Local stability analysis of differential equations with state-dependent delay

In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.


1.
Introduction. Let h > 0 and n ∈ N be fixed. Further, after choosing some norm · R n in R n , let us denote by C the Banach space of all continuous functions ϕ : [−h, 0] → R n provided with the norm ϕ C := sup s∈[−h,0] ϕ(s) R n . Similarly, C 1 denotes the Banach space of all continuously differentiable ϕ : [−h, 0] → R n with the norm given by ϕ C 1 := ϕ C + ϕ C . For any continuous function x : I → R n defined on some interval I ⊂ R and any real t ∈ R with [t − h, t] ⊂ I, let x t ∈ C denote the so-called segment of x at t, that is, the function x t : [−h, 0] → R n defined by x t (θ) := x(t + θ) for all −h ≤ θ ≤ 0.
In what follows, we consider the functional differential equation defined by some map f : U → R n from an open neighborhood U ⊂ C 1 of the origin in C 1 into R n with f (0) = 0. In doing so, we have in mind that Eq. (1) represents a differential equation with a state-dependent delay in a more abstract form. In order to clarify this point, consider for simplicity the differential equation

EUGEN STUMPF
we see that Eq. (2) can be written in the more abstract form x (t) =g(x(t − r(x(t)))) =g(x t (−r(x t (0)))) =f (x t ) of Eq. (1). Hence, instead of studying the original differential equation (2), we may as well study Eq. (1). The proposed transformation works also for many other differential equations with state-dependent delay. In addition, observe that in the discussed example the mapf could have been defined on the greater Banach space C and not on C 1 . Then Eq. (1) would form a so-called retarded functional differential equation as considered for example in Diekmann et al. [3]. But in contrast to the constant delay case, the theory of retarded functional differential equations is in general not applicable in the presence of a state-dependent delay (see for instance Walther [15]).
A solution of Eq. (1) is either a C 1 -smooth function x : [t 0 − h, t e ) → R n , t 0 < t e ≤ ∞, such that x t ∈ U for all t 0 ≤ t < t e and x satisfies (1) as t 0 < t < t e , or a C 1 -smooth function x : R → R n such that x t ∈ U for each t ∈ R and Eq. (1) is satisfied everywhere in R. For instance, x : R t → 0 ∈ R n is a solution of Eq.
In order to get further solutions of Eq. (1), we shall make two standing smoothness assumptions on the map f under consideration: (S1) f is continuously differentiable, and (S2) for each ϕ ∈ U the derivative Df (ϕ) : C 1 → R n extends to a linear map D e f (ϕ) : C → R n such that is continuous. In particular, these conditions are typically satisfied in cases where f represents the right-hand side of a differential equation with state-dependent delay. Provided that f satisfies (S1) and (S2), the results in Walther [15] show that for each ϕ ∈ X f with X f defined by X f := {ψ ∈ U | ψ (0) = f (ψ)} , there is a uniquely determined t + (ϕ) > 0 and a (in the forward t-direction) noncontinuable solution x ϕ : [−h, t + (ϕ)) → R n of Eq. (1) with initial value x ϕ 0 = ϕ. Moreover, all segments x ϕ t , 0 ≤ t < t + (ϕ) and ϕ ∈ X f , are contained in the solution manifold X f and the relations F (t, ϕ) := x ϕ t define a continuous semiflow F : Ω → X f with domain Ω := {(t, ψ) ∈ [0, ∞) × X f | 0 ≤ t < t + (ψ)} and continuously differentiable time-t-maps In the context of the semiflow F , the trivial solution x : R t → 0 ∈ R n of Eq. (1) is the equivalent of the stationary point ϕ 0 := 0 ∈ X f of F as we clearly have F (t, 0) = 0 for all t ∈ R. In order to describe the qualitative behavior of some other solutions of Eq. (1) in close vicinity of the trivial one, it is natural to analyze the stability properties of the stationary point ϕ 0 of F . Recall that ϕ 0 is called stable if for each ε > 0 there is some constant δ(ε) > 0 such that for all ϕ ∈ X f with ϕ − ϕ 0 C 1 = ϕ C 1 < δ(ε) it follows that t + (ϕ) = ∞ and that F (t, ϕ) − F (t, ϕ 0 ) C 1 = F (t, ϕ) C 1 < ε for all t ≥ 0. Otherwise, we call ϕ 0 unstable. So, in the situation of stability of ϕ 0 , each sufficiently small initial value ϕ ∈ X f leads to a solution x ϕ of Eq. (1) which exists and remains small for all non-negative t ∈ R. On the other hand, if ϕ 0 is unstable then there exists an open neighborhood V of 0 ∈ X f with the property that for any δ > 0 we find an initial value ϕ ∈ V with ϕ − ϕ 0 C 1 = ϕ C 1 < δ such that the associated trajectory One of the most common methods for the stability analysis of stationary points of flows or semiflows is based on the study of the linearization and its spectrum. In the situation of the semiflow F and the stationary point ϕ 0 considered here, the linearization is given by the C 0 -semigroup T := {T (t)} t≥0 of bounded linear operators T (t) := D 2 F (t, ϕ 0 ) = D 2 F (t, 0) acting on the Banach space which is equipped with the norm · C 1 of the larger space C 1 . For the action of an operator T (t) on some Now, the spectrum σ(G) ⊂ C of G determines not only the stability properties of the trivial stationary point of the linearization T of F but in certain situations also the stability properties of the trivial stationary point ϕ 0 = 0 of F . To make it more apparent, observe that by using the linear operator L := Df (0) ∈ L(C 1 , R n ) and the generally nonlinear map g : U ϕ → f (ϕ) − Lϕ ∈ R n , we may rewrite Eq.
(1) into the form Both, L and g, inherit properties (S1) and (S2) from f , and we have g(0) = 0 ∈ R n and Dg(0) = 0 ∈ L(C 1 , R n ). So, in close vicinity of 0 ∈ C 1 the map g is small in a sense and, under certain conditions on σ(G), the linear part on the righthand side of Eq. (4) has such a strong impact on the local dynamic near the origin such that the trivial solution of Eq. (4), and so of Eq. (1), has the same stability properties as the trivial solution of the linearized variational equation (3). However, before we will discuss this point in length, we shall point out that the semigroup T and its generator G are closely related to another strongly continuous semigroup and the associated infinitesimal generator. For this purpose, recall that due to assumption (S2) the operator Df (0) : C 1 → R n extends to a bounded linear operator L e := D e f (0) : C → R n . In particular, L e defines the linear retarded functional differential equation The corresponding Cauchy problem v (t) = L e v t , v 0 = χ, has for each χ ∈ C a uniquely determined solution; that is, for each χ ∈ C there is a unique continuous function v χ : [−h, ∞) → R n which is continuously differentiable on (0, ∞), satisfies the linear retarded functional differential equation for all t > 0, and the segment of v χ at t = 0 coincides with the initial value χ. The relations T e (t)χ = v χ t for χ ∈ C and t ≥ 0 induce a C 0 -semigroup T e := {T e (t)} t≥0 on the Banach space C. Its infinitesimal generator is given by The last set clearly coincides with T 0 X f . Moreover, as discussed in Hartung et al. [5], we have T (t)ϕ = T e (t)ϕ for all ϕ ∈ D(G e ) and all t ≥ 0, and the two spectra σ(G e ), σ(G) ⊂ C of the generators G e , G, respectively, are identical. The spectrum σ(G e ), and so as well the spectrum σ(G), is given by the zeros of a familiar characteristic equation. It is discrete and contains only eigenvalues whose generalized eigenspaces are finite-dimensional. In addition, for any β ∈ R the intersection {λ ∈ C | (λ) > β} ∩ σ(G e ) is either empty or finite. But let us return to the question of stability of the stationary point ϕ 0 = 0 of the semiflow F . Suppose that spectrum σ(G e ) and so σ(G) contains at least one eigenvalue with positive real part. Then, by the principle of linearized instability, ϕ 0 is an unstable stationary point of the semiflow F . On the other hand, assume that all eigenvalues of G e have negative real part. Then, by the principle of linearized stability, ϕ 0 is a stable stationary point of the semiflow F . To be more precisely, in this situation ϕ 0 is even locally asymptotically stable; that is, it is stable and attractive. Here, the last point means that there is some ε > 0 such that for all ϕ ∈ X f with ϕ − ϕ 0 C 1 = ϕ C 1 < ε we have t + (ϕ) = ∞ and So, each sufficiently small initial data in X f does lead to a solution of Eq. (1) that does not only exist and stay small for all t ≥ 0 but also converges to 0 as t → ∞.
In summary, we obtain the following theorem about local stability analysis of the semiflow F at the stationary point ϕ 0 = 0 via the spectrum of its linearization.
Theorem 1.1. Let f : U → R n defined on some open neighborhood U ⊂ C 1 of 0 ∈ C 1 with f (0) = 0 be given and suppose that f satisfies the two smoothness assumptions (S1) and (S2).
A detailed proof of assertion (ii) is contained in Hartung et al. [5,Theorem 3.6.1] whereas a proof of statement (i) can be found in [10,Proposition 1.4]. Further, we shall mentioned two points related to the theorem above. 2. The assertion of the principle of the linearized stability, that is, part (ii) of Theorem 1.1, goes even further than only local asymptotic stability of the stationary point ϕ 0 . In fact, the rate of the attraction is exponential. More precisely, we find reals ε > 0, K > 0 and ω > 0 such that for all ϕ ∈ X f with ϕ−ϕ 0 C 1 = ϕ C 1 < ε we have t + (ϕ) = ∞ and The new ingredient of this paper is now the study of the situation where, under the standing smoothness assumptions (S1) and (S2) on f , an application of Theorem 1.1 fails in order to draw any conclusions about the local stability properties of ϕ 0 from the linearized differential equation and its spectrum. This clearly occurs when the spectrum σ(G e ) of the linearization does not have any eigenvalue with positive real part but at least one eigenvalue on the imaginary axis. In our main result Theorem 3.1 we show that, under the described conditions, ϕ 0 has the same local stability behavior as the zero solution of the ordinary differential equation obtained by the reduction of Eq. (1) to a local center manifold of F at ϕ 0 .
Note that Theorem 3.1 is completely in analogy with the theory of ordinary differential equations where the analog statement is known as the Pliss reduction principle (compare Pliss [8] and Vanderbauwhede [14,Theorem 5.18]). Moreover, in order to show Theorem 3.1 we follow the proof of the Pliss reduction principle given in Vanderbauwhede [14] and at the first glance we will need only negligible modifications. But observe that the key ingredient of the approach is an attraction property of so-called local center-unstable manifolds, and the proof of this attraction property in case of Eq. (1) differs in some parts essentially from the one in the situation of an ordinary differential equation. A reason for that is the fact that, in contrast to an ordinary differential equation, a solution of Eq. (1) may generally not be continued in the backward time direction. However, the attraction property used in this paper is stated in Proposition 2.1 and it is a consequence of the main results in [13].
The rest of this paper is organized as follows. The next section is devoted to local invariant manifolds of the semiflow F at the stationary point ϕ 0 = 0. At first, we recap the existence and some properties of so-called local center-unstable and local center manifolds. After that we show, that under certain assumption on the spectrum σ(G e ), the two classes of local invariant manifolds coincide in the sense that each local center-unstable manifold is also a local center manifold and vice versa. Then we proceed with the discussion of the possibility to reduce the dynamic of the semiflow F near ϕ 0 = 0 to such an invariant manifold. This point will be essential for the formulation of our main result, which we will state and prove in Section 3. In the final section, we close the present paper with the discussion of a concrete example for the application of Theorem 1.1 as well as of Theorem 3.1.

2.
Local center and center-unstable manifolds. Recall that the spectrum σ(G e ) is given by the zeros of a characteristic equation, it is discrete and it consists only of eigenvalue with finite dimensional generalized eigenspaces. In addition, we have the decomposition where σ u (G e ), σ c (G e ) and σ s (G e ) are subsets of σ(G e ) with eigenvalues with positive, zero, and negative real part, respectively. Since for every β ∈ R the set {λ ∈ C | (λ) > β} ∩ σ(G e ) is either empty or finite, each of the sets σ u (G e ) and σ c (G e ) is either empty or finite as well. In particular, the associated realified generalized eigenspaces C u ⊂ C and C c ⊂ C, which are called the unstable and the center space, respectively, are both finite dimensional and contained in D(G e ) ⊂ C 1 . On the other hand, the stable space C s ⊂ C, which is the realified generalized eigenspace associated with σ s (G e ), is infinite dimensional. However, each of these three subspaces is invariant under the generator G e and altogether they provide the decomposition of the Banach space C. Moreover, the intersection C 1 s := C s ∩ C 1 is closed in C 1 such that we also obtain the decomposition Assume now that, apart from our assumptions on f so far, C cu := C c ⊕C u = {0}. Then the main results in [11,compare Theorems 1 & 2] show that in close vicinity of the origin in X f we find a so-called local center-unstable manifold W cu of F at the stationary point ϕ 0 = 0; that is, there exist open neighborhoods C cu,0 of 0 in C cu and C 1 s,0 of 0 in C 1 s with N cu := C cu,0 + C 1 s,0 ⊂ U and a continuously differentiable map w cu : C cu,0 → C 1 s,0 with w cu (0) = 0 and Dw cu (0) = 0 such that the graph which clearly contains ϕ 0 = 0, has the properties below.
with γ(t) ∈ N cu for all t ≤ 0. As proven in [13, Theorem 1.2] such a local center-unstable manifold W cu is also attractive in the following sense: For each ϕ ∈ X f with t + (ϕ) = ∞ and with F (t, ϕ) being sufficiently small for all t ≥ 0 there is some ψ ∈ X f with t + (ψ) = ∞ such that F (t, ψ) ∈ W cu as t ≥ 0 and such that F (t, ψ) − F (t, ϕ) → 0 exponentially for t → ∞. In other words, W cu attracts the segments of all solutions of Eq. (4) which exist and remain sufficiently close to the stationary point ϕ 0 = 0 for all t ≥ 0. But even more is true as we shall see in the next proposition that forms a local version of Corollary 5.11 in [13].
Proof. The assertion follows by application of Corollary 5.11 in [13] and subsequent restriction of the statement to a neighborhood of the stationary point ϕ 0 = 0. In order to be more precisely, recall from [13] that by construction W cu is the subset of a global center-unstable manifold W η , η > 0, that is contained in some open neighborhood, say O, of 0 in U . Moreover, the manifold W η is attractive in the sense of Theorem 4.1 in [13] and this attraction property is formulated by making use of a continuous semiflow F δ : [0, ∞) × X δ → X δ on a state space X δ ⊂ C 1 with 0 ∈ X δ and of a continuous map H η cu : X δ → W η with H η cu (0) = 0.
Next, observe that in each sufficiently small neighborhood of 0 in U , the state space X δ coincides with X f and each time-t-map F δ (t, ·) takes the same values as Then H is clearly continuous and satisfies H(0) = 0. Further, we claim that we also may assume that H(D) ⊂ W cu . Indeed, in other case we could as the new domain of the map H. Now, given ε A > 0, by Corollary 5.11 in [13] we find some δ A > 0 such that ηt for all t ≥ 0 and all ϕ ∈ X δ with ϕ C 1 < δ A . Consider now any ϕ ∈ D satisfying ϕ C 1 < δ A and suppose that for The proposition above will be essential for the proof of our main result although the last one actually will concern the dynamic of the semiflow F induced on a socalled local center manifold. In order to clarify this point in some detail, suppose that, in addition to the hypothesis on f , dim C c ≥ 1, that is,  [7] show that we find open neighboorhoods C c,0 of 0 in C c and C 1 su,0 of 0 in C 1 s ⊕ C u with N c = C c,0 + C 1 su,0 ⊂ U , and a C 1 -smooth map w c : C c,0 → C 1 su with w c (0) = 0 and Dw c (0) = 0 such that for the set which contains ϕ 0 = 0 and is called a local center manifold of F , the following holds: (ii) W c is locally positively invariant with respect to F relative to N c . (iii) W c contains the image γ(R) of any globally defined trajectory γ : R → X f of F with γ(t) ∈ N c for all t ∈ R. So, in particular, W c contains the segments of all globally defined and sufficiently small solutions of Eq. (4), and so of Eq. (1). In addition, observe that we also may assume that the derivative of the map w c is bounded on its domain. That is a simple consequence of the C 1 -smoothness of w c in combination with the equations w c (0) = 0 and Dw c (0) = 0 as shown below.

Corollary 2.2.
There is no restriction of generality in assuming that for the map w c : C c,0 → C 1 su , whose graph defines the local center manifold W c , sup If now the spectral part σ u (G e ) is not empty then, in view of the statements about the dimension, it is clear that the local manifolds W cu and W c differ from each other. But in our main result we will treat the situation where the linearization does not have any unstable direction, that is, where σ u = ∅. And in this case, it may be assumed that both W c and W cu coincide as discussed below.  respectively. Then recall that in the main the construction of local center manifolds in [5] runs as follows. After fixing appropriate real η > 0 and small enough δ > 0, one considers a specific parameter-dependent contraction G η,δ c : with G η,δ c (0, 0) = 0 such that for each ϕ ∈ C c the equation u = G η,δ c (u, ϕ) has a uniquely determined solution u(ϕ) ∈ C 1 η,R . This leads to a continuous map u η,δ c : C c ϕ → u(ϕ) ∈ C 1 η , and a local center manifold of F at ϕ 0 = 0 is then defined as that subset of W η,δ c := {u η,δ c (ϕ)(0) | ϕ ∈ C c } where the parameters ϕ ∈ C c are contained in the open ball of radius δ about 0 in C c .
Similarly, the local center-unstable manifolds in [11] are constructed by considering, for the same η > 0 as in the case of local center manifolds and sufficiently small δ > 0, a parameter-dependent contraction G η,δ cu : satisfying G η,δ cu (0, 0) = 0 and leading for each fixed ϕ ∈ C cu to a uniquely determined solution u(ϕ) of the equation u = G η,δ cu (u, ϕ). This results in a continuous mapping u η,δ cu : C cu ϕ → u(ϕ) ∈ C 1 η,(−∞,0] , and the restriction of W η,δ cu := {u η cu (ϕ)(0) | ϕ ∈ C cu } to parameters ϕ ∈ C cu in the open ball of radius δ about 0 in C cu defines a local center-unstable manifold of F at ϕ 0 = 0. Now, observe that by assumptions we have C u = {0} ⊂ C 1 and so C c = C cu . Therefore, a careful comparison of the definition of G η,δ c in [5] and the one of G η,δ cu in [11] leads to the conclusion that for all sufficiently small δ > 0 we have for all ϕ ∈ C c = C cu and all t ≤ 0. In particular, u η,δ c (ϕ)(0) = u η,δ cu (ϕ)(0) for each ϕ ∈ C c . It follows that W η,δ c = W η,δ cu , which implies the assertion.
From now on and until the end of the next section, we assume that the assumptions of the last result hold and we set W c = W cu , w c = w cu , and d := dim W c > 0. Further, let P c : C 1 → C c denote the continuous projection of C 1 along C 1 s onto C c = C cu .
Our next goal is to derive an ordinary differential equation describing the dynamics on W c induced by solutions of Eq. (4). For this purpose, choose a basis {ϕ 1 , . . . , ϕ d } ⊂ C 1 of the center space C c and introduce the row vector Φ c := (ϕ 1 , . . . , ϕ d ).
Then each ϕ ∈ C c has clearly a uniquely determined representation as with a column vector c(ϕ) := (c 1 (ϕ), · · · , c d (ϕ)) T ∈ R d . Thus, using the notation Γ c : C c → R d for the bounded linear map assigning each ϕ ∈ C c the coefficient vector c Next, observe that, in consideration of the invariance of C c under G e , we find some matrix B c ∈ R d×d such that with the row vector G e Φ c := (G e ϕ 1 , . . . , G e ϕ d ). The eigenvalues of the matrix B c coincide with σ c (G e ), that is, B c has the same eigenvalues as the restriction of G e to C c . Now, as discussed in [10, Chapter 2.6], we find an open neighborhood V ⊂ R d of 0 ∈ R d and a continuously differentiable function h : V → R d with h(0) = 0 and Dh(0) = 0 such that the center manifold reduction of F to W c reads In other words, there are V and h as described above such that on the one hand, given any solution x : I + [−h, 0] → R n , I ⊂ R an interval, of Eq. (4) with x t ∈ W c for all t ∈ I, the function z : I t → Γ c P c x t ∈ R d forms a solution of Eq. (5). And on the other hand, for any solution z : I → R d , I ⊂ R an interval, of Eq. (5) we find a solution x : 3. The reduction principle and its proof. After all the preparatory work we are now in the position to state our main result.
Theorem 3.1. Let f be as in Theorem 1.1 and suppose that σ u (G e ) = ∅ but σ c (G e ) = ∅. If z : R t → 0 ∈ R d is unstable / stable / locally asymptotically stable as a solution of Eq. (5), then ϕ 0 = 0 is unstable / stable / locally asymptotically stable as a stationary point of the semiflow F . Remark 3.2. As already mentioned in the introduction, the above result is completely similar to the so-called Pliss reduction principle from the theory of ordinary differential equations. For more details, we refer the reader to Pliss [8] and Vanderbauwhede [14].
The statement of Theorem 3.1 consists of three parts and we show them in a series of propositions. But before doing so, we prove the following auxiliary result which is of similar type as Theorem 1.6 in Getto & Waurick [4]: To begin with, recall from Corollary 1 in Walther [16] that there is some open neighborhood U B about 0 in U on which the map D e f : U → L(C, R n ) is bounded. Next, note that by the continuity of the semiflow F we also find some } are bounded with respect to the norm of C and equicontinuous. This is done in the following.
First, recall that by definition x ψ t C 1 = x ψ t C + (x ψ t ) C as 0 ≤ t < t + (ψ). Therefore, the assumption x ψ t C 1 ≤ b for all 0 ≤ t < t + (ψ) clearly implies the boundedness of both O and O with respect the · C -norm. Furthermore, and therefore for all 0 ≤ t < t + (ψ) and all s, u ∈ [−h, 0]. Thus, O is also equicontinuous. The only point remaining concerns the equicontinuity of O . In order to see this, let ε > 0 be given. Recall from the beginning of the proof that we have t + (ψ) > h. As (x ψ ) is continuous, its restriction to the interval [−h, h] is uniformly continuous. For this reason, there is some δ 1 > 0 such that for all u, s ∈ [−h, h] the implication holds. In particular, given 0 ≤ t ≤ h and reals s, u ∈ [−h, 0] with |s − u| < δ 1 , we have (x ψ t ) (s) − (x ψ t ) (u) R n < ε. Next, consider any fixed t ≥ h. Then, for all s, u ∈ [−h, 0], we get and so, in view of the boundedness of ϕ → D e f (ϕ) on U B and estimate (6), In particular, it follows that ( Now, choosing 0 < δ < min{δ 1 , δ 2 }, we see that for all t ≥ 0 and all s, u ∈ [−h, 0] with |s − u| < δ we have (x ψ t ) (s) − (x ψ t ) (u) R n < ε. This shows the equicontinuity of O which finally finishes the proof of the proposition. Now, we return to Theorem 3.1 and prove the assertion that the stability of the zero solution of the reduced differential equation implies the stability of ϕ 0 . Proposition 3.4. Consider f and σ(G e ) as in Theorem 3.1 and suppose that z : R t → 0 ∈ R d is stable as a solution of Eq. (5). Then ϕ 0 = 0 ∈ X f is a stable stationary point of the semiflow F .
Proof. 1. Let ε > 0 be given. We have to find some constant δ > 0 with the property that for each ψ ∈ X f with ψ C 1 < δ it follows t + (ψ) = ∞ and F (t, ψ) C 1 < ε for all t ≥ 0. For this purpose, let · R d denote any norm in R d and k ≥ 0 some real ensuring d j=1 |v i | ≤ k v R d for all v ∈ R d . Further, choose any 0 <ε ≤ ε such that for all ϕ ∈ X f with ϕ C 1 <ε we have ϕ ∈ V where V is the open neighborhood of 0 in X f from Proposition 2.1. Observe that additionally we may assume thatε satisfiesε < b with constant b introduced in Proposition 3.3.
2. Given ψ ∈ W c , suppose that for the associated solution x ψ of Eq. (4) we have t + (ψ) = ∞ and x ψ t ∈ W c for all t ≥ 0. Then, by definition, each segment x ψ t , t ≥ 0, may be written in the form x ψ t = P c x ψ t + w c (P c x ψ t ). Moreover, the function z ψ : [0, ∞) t → Γ c P c x ψ t ∈ R d is a solution of Eq. (5). Next, observe that for each t ≥ 0 we have and therefore with M := k (1 + sup being assumed to be bounded due to Corollary 2.2. Now, recall that by assumption z : R t → 0 ∈ R d is stable; that is, for any fixed ε R > 0 there is some δ R (ε R ) > 0 such that for eachz ∈ R d with z R d < δ R (ε R ) the solution z(·;z) of Eq. (5) with z(0;z) =z does exist for all 0 ≤ t < ∞ and satisfies z(t;z) R d < ε R as t ≥ 0. Set ε R :=ε 2M and then fix We claim that for eachψ ∈ W c with ψ C 1 < δ we have both t + (ψ) = ∞ and F (t,ψ) = xψ t C 1 < ε/2 as t ≥ 0. In order to see this, set z(ψ) := Γ c P cψ and observe that Hence, the solution z(·; z(ψ)) of Eq. (5) does exist for all t ≥ 0 and additionally satisfies z(t; z(ψ)) R d < ε R as t ≥ 0. But then we find a solution x : [−h, ∞) → R n of Eq. (4) with x t = Φ c z(t; z(ψ)) + w c (Φ c z(t; z(ψ))) ∈ W c . As In particular, xψ t ∈ W c for all t ≥ 0 such that, in view of estimate (7), we finally get for each t ≥ 0 as claimed. Note that, due to the choice ofε > 0 in the first part, the last estimate also implies F (t,ψ) ∈ V for all t ≥ 0. 3. Consider now the open neighborhoods V, D of ϕ 0 = 0 in X f , the real η A > 0, and the map H : D → W cu = W c from Proposition 2.1, and choose some fixed 0 < ε 1 < min{δ,ε/2}. As V, D are open, the map H continuous, and H(0) = 0, there clearly is some real 0 < δ 1 (ε 1 ) < 3ε/4 such that for all ϕ ∈ X f with ϕ C 1 < δ 1 (ε 1 ) we have ϕ ∈ V ∩ D and H(ϕ) C 1 < ε 1 . Observe that for those ϕ it particularly follows that H(ϕ) ∈ W c and H(ϕ) C 1 < δ. Hence, given any ϕ ∈ X f with ϕ C 1 < δ 1 (ε 1 ), from the last part we conclude that t + (H(ϕ)) = ∞ and that for each t ≥ 0 we have both F (t, H(ϕ)) C 1 <ε/2 and F (t, H(ϕ)) ∈ V.
Next, we extend the arguments in the last proof in order to show that ϕ 0 is locally asymptotically stable provided the zero solution of the reduced differential equation is so.
Proposition 3.5. Suppose that, under the assumptions of Proposition 3.4, the function z is not only stable but locally asymptotically stable as a solution of (5). Then ϕ 0 = 0 ∈ X f is a locally asymptotically stable stationary point of F .
Proof. Revisit the proof of Proposition 3.4 and suppose that the solution z is not only stable but also attractive, that is, suppose that there is some A o > 0 such that for allz ∈ R d with z R d ≤ A o the solution z(·;z) of Eq. (5) does exist for all t ≥ 0 and converges to 0 ∈ R d as t → ∞. Then we may assume that A o < δ R (ε R ) holds, since otherwise we could take the real δ R (ε R ) instead of A o for the definition of an attraction region of z. Further, combining the continuity of the map H with H(0) = 0, we find some 0 < A d < δ 1 (ε 1 ) such that for all ϕ ∈ X f with ϕ C 1 < A d it follows that H(ϕ) C 1 < A o / Γ c P c . Consider now any ψ ∈ X f with ψ C 1 < A d , and setψ := H(ψ) ∈ W cu = W c and z(ψ) := Γ c P cψ . By assumption, ψ C 1 < δ 1 (ε 1 ) and so ψ C 1 < ε 1 < δ. Therefore, both solutions x ψ and xψ of Eq. (4) do exist for all t ≥ 0. Furthermore, all segments of xψ belong to W c such that from the estimates (11) and (7) we conclude that for all t ≥ 0. As η A > 0 and it follows that the right-hand side of (13) converges to 0 ∈ R as t → ∞. But then we clearly also have F (t, ψ) → 0 for t → ∞, and this proves the assertion.
Finally, we complete the proof of Theorem 3.1 by showing that in the case of an unstable zero solution of the reduced differential equation the trivial stationary point ϕ 0 of F is unstable as well.
Proposition 3.6. Given f and σ(G e ) as in Theorem 3.1, assume that the zero function z(t) = 0 ∈ R d , t ∈ R, is unstable as a solution of Eq. (5). Then ϕ 0 = 0 ∈ X f is an unstable stationary point of F . Proof. Suppose that ϕ 0 = 0 is a stable stationary point of F . Then ϕ 0 = 0 is clearly as well a stable stationary point of the dynamics induced by F on W c . But that particularly means that the zero function is a stable solution of Eq. (5). Hence, by the contrapositive, if z(t) = 0 ∈ R d , t ∈ R, is unstable as a solution of Eq. (5) then ϕ 0 = 0 is an unstable stationary point of F , and that is exactly the assertion of the proposition. 4. Example. In this final section, we give a concrete example to illustrate the application of Theorem 1.1 and, especially, of Theorem 3.1. For doing so, set h = 1 and n = 1 in the definitions of the Banach spaces C and C 1 , and consider the scalar differential equation with a real parameter a > 0 and a delay r > 0. This equation represents a mathematical model to describe short-term fluctuations of exchange rates. Originally, it was motivated in the case of the constant delay r = 1 and a thorough discussion of Eq. (14) and the behavior of its solutions in this situation is contained in Brunovský et al. [2]. Here, we consider the situation of a state-dependent delay r = r(x(t)) > 0, that is, which is studied in [10,12]. For the delay function r : R → R under consideration, it is assumed that the following hypotheses hold: Observe that different results in [10,12] -in particular, the main result in [12] require the additional assumption (DF5): |r (s)| < 1/(4a 2 ) for all −2a ≤ s ≤ 2a on the delay function r, where the real a > 0 is just the parameter involved in Eq. (15). However, for the application of Theorem 1.1 as well as of Theorem 3.1 discussed in the following, the restriction (DF5) on r is not needed. Therefore, unless otherwise stated, we consider Eq. (15) under the assumption that r does only satisfy conditions (DF1)-(DF4), and begin our discussion with repeating some relevant material from [10,12] without proofs below.
For each ψ ∈ C this linear equation has a unique solution v ψ : [−1, ∞) → R satisfying v ψ 0 = ψ. The associated solution semigroup T e := {T e (t)} t≥0 is defined by T e (t) : C ψ → v ψ t ∈ C and recall that it is closely related to the linearization T := {DF t (0)} t≥0 of the semiflow F at ϕ 0 = 0 ∈ X f . In particular, we have σ(G e ) = σ(G) for the spectra of the generators G e and G of the two semigroups T e and T , respectively. Using the ansatz z(t) = e λt with λ ∈ C for a solution of Eq. (18), we find the characteristic equation (2) For all a > 0 with a = 1, Eq. (19) has a unique non-zero root λ = κ ∈ R. The root λ = κ is simple, and κ < 0 for 0 < a < 1 and κ > 0 for a > 1. (3) Apart from the real roots from (1) and (2), all other roots of Eq. (19) for parameter a > 0 occur in conjugate complex pairs µ ± iν with µ < 0 and ν = 0. By combining the statement (2) about σ(G e ) with part (i) of Theorem 1.1, we immediately get our first stability result: Under the conditions (DF1)-(DF5) on r, the last result was already shown in [10,Corollary 4.11]. However, for 0 < a ≤ 1 the application of Theorem 1.1 fails due to the presence of the zero root of Eq. (19), and the article [10] contains only the conjecture -compare page 109 in [10] -that in this situation the zero solution of Eq. (15) should be locally asymptotically stable. Below, we prove this conjecture at least for 0 < a < 1 rigorously.