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On time transformations for differential equations with state-dependent delay

Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.


Introduction
Delay differential equations successfully used to model and study a number of applied problems in physics, biology, chemistry. Basic mathematical theory of constant delay equations can be found in the classical monographs [5,6] and references therein.
Differential equations with state-dependent delay (SDD) attracted much attention during last decades and many deep results have been obtained so far (see [9,11,15,10] and references therein). Such equations with discrete state-dependent delays are always nonlinear by their nature. As described in [9], this type of delay brings additional difficulties in proving such basic properties of solutions as uniqueness and continuous dependence on initial data. The main approach to get the well-posed initial-value problem is to restrict the set of initial functions and hence the set of solutions to C 1 -functions [9]. For an alternative approach where an additional condition on the SDD is used to get a well-posed initial-value problem in the space of continuous functions see [12,13]. In this note we rely on the classical approach [9] and compare the SDD problem with another one with a constant delay. This constant delay problem is constructed by using the so-called time transformations [2,3]. This transformation could be applied to any particular solution along which the deviating argument is monotone. To be assured that the monotonicity holds for all solutions we concentrate on the system when the state-dependent delay is governed by an additional differential equation and provide a sufficient condition for the monotonicity of the deviating argument. This type of equations is used to describe some models of population dynamics, see [1] and references therein. In [1] one could also find motivations to study this type of SDD and comparison with the frequently used case when SDD is presented by explicit or implicit functionals.
Our main goal in this note is to compare the asymptotic properties of the SDD system with the corresponding ones of the system after the time transformation. We try to find conditions which guarantee that such properties as stability, boundedness and compactness of the initial SDD problem survives under the time transformations.

Time transformations
We study the following non-autonomous system with state-dependent delay (see the autonomous case in [ with initial data Here y ∈ R m , η ∈ R, η 0 , η > 0. The function η is a state-dependent delay since it is a solution of the equation (2) where there is a dependence on y.
In the sequel we will denote h ≡ 2 η and also X ≡ C 1 ([−h, 0]; R m ) × R with the natural norm.
Proof of lemma 1. The existence of solutions is simple since the righthand sides of equations (1), (2) are continuous. Solutions are global due to Lipschitz properties of f and G. The uniqueness follows from the well-known results on the state-dependent delay equations (see e.g. [9]) since we consider Lipschitz initial function g.
Using the property |G(y)| ≤ µ η, one can easily show that (for any y) any solution of (2), (4) satisfies Now we show the continuous dependence on initial data. For the simplicity of presentation we put t 0 = 0. Let us consider a pair (ḡ;η 0 ) ∈ C 1 ([−h, 0]; R m ) × [0, h] ⊂ X and an arbitrary sequence (g n ; η 0,n ) such that ||(g n ; η 0,n ) − (ḡ;η 0 )|| X → 0 as n → ∞. We rewrite the system (1)-(4) in the integral form Similar equations are for initial data (ḡ;η 0 ). For the differences of solutions, using the Lipschitz properties of f and G (the corresponding Lipschitz constants are denoted by L f and L G ) we have Let us fix any T > 0. Since all solutions are C 1 in time (see e.g. [9])) for t > 0 we can denote by We apply the Gronwall's inequality and get for all t ∈ [0, T ] The last convergence and equations (1), (2) give a similar estimate (convergence) for the time derivatives, so we finally get This gives the continuous dependence on initial data and completes the proof of lemma 1.
For any solution (y; η) of the system (1)-(4) we call the function σ given by the deviating argument of (y; η).
Our goal is to investigate properties connected to the time transformation approach introduced in [2,3]. We are going to use a function t = α(s) called time transformation [3] to convert a particular solution (y; η) of the system (1)-(4) into a solution (z; χ; α) of constant delay system where α satisfies the algebraic equation Here ω : [s 0 − h, s 0 ] → R is an arbitrary C 1 -function with positive derivative and such that ω( Remark. We notice that equation (9) is different from the corresponding rules used in [3] and [2] since here we have no a given lag function.
The last property together with s n k →ŝ and differentiability of all γ n k imply that the derivatives d ds γ n k are unbounded in a neighborhood ofŝ. This contradicts the property d dt σ n (t) ≥δ > 0 (see (12)) since Remark. We should notice that (8), (9) is the system of coupled differential and algebraic equations. It is necessary to comment on how to solve it. The way is different from the one of [3,2] since we have no a given lag function (c.f. [3, Section 2.1]). Using (9), we write for s ∈ [0, h] (and than continue step by step with the step h): α(s) = χ(s) + ω(s − h). Then we substitute it into the differential equation for χ in (8) Now to rewrite the first equation in (8) It easy to see that the last system gives solution (z(s), χ(s)) for s ∈ [0, h]. Then the time transformation α is found by α(s) = χ(s) + ω(s − h). In general it reads as and can be solved step by step.

Connection between asymptotic properties of systems (1)-(4) and (8)-(9).
In this section we discuss how to determine if some qualitative properties of solutions of the initial state-dependent delay system (1)-(4) survive the time transformation i.e. still be valid for the corresponding solutions of constant delay system (8)- (9). We are also interested to connect the known properties of solutions of (8)-(9) with the ones of (1)-(4).
Let us start with the discussion of the property of the exponential stability. For the simplicity of presentation we assume that the zero function y(t) ≡ 0 is a solution of (1)-(4). Hence, by (11), z(s) ≡ 0 will be also a solution of (8)- (9).
Assume one has Hence, by (11), we get It is easy to see that if (and only if) e D 2 t−D 1 α −1 (t) is bounded, then we have The above considerations show that the exponential stability of the zero solution z(s) ≡ 0 implies the exponential stability of the zero solution y(t) ≡ 0 provided there are positive constants D 1 , D 2 such that Similar estimates give the inverse implication i.e. the exponential stability of the zero solution y(t) ≡ 0 implies the exponential stability of the zero solution z(s) ≡ 0 provided there are positive constants C 1 , C 2 such that C 2 s − C 1 α(s) ≤ 0 for all s ≥ s 0 . Since by (11) t = α(s), we arrive to the following Definition 1. We say that "s-time" and "t-time" are equivalent if there are constants Remark A. It is evident that in this case we also have 1 We saw that in the case of the equivalent "s-time" and "t-time" the property of the exponential stability survives under the time transformation. Another consequence of the equivalence is that t → +∞ if and only if s → +∞, which is clearly important for the study of long-time asymptotic behavior of solutions.
The last result suggests to study the notion of time-equivalence in detail. Let us try to find if in our case we have the equivalence. The rule (10) and (7) show that we need to analyse the function σ −1 . Using the property σ(t) ≥ t−h (bounded delay) and invertibility of σ, we get σ −1 (τ ) ≤ τ +h. Hence, by (10), one has α(s) Hence the property α(kh) ≤ α(0) + kh and the strict increase of α give the following estimate Since α(s) = t, the estimate (17) means the lower bound in definition 1 and the upper bound in remark A (with A 1 = 1, The supplemented bounds in definition 1 and remark A are less obvious. For the moment we do not claim that it is true in general case, but present an additional assumption which guarantees the bounds. Let us assume that the value of delay is bounded from below by a positive constant, say h 1 > 0. More precisely, η(t) ≥ h 1 ∈ (0, η] for all t ≥ 0. A sufficient condition for the last property is |G(y)| ≤ µ| η − h 1 | for all y ∈ R m . Under the above condition we have σ −1 (τ ) ≥ τ + h 1 . By (10), one has Hence the property α(kh) ≤ α(0) + kh 1 and the strict increase of α give the following estimate Combining (17) and (18) we conclude that "s-time" and "t-time" are equivalent (with A 1 = 1, in definition 1 and remark A). Having the equivalence proved we can use it to compare the asymptotic behavior of the corresponding dynamical systems (process), constructed by solutions of systems before and after the time transformations.
Let us consider an autonomous case of (1) i.e. the system (c.f. (1)-(4)) with initial data We can restrict our study (using lemma 1) to the set of initial data The set X f a is an analogous to the solution manifold used in [15] (see also [9]). We notice that the reason for this restriction is to have C 1 smoothness of solution at zero i.e.ẏ(0−) =ġ(0−) =ẏ(0+) = f a (g(0), g(−η 0 )). It is easy to see that X f a is invariant. We define the evolution operator S a (t) : X f a → X f a , associated to the system (19)-(22), by the formula S a (t)(g; η 0 ) = (y t ; η(t)), where (y; η) is the unique solution of (19)-(22). It is easy to see that under our assumptions the pair (S a , X f a ) constitutes a dynamical system (in other words, the IVP (19)-(22) is well-posed in X f a ). For more definitions and details on dynamical systems see e.g. [8,14,4].
Let us continue to discuss which asymptotic properties of (1)-(4) survive the time transformation i.e. still be valid for the corresponding solutions of constant delay system (8)-(9).
One of the important properties of dynamical systems and processes are boundedness of solutions and their compactness (asymptotic compactness) see e.g. [8,14,4]. Below we always assume that t → +∞ if and only if s → +∞, which is true, for example, in the case of the equivalence of t−time and s−time. One can easily see, by (11), that ||y(t)|| ≤ C, t ≥ t 1 is equivalent to ||z(s)|| ≤ C, s ≥ s 1 . Hence, the existence of a bounded absorbing set for z s -coordinate is equivalent to the existence of a bounded absorbing set for y t -coordinate, both in the space C([−h, 0]; R m ). To go further, let us discuss the following additional assumptions on the time transformation α: (A1) ∃C 1,α > 0, ∀s ≥ s 1 ⇒α(s) ≤ C 1,α . (A1 ′ ) ∃C 2,α > 0, ∀t ≥ t 1 ⇒α −1 (t) ≤ C 2,α . Using (11), we haveż(s) =ẏ(α(s))α(s). Hence, the existence of a bounded absorbing set for the y t -coordinate in the space C 1 ([−h, 0]; R m ) implies the existence of a bounded absorbing set for the z s -coordinate, provided (A1) is satisfied. The inverse implication is valid provided (A1 ′ ) is satisfied. For