1) are monotonic.
) ∈ R is continuously differentiable and has property (e), and for each φ ∈ X f the delayed argument function

is strictly increasing.
Here C 2 h denotes the Banach space of twice continuously differentiable functions ψ : [−h, 0] → R, with the norm given by |ψ| 2,r = 2 k=0 max −r ≤t≤0 |ψ (k) (t)|. A different result on complicated motion caused by state-dependent delay with monotonic delayed argument functions has recently been obtained in [5].
The proof of Theorem -the delayed argument function [0, t 5 ] t → t − n (t) ∈ R along the delay function n is strictly increasing, -on some subinterval of length h in [0, t 5 ] the function x (n) coincides with a translate of a member p n of a sequence which is dense in A, -on some subinterval of length 2s in [0, t 5 ] the function x (n) coincides with a translate of κ n = κ s,n .
In Sect. 5 shifted copies of the functions n and of the functions ±x (n) are concatenated, respectively, and yield a twice continuously differentiable function x : [t b , ∞) → R and a continuously differentiable delay function on [0, ∞) which is bounded by some r > max{h, −t b }. A twice continuously differentiable extension of the function x to the ray [−r , ∞) → R satisfies the linear equation converts the delay function into a delay functional d on the trace {x t ∈ C 2 r : t ≥ r }. Sections 6, 7, and 8 prepare the extension of this functional to an open neighbourhood N of the trace {x t ∈ C 2 r : ( j r − 1)t 5 ≤ t} in the space C 1 r , with an integer j r ≥ 2 so that r < ( j r − 1)t 5 . Section 6 contains an ingredient of the construction which will be used in the final Sect. 9, namely, separation of nonadjacent arcs The separation result is based on the properties of the functions κ s,n from Sect. 3 whose translates appear as restrictions of x on a sequence of mutually disjoint intervals tending to infinity.
The constructions in Sects. 2, 3, 4, 5, and 6 are to some extent parallel to constructions in [14]. The next steps in Sects. 7 and 8 are rather different from their counterparts in [14]. The new tool, introduced in Sect. 7, is a bundle of transversal hyperplanes K t , t > 0, along the curve (0, ∞) t → x t ∈ C 0 r . Working with the bundle allows for an extension of the delay functional from an arc {x t ∈ C 2 r : (k − 1)t 5 ≤ t ≤ kt 5 }, j r ≤ k ∈ N, to a kind of tubular neighbourhood U k ⊂ C 0 r (Sect. 8), and for the arrangement of compatibility relations on overlapping domains U k ∩ U k+1 , in ways which are simpler than corresponding procedures in [14].
Section 9 begins with the definition of the domain N ⊂ C 1 r and the functional d : N → (0, r ), and completes the proof of Theorem 1.1. The verification that the functional f : N → R in Theorem 1.1 has property (e) uses that the delay functional d : N → (0, r ) has property (e). The latter is achieved by means of the following proposition whose statement involves the injective linear continuous inclusion map Notation, preliminaries. A sequence in a metric space is called dense if each point of the metric space is an accumulation point of the sequence. A metric space is called separable if it contains a dense sequence.
For  In case a = −r and b = 0, the abbreviations C j r = C j −r ,0 and | · | j,r = | · | j,−r ,0 are used. If functions φ ∈ C 2 r and φ ∈ C 1 r are considered as elements of the ambient space C 0 r then we use φ ∈ C 0 r or J φ ∈ C 0 r , depending on which form makes an argument more transparent.
For r > 0 the evaluation map is continuous but not locally Lipschitz continuous, and the evaluation map , see e. g. [4,8].
In Sect. 8 below the following is used.
Proof By continuity there exists t a ∈ (a, t) with c([t a , t]) ⊂ U /2 (c(t)). The compact sets c([a, t a ]) and c ([t, b]) are disjoint, which gives Choose ρ ∈ 0, 2 with The assumption u a < t a yields a contradiction to the inequality which means z ∈ U (c(t)).

Separability
Let h > 0 be given. The restrictions of polynomials R → R to the interval [−h, 0] are dense in C 2 h , which is an easy consequence of the Weierstraß approximation theorem. Let P 5 ⊂ C 2 h denote the subspace of restrictions of polynomials of degree not larger than 5 and let C 2 h−0 ⊂ C 2 h denote the closed subspace given by the equations Then dim P 5 = 6 and which follows from the fact that given φ ∈ C 2 h there exists a unique p ∈ P 5 satisfying

Proposition 2.1 Let an open set U
Proof The restricted polynomials with rational coefficients form a sequence which is dense in C 2 h . Projection along P 5 onto C 2 h−0 yields a sequence which is dense in C 2 h−0 , and translation by adding p * results in a sequence which is dense in p * +C 2 h−0 . The members of this sequence which belong to U form a sequence which is dense in A.

Example 2.2 For given reals
Notice that We add the obvious fact that the dense sequence provided by Proposition 2.1 is dense in

Differentiable Functions with Separated Shifted Copies
Let s > 0 be given. We construct a sequence of functions κ n ∈ C 2 −s,s , n ∈ N, so that shifted copies of these functions keep a positive minimal distance from each other with espect to the norm | · | 1,−s,s .
Let also positive reals a, ξ, η be given and choose ∈ 0, a 4 . There exists Proof Let positive integers n = k and t ∈ − s 2 , 0 be given. In case n > k consider In case k > n set u = −t + s 2 n+1 . Then and observe that Using Proposition 3.1 and < a 4 we get the following result.

The Delay Function on a Compact Interval
In this section we find which in the next section will be used to form a solution of Eq. (1.2) whose short segments are dense in the set A ∪ (−A). Choose reals such that there exists t 2 > 1 with bt 2 > ξ > at 2 , Choose we can choose v in such a way that also b + Fig. 2 The function x ∈ C 2 The equation so that x(t 2 ) = −ξ , and let t 1 ∈ (0, t 2 ) be given by The functions in A are negative and strictly decreasing, with the derivative strictly increasing. Proposition 2.1 guarantees a sequence ( in case n odd, Obviously, It follows that the equation In particular, .
Fix some s > 0 and recall κ n ∈ C 2 −s,s from Sect. 3, with a, ξ, η from the present section. Then and define an extension of x (n) to a map in By the symmetry of κ n , It follows that the equation Setting we get an extension of n ∈ C 1 0,t 2 to a nonnegative map in C 1 0,t 3 , with for example, t 4 = t 3 + 1.
Proof Consider the discontinuous function g 0 : [t 3 , t 4 ] → R given by g 0 (t 3 ) = t 1 and g 0 (t) = t 3 for t 3 < t ≤ t 4 . There is a sequence of functions g j ∈ C 1 which converge pointwise to g 0 . For every j ∈ N, , and the Lebesgue dominated convergence theorem yields Similarly there is a sequence of functions h j ∈ C 1 t 3 ,t 4 with the same properties as g j which converge pointwise to h 0 : [t 3 , t 4 ] → R given by h 0 (t 4 ) = t 3 and h 0 (t) = t 1 for t 3 ≤ t < t 4 , and The limits satisfy due to the choice of t 4 . So there exists j ∈ N with The function is continuous. Using the intermediate value theorem we find some θ ∈ (0, 1) with Notice that the convex combination k(θ, ·) ∈ C 1 t 3 ,t 4 shares the properties of g j and h j . Define δ n * by t − δ n * (t) = k(θ, t).
It follows that the equation defines a continuation of n ∈ C 1 0,t 4 to a nonnegative function in C 1 0,t 5 so that we have and

Concatenation
All functions x (n) ∈ C 2 t b ,t 5 , n ∈ N, coincide on the set we have t 4 = t 5 + t b , and for every n ∈ N, Moreover, for every n ∈ N the nonnegative function n ∈ C 1 0,t 5 satisfies and we have Therefore the relations The short segments x (n−1)t 5 +t d ,short = p n+1 2 ∈ C 2 h , n ∈ N odd, which are given by for each n ∈ N and set The curvex r for all t > 0, compare [13, Proposition 4.1]. As t 2 +t 3 2 is the only zero of (x (n) ) : [t b , t 5 ] → R, for any n ∈ N, we have

Proposition 5.1 The restriction of the curvex to the ray [r , ∞) is injective.
Proof Assume r ≤ t ≤ u andx(t) =x(u). Then There are n ∈ N and k ∈ N with (n − 1)t 5 ≤ t < nt 5 and (k − 1)t 5 ≤ u < kt 5 .
From t 5 < r ≤ t we have n ≥ 2, and from t ≤ u we have n ≤ k.
As the interval (u − t 5 , u] contains exactly one zero of x, situated at (k − 1)t 5 , we get u + w = (k − 1)t 5 , hence 2. The case (n − 1)t 5 + t 3 ≤ t (< nt 5 ). Using Part 1 of the proof we get For every w ∈ [−s, s] we obtain and it follows that n = k. By Part 1, t = u.

Delay Functionals on C 0 r -Neighbourhoods of Compact Arcs
The curvex We have with the projections onto the first and second component, respectively, with the continuous linear evaluation maps and with the multiplication m : which follows from the fact that the zeros of x in [t b , ∞) are given by 1 In the sequel we show that every compact arc Jx ([u, v]) ⊂ C 0 r , r < u < v, has a neighbourhood U in C 0 r on which the representation φ = x t + κ with κ ∈ K t , t close to [u, v], and κ = φ − x t small in C 0 r is unique. Knowing this we shall define a delay functional d U : Then d is constant along each fibre (x t + K t ) ∩ U , with t close to [u, v]. Obviously, for all φ ∈ C 0 r and all σ > 0.
For every φ ∈ U (x t ) and for σ = τ (φ), Proof Let t > 0 be given. The map is continuously differentiable and satisfies f (t, x t ) = 0. Using the formula defining the map L we infer hence Apply the Implicit Function Theorem and obtain δ ∈ (0, t), > 0, and a continuously differentiable map τ with the properties stated in the first sentence of the proposition. Notice that one can achieve ≤ δ. For φ ∈ U (x t ) and σ = τ (φ) we get Proposition 7.2 (Fibre representation along compact arcs) Let reals u < v in (r , ∞) and n ∈ N be given. There exist positive ρ = ρ(u, v, n) ≤ 1 n so that for every φ ∈ U ρ (Jx ([u, v])) there is one and only one |x w | 0,r .
2. Apply Proposition 7.1 to each w ∈ [u, v], and obtain = w and δ = δ w and τ = τ w according to Proposition 7.1. Notice that one my assume Using the compactness of Jx ([u, v]) ⊂ C 0 r one finds a strictly increasing finite sequence (w j )¯j 1 in [u, v] so that the associated neighbourhoods U w j (x(w j )), j ∈ {1, . . . ,j}, form a covering of Jx ([u, v]). There exists a positive real number Notice that For every φ ∈ U ρ (Jx ([u, v]) we obtain (at least one) Or, the set R n ⊂ (0, ∞) of all ρ ∈ 0, 1 n such that for every φ ∈ U ρ (Jx ([u, v])) there exist and |φ − x σ | 0,r ≤ 1 n is unbounded. We derive a contradiction. The elements of I form a strictly increasing sequence (n k ) ∞ 1 . For every k ∈ N select some φ k in U ρ (Jx ([u, v])) with ρ = ρ n k and σ (1) k < σ (2) Using the compactness of, say, [0, v + 1], and successively choosing subsequences we find a strictly increasing sequence (k κ ) ∞ 1 so that the equations define two sequences which converge to z (1) ≤ z (2) in [0, v + 1], respectively. Necessarily, (2) . Apply Proposition 7.1 to t = z (1) = z (2) and choose positive ≤ δ according to this proposition. For κ ∈ N sufficiently large we have κ belong to (t − δ, t + δ), and This yields a contradiction to the first part of Proposition 7.1. 4. Combining the results of Parts 1 and 2 we obtain n(u, v) ∈ N such that for every integer n ≥ n(u, v) and for every φ ∈ U ρ n (Jx ([u, v])) there exists one and only one n . Now the assertion of Proposition 7.2 follows easily. Proposition 7.2 yields that for u < v in (r , ∞) and n ∈ N there exists ρ ≤ 1 n so that the relations  (Jx([u, v])) and for every σ ∈ u − 1 n , v + 1 n ∩ (0, ∞), The construction is iterative. We carry out the initial step and the step thereafter. This second step is the model for the step from statements for general k ≥ j to statements for k + 1. 1. The initial step for k = j. 1.1. Apply Proposition 7.1 with t = jt 5 atx(t), choose δ = δ( j) > 0, = ( j) ∈ (0, δ], and a map τ = τ j from U (x(t)) ⊂ C 0 r into (t − δ, t + δ) accordingly. By continuity there are n = n( j) ∈ N witĥ n .
An application of Proposition 1.
Set U j = U η (J X j ) and s j = s u,v,η .
Proof We have which shows that f is continuously differentiable. Recall D 1 ev 1 r (φ, t)φ =φ(t) and D 2 ev 1 r (φ, t)t =tφ (t). The chain rule yields For φ ∈ N the equation defines a linear extension D e f (φ) : C 0 r → R of the derivative D f (φ) : C 1 r → R. Using the continuity of the evaluation map C 0 r × [−r , 0] (χ, t) → χ(t) ∈ R and property (e) of d one finds that the map N × C 0 r (φ, χ) → D e f (φ)χ ∈ R is continuous.
For t ≥ j r t 5 we have x t ∈ N and, due to Eq. (9.1), This implies that the twice continuously differentiable function Finally we show that for each φ ∈ X f the delayed argument function is strictly increasing. Let φ ∈ X f and t ∈ (0, t φ ) be given and set y = x φ . As y : [−r , t φ ) → R is continuously differentiable the curveỹ : [0, t φ ) t → J y t ∈ C 0 r is continuously differentiable with Dỹ(u)1 = y u for all u > 0, compare [13,Proposition 4.1]. The segment y t ∈ X f ⊂ N is contained in N k for some integer k ≥ j r . By continuity of the flowline [0, t φ ) u → y u ∈ X f ⊂ N ⊂ C 1 r , there is > 0 with y u ∈ N k for all u ∈ (t − , t + ). Then d(y u ) = d k (J y u ) = d k (ỹ(u)) on (t − , t + ). It follows that the curve is differentiable with derivatives given by Dd k (J y u )y u < 1. This implies that on (0, t φ ) the delayed argument function is differentiable with positive derivative, from which the assertion follows.
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